# On $\rho$-conjugate Hopf-Galois structures

Paul J. Truman  School of Computing and Mathematics
Keele University
Staffordshire
ST5 5BG
UK<EMAIL_ADDRESS>
###### Abstract.

The Hopf-Galois structures admitted by a given Galois extension of fields
$L/K$ with Galois group $G$ correspond bijectively with certain subgroups of
$\operatorname{\mbox{Perm}}(G)$. We use a natural partition of the set of such
subgroups to obtain a method for partitioning the set of corresponding Hopf-
Galois structures, which we term $\rho$-conjugation. We study properties of
this construction, with particular emphasis on the Hopf-Galois analogue of the
Galois correspondence, the connection with skew left braces, and applications
to questions of integral module structure in extensions of local or global
fields. In particular, we show that the number of distinct $\rho$-conjugates
of a given Hopf-Galois structure is determined by the corresponding skew left
brace, and that if $H,H^{\prime}$ are Hopf algebras giving $\rho$-conjugate
Hopf-Galois structures on a Galois extension of local or global fields $L/K$
then an ambiguous ideal $\mathfrak{B}$ of $L$ is free over its associated
order in $H$ if and only if it is free over its associated order in
$H^{\prime}$. We exhibit a variety of examples arising from interactions with
existing constructions in the literature.

###### Key words and phrases:

Hopf-Galois structure, Hopf-Galois theory, Skew left braces, Galois module
structure, Associated order

###### 2020 Mathematics Subject Classification:

Primary 16T05; Secondary 12F10, 11S33, 20N99

##  1\. Introduction

A Hopf-Galois structure on a finite extension of fields $L/K$ consists of a
cocommutative $K$-Hopf algebra $H$ making $L$ into an $H$-module algebra
satisfying a certain non-degeneracy condition (see [6, (2.7) Definition] for
the full definition). The motivating example is a Galois extension $L/K$ with
Galois group $G$: in this case the group algebra $K[G]$ (with its usual action
on $L$) gives a Hopf-Galois structure on the extension. Hopf-Galois structures
allow many results from Galois theory to be generalized to extensions which
are inseparable or non-normal, but there is also considerable interest in the
applications to Galois extensions: various authors have studied the number of
Hopf-Galois structures admitted by particular extensions, the structure of the
Hopf algebras involved, and questions concerning the Hopf-Galois analogue of
the Galois correspondence. We highlight two applications in particular.
Firstly, it has recently been discovered that each Hopf-Galois structure on a
Galois extension yields a skew left brace; these objects are intensively
studied because they can be used to generate set-theoretic solutions of the
Yang-Baxter equation. The translation of existence, classification, and
structural results between Hopf-Galois theory and skew brace theory has
enriched both disciplines. Secondly, Hopf-Galois theory has been applied
fruitfully to questions originating in Galois module theory: if $L/K$ is a
finite extension of local or global fields and $H$ is a Hopf algebra giving a
Hopf-Galois structure on the extension then we may study each fractional ideal
$\mathfrak{B}$ of $L$ as a module over its associated order in $H$. This
raises the possibility of comparing the structure of $\mathfrak{B}$ as a
module over its associated orders in the various Hopf-Galois structures
admitted by the extension. We refer the reader to [8] for a recent survey of
these and other related topics.

A theorem of Greither and Pareigis [11] classifies the Hopf-Galois structures
admitted by a finite separable extension of fields $L/K$. In the special case
of Galois extension with Galois group $G$ their theorem implies that Hopf-
Galois structures admitted by $L/K$ correspond bijectively with regular
subgroups $N$ of $\operatorname{\mbox{Perm}}(G)$ that are normalized by the
image of the left regular representation
$\lambda:G\hookrightarrow\operatorname{\mbox{Perm}}(G)$. In [14] it is noted
that such a subgroup $N$ need not be normalized by the image of right regular
representation $\rho:G\hookrightarrow\operatorname{\mbox{Perm}}(G)$, and that
for each $g\in G$ the subgroup $\rho(g)N\rho(g)^{-1}$ is again regular and
normalized by $\lambda(G)$, and therefore corresponds to a Hopf-Galois
structure on $L/K$. Following [17] we call this method of partitioning the set
of Hopf-Galois structures on $L/K$ $\rho$-conjugation. We review existing
results concerning $\rho$-conjugate Hopf-Galois structures in more detail in
Section 2 of this paper.

In Section 3 we study the Hopf-Galois analogue of the Galois correspondence,
showing that the lattices of subfields realized by $\rho$-conjugate Hopf-
Galois structures are intimately related to one another. Section 4 concerns
the skew braces corresponding to $\rho$-conjugate Hopf-Galois structures; we
show that the number of distinct $\rho$-conjugates of a given Hopf-Galois
structure is determined by the corresponding skew brace, and obtain a
criterion for a Hopf-Galois structure to have no nontrivial $\rho$-conjugates.

In Section 5 we suppose that $L/K$ is a separable extension of local or global
fields and study questions of integral module structure. We show that if an
ambiguous ideal $\mathfrak{B}$ of $L$ is free over its associated order in a
particular Hopf-Galois structure then it is free over its associated order in
each of the Hopf-Galois structures that are $\rho$-conjugate to the original
one. Finally, in Section 6 we study how $\rho$-conjugation interacts with
other constructions in the literature, including Hopf-Galois structures
arising from fixed point free pairs of homomorphisms (due to Byott and
Childs), those arising from abelian maps (instigated by Childs and generalized
by Koch), and induced Hopf-Galois structures (due to Crespo, Rio, and Vela).

##  2\. $\rho$-conjugate Hopf-Galois structures

In this section we recall the definition of $\rho$-conjugate Hopf-Galois
structures and some of their known properties.

Let $L/K$ be a Galois extension of fields with Galois group $G$. As mentioned
in Section 1, a theorem of Griether and Pareigis [11] implies that the Hopf-
Galois structures admitted by $L/K$ correspond bijectively with regular
subgroups $N$ of $\operatorname{\mbox{Perm}}(G)$ that are normalized by the
image of $G$ under the left regular representation
$\lambda:G\hookrightarrow\operatorname{\mbox{Perm}}(G)$. The normalization
condition is equivalent to requiring that $N$ is stable under the action of
$G$ on $\operatorname{\mbox{Perm}}(G)$ given by
$g\ast\eta=\lambda(g)\eta\lambda(g)^{-1}$ for all
$\eta\in\operatorname{\mbox{Perm}}(G)$ and all $g\in G$; accordingly, we shall
refer to the subgroups of interest as $G$-stable regular subgroups.

In general, a $G$-stable regular subgroup $N$ of
$\operatorname{\mbox{Perm}}(G)$ need not be normalized by the image of the
right regular representation
$\rho:G\hookrightarrow\operatorname{\mbox{Perm}}(G)$. This motivates the
following (see also [14, Example 2.7]).

######  Proposition 2.1.

Let $N$ be a $G$-stable regular subgroup of $\operatorname{\mbox{Perm}}(G)$,
and let $g\in G$. Then $N_{g}:=\rho(g)N\rho(g)^{-1}$ is a $G$-stable regular
subgroup of $\operatorname{\mbox{Perm}}(G)$.

###### Proof.

Clearly $N_{g}$ is a subgroup of $\operatorname{\mbox{Perm}}(G)$ of the same
order as $N$. Since $N$ acts transitively on $G$, we have

$N_{g}[g]=(N[e_{G}])g^{-1}=Gg^{-1}=G,$

so $N_{g}$ acts transitively on $G$. Therefore $N_{g}$ is a regular subgroup
of $\operatorname{\mbox{Perm}}(G)$. To show $G$-stability, we note that
$\lambda(G)$ and $\rho(G)$ centralize one another inside
$\operatorname{\mbox{Perm}}(G)$; hence for $h\in G$ and $\eta\in N$ we have

$\displaystyle h\ast\rho(g)\eta\rho(g)^{-1}$ $\displaystyle=$
$\displaystyle\lambda(h)\rho(g)\eta\rho(g)^{-1}\lambda(h)^{-1}$
$\displaystyle=$
$\displaystyle\rho(g)\lambda(h)\eta\lambda(h)^{-1}\rho(g)^{-1}$
$\displaystyle=$ $\displaystyle\rho(g)(h\ast\eta)\rho(g)^{-1}.$

We have $h\ast\eta\in N$ because $N$ is $G$-stable; hence
$\rho(g)(h\ast\eta)\rho(g)^{-1}\in N_{g}$, and so $N_{g}$ is $G$-stable. ∎

######  Definition 2.2.

We shall call two $G$-stable regular subgroups $N,N^{\prime}$ of
$\operatorname{\mbox{Perm}}(G)$ $\rho$-conjugate to mean that
$N^{\prime}=N_{g}$ for some $g\in G$. In this case we shall also say that the
Hopf-Galois structures corresponding to $N,N^{\prime}$ are $\rho$-conjugate to
one another.

It is clear that $\rho$-conjugation is an equivalence relation on the set of
$G$-stable regular subgroups of $\operatorname{\mbox{Perm}}(G)$, and so yields
an equivalence relation on the set of Hopf-Galois structures admitted by
$L/K$. Given a $G$-stable regular subgroup $N$, we have $N_{g}=N_{h}$ if and
only if
$\rho(g^{-1}h)\in\mathrm{Norm}_{\scriptsize{\operatorname{\mbox{Perm}}(G)}}(N)$.
In particular, if $G$ is abelian then $\rho(G)=\lambda(G)$, and since $N$ is
$G$-stable we have $N_{g}=N$ for all $g\in G$. We shall return to the question
of determining the number of distinct $\rho$-conjugates of a given $G$-stable
regular subgroup in Section 4.

######  Example 2.3.

Let $p,q$ be prime numbers with $p\equiv 1\pmod{q}$ and let $L/K$ be a Galois
extension whose Galois group $G$ is isomorphic to the metacyclic group of
order $pq$:

$G=\langle s,t\mid s^{p}=t^{q}=e,\;tst^{-1}=s^{d}\rangle,$

where $d$ is a positive integer of multiplicative order $q$ modulo $p$. Let
$\eta=\lambda(s)\rho(t)\in\operatorname{\mbox{Perm}}(G)$ and
$N=\langle\eta\rangle$. It is routine to verify that $N$ is $G$-stable and
regular. We find that $N_{t}=N$ and that

$N_{s^{i}}=\langle\lambda(s)\rho(s^{i(1-d)}t)\rangle$

for each $i=0,\ldots,p-1$. These subgroups are all distinct, and so we obtain
a family of $p$ mutually $\rho$-conjugate Hopf-Galois structures on $L/K$. By
[4, Theorem 6.2], these are all the Hopf-Galois structures admitted by $L/K$
for which the corresponding subgroup of $\operatorname{\mbox{Perm}}(G)$ is
cyclic (that is: those of cyclic type).

######  Example 2.4.

Let $n$ be an even natural number and let $L/K$ be a Galois extension whose
Galois group $G$ is isomorphic to the dihedral group of order $2n$:

$G=\langle r,s\mid r^{n}=s^{2}=e,\;srs^{-1}=r^{-1}\rangle.$

Let $N=\langle\lambda(r)\rho(s),\lambda(s)\rangle$. Then $N$ is $G$ stable,
regular, and isomorphic to $D_{2n}$. We find that $N_{s}=N$ and that

$N_{r^{k}}=\langle\lambda(r)\rho(r^{2k}s),\lambda(s)\rangle$

for each $k=0,\ldots,n/2-1$. These subgroups are all distinct, and so we
obtain a family of $n/2$ mutually $\rho$-conjugate Hopf-Galois structures on
$L/K$.

We shall shed more light on these examples, and construct others, in Section
6.

In Greither and Pareigis’s original paper classifying Hopf-Galois structures
on separable extensions [11] they show that if $N$ is a $G$-stable regular
subgroup of $\operatorname{\mbox{Perm}}(G)$ then so too is

$N^{opp}=\mathrm{Cent}_{{\tiny\operatorname{\mbox{Perm}}(G)}}(N).$

In [16] the Hopf-Galois structure corresponding to $N^{opp}$ is called the
opposite of the one corresponding to $N$, and [17, Corollary 7.2] shows that
for each $g\in G$ we have $(N_{g})^{opp}=(N^{opp})_{g}$. We shall see fruitful
applications of this interaction in Section 5.

######  Example 2.5.

Recall the hypotheses and notation of Example 2.4 above. The opposite of the
subgroup $N$ constructed in that example is the subgroup generated by
$\rho(s)$ and the permutation $\mu_{0}$ defined by

$\mu_{0}[r^{i}s^{j}]=r^{i+(-1)^{i+j}}s^{j}.$

Exploiting the interaction between opposite subgroups and $\rho$-conjugation,
we find that for each $k=0,\ldots,n/2-1$ the opposite of the subgroup
$N_{r^{k}}$ is generated by $\rho(s)$ and the permutation $\mu_{k}$ defined by

$\mu_{k}[r^{i}s^{j}]=r^{i+(-1)^{i+j+k}}s^{j}.$

The Hopf algebra giving the Hopf-Galois structure corresponding to a
$G$-stable regular subgroup $N$ is the fixed ring $L[N]^{G}$, where $G$ acts
on $L$ via the usual Galois action and on $N$ via
$g\ast\eta=\lambda(g)\eta\lambda(g)^{-1}$. It is possible for two distinct
Hopf-Galois structures on $L/K$ to involve isomorphic Hopf algebras, and it is
well known (see [14, Corollary 2.3], for example) that we have $L[N]^{G}\cong
L[N^{\prime}]^{G}$ if and only if there is a group isomorphism
$\phi:N\xrightarrow{\sim}N^{\prime}$ such that
$h\ast\phi(\eta)=\phi(h\ast\eta)$ for all $\eta\in N$ and all $h\in G$. The
proof of Proposition 2.1 shows that the isomorphism $\phi:N\rightarrow N_{g}$
defined by $\phi(\eta)=\rho(g)\eta\rho(g)^{-1}$ has this property, and so
$L[N]^{G}\cong L[N_{g}]^{G}$ as $K$-Hopf algebras (see [14, Example 2.7]).
More explicitly: $\phi$ induces an isomorphism of $L$-Hopf algebras
$\phi:L[N]\xrightarrow{\sim}L[N_{g}]$, which descends to an isomorphism of
$K$-Hopf algebras $\phi:L[N]^{G}\xrightarrow{\sim}L[N_{g}]^{G}$, as follows

$\phi\left(\sum_{\eta\in N}c_{\eta}\right)=\sum_{\eta\in
N}c_{\eta}\rho(g)\eta\rho(g)^{-1}.$ (1)

##  3\. The Hopf-Galois correspondence

If a Hopf algebra $H$ gives a Hopf-Galois structure on a finite extension of
fields $L/K$ then each Hopf subalgebra $H^{\prime}$ of $H$ has a corresponding
fixed field

$L^{H^{\prime}}=\\{x\in L\mid z\cdot x=\varepsilon(z)x\mbox{ for all }z\in
H^{\prime}\\},$

where $\varepsilon:H\rightarrow K$ denotes the counit map of $H$. In analogy
with the classical Galois correspondence, we have
$[L:L^{H^{\prime}}]=\dim_{K}(H^{\prime})$. The correspondence from Hopf
subalgebras of $H$ to intermediate fields of the extension $L/K$ is inclusion
reversing and injective, but not surjective in general; we say that an
intermediate field $M$ is realizable with respect to $H$ to mean that there is
a Hopf subalgebra $H^{\prime}$ of $H$ such that $L^{H^{\prime}}=M$.

Specializing to the case in which $L/K$ is a Galois extension, we have seen
that each Hopf algebra giving a Hopf-Galois structure on $L/K$ has the form
$L[N]^{G}$ with $N$ some $G$-stable regular subgroup of
$\operatorname{\mbox{Perm}}(G)$; in this case the Hopf subalgebras of
$L[N]^{G}$ are precisely the sets $L[P]^{G}$ with $P$ a subgroup of $N$ that
is itself $G$-stable (see [9, Theorem 2.3] or [8, Theorem 7.2]). Abusing
notation, we write $L^{P}$ in place of of $L^{L[P]^{G}}$; since
$\dim_{K}(L[P]^{G})=|P|$, we then have $[L:L^{P}]=|P|$. Now let $g\in G$ and
consider the $G$-stable regular subgroup $N_{g}$ and the corresponding Hopf-
Galois structure, with Hopf algebra $L[N_{g}]^{G}$. By virtue of the Hopf
algebra isomorphism $\phi:L[N]^{G}\xrightarrow{\sim}L[N_{g}]^{G}$ given in
Equation 1, the lattices of Hopf subalgebras of $L[N]^{G}$ and $L[N_{g}]^{G}$
are isomorphic; hence the lattices of intermediate fields of $L/K$ realizable
with respect to these Hopf-Galois structures are isomorphic. We can give a
more precise statement by studying the actions of $L[N]^{G}$ and
$L[N_{g}]^{G}$ on $L$ in more detail.

It follows from the theorem of Greither and Pareigis that the action of an
element $\sum_{\eta\in N}c_{\eta}\eta\in L[N]^{G}$ on $L$ is given by

$\left(\sum_{\eta\in N}c_{\eta}\eta\right)\cdot x=\sum_{\eta\in
N}c_{\eta}\eta^{-1}[e_{G}](x)\mbox{ for all }x\in L$ (2)

(see [8, Proposition 4.14], for example). The action of $L[N_{g}]^{G}$ on $L$
is given by an analogous formula. We describe how the isomorphism $\phi$
interacts with these actions.

######  Proposition 3.1.

Let $x\in L$. Then for all $z\in L[N]^{G}$ we have

$\phi(z)\cdot x=g(z\cdot g^{-1}(x)).$

###### Proof.

Write $z=\sum_{\eta\in N}c_{\eta}\eta$ with $c_{\eta}\in L$. Since $z\in
L[N]^{G}$ we have $z=h\ast z$ for all $h\in G$. In particular,

$z=g\ast z=\sum_{\eta\in N}g(c_{\eta})\lambda(g)\eta\lambda(g)^{-1},$

and so

$\displaystyle\phi(z)$ $\displaystyle=$ $\displaystyle\sum_{\eta\in
N}g(c_{\eta})\rho(g)\lambda(g)\eta\lambda(g)^{-1}\rho(g)^{-1}$
$\displaystyle=$ $\displaystyle\sum_{\eta\in
N}g(c_{\eta})\lambda(g)\rho(g)\eta\rho(g)^{-1}\lambda(g)^{-1}\mbox{ by
Proposition \ref{prop_rho_conjugate_G_stable_regular}.}$

Therefore

$\displaystyle\phi(z)\cdot x$ $\displaystyle=$
$\displaystyle\left(\sum_{\eta\in
N}g(c_{\eta})\lambda(g)\rho(g)\eta\rho(g)^{-1}\lambda(g)^{-1}\right)\cdot x$
$\displaystyle=$ $\displaystyle\sum_{\eta\in
N}g(c_{\eta})\lambda(g)\rho(g)\eta^{-1}\rho(g)^{-1}\lambda(g)^{-1}[e_{G}](x)$
$\displaystyle=$ $\displaystyle\sum_{\eta\in
N}g(c_{\eta})\lambda(g)\rho(g)\eta^{-1}[g^{-1}g](x)$ $\displaystyle=$
$\displaystyle\sum_{\eta\in N}g(c_{\eta})g\eta^{-1}[e_{G}]g^{-1}(x)$
$\displaystyle=$ $\displaystyle g\left(\sum_{\eta\in
N}c_{\eta}\eta^{-1}[e_{G}]g^{-1}(x)\right)$ $\displaystyle=$ $\displaystyle
g\left(\sum_{\eta\in N}c_{\eta}\eta\right)\cdot g^{-1}(x)$ $\displaystyle=$
$\displaystyle g\left(z\cdot g^{-1}(x)\right),$

as claimed. ∎

Now we have the following:

######  Theorem 3.2.

An intermediate field $M$ of $L/K$ is realizable with respect to $L[N]^{G}$ if
and only if the intermediate field $g(M)$ is realizable with respect to
$L[N_{g}]^{G}$.

###### Proof.

Suppose that $M$ is realizable with respect to $L[N]^{G}$. Then $M=L^{P}$ for
some $G$-stable subgroup $P$ of $N$. We have $\phi(L[P]^{G})=L[P_{g}]^{G}$, a
Hopf subalgebra of $L[N_{g}]^{G}$, and for $x\in L$ we have

$\displaystyle x\in L^{P_{g}}$ $\displaystyle\Leftrightarrow$
$\displaystyle\phi(z)\cdot x=\varepsilon(z)x\mbox{ for all }z\in L[P]^{G}$
$\displaystyle\Leftrightarrow$ $\displaystyle g(z\cdot
g^{-1}(x))=\varepsilon(z)x\mbox{ for all }z\in L[P]^{G}\mbox{ by Proposition
\ref{prop_conjugation_action}}$ $\displaystyle\Leftrightarrow$ $\displaystyle
z\cdot g^{-1}(x)=\varepsilon(z)g^{-1}(x)\mbox{ for all }z\in L[P]^{G}$
$\displaystyle\Leftrightarrow$ $\displaystyle g^{-1}(x)\in L^{P}=M$
$\displaystyle\Leftrightarrow$ $\displaystyle x\in g(M).$

Hence $g(M)$ is realizable with respect to $L[N_{g}]^{G}$. For the converse
statement we replace $N$ by $N_{g}$ and $g$ by $g^{-1}$ in the argument above.
∎

######  Example 3.3.

Recall the hypotheses and notation of Example 2.3 above. Each of the subgroups
$N_{s^{i}}$ is cyclic of order $pq$, and therefore has a unique subgroup
$P_{i}$ of order $p$ and a unique subgroup $Q_{i}$ of order $q$; by
uniqueness, each of these is $G$-stable. Since $N_{s^{i}}$ is generated by
$\lambda(s)\rho(s^{i(1-d)}t)$, we see that $P_{i}=\langle\lambda(s)\rangle$
regardless of $i$, and that $Q_{i}=\langle\rho(s^{i(1-d)}t)\rangle$. Therefore
$L^{P_{i}}=L^{\langle s\rangle}$ (the unique intermediate field of degree $p$
over $K$) and $L^{Q}=L^{\langle s^{i(1-d)}t\rangle}$. Thus each of this family
of $\rho$-conjugate Hopf-Galois structures realizes the intermediate fields
$L,K,L^{\langle s\rangle}$, and a unique intermediate field of degree $q$ over
$K$, and each intermediate field of degree $q$ over $K$ is realized by exactly
one Hopf-Galois structure in this family.

##  4\. Skew braces

A (left) skew brace is triple $(B,\star,\circ)$ where $B$ is a set and
$\star,\circ$ are binary operations on $B$, each making $B$ into a group, such
that

$x\circ(y\star z)=(x\circ y)\star x^{-1}\star(x\circ z)\mbox{ for all
}x,y,z\in B.$ (3)

Here $x^{-1}$ denotes the inverse of $x$ with respect to $\star$; we denote
the inverse of $x$ with respect to $\circ$ by $\overline{x}$. We call Equation
3 the (left) skew brace relation; the right skew brace relation is analogous.
We call a skew brace two-sided if both the left and right skew brace relations
hold for all triples of elements of $B$. Skew braces with underlying set $B$
yield solutions of the set theoretic Yang-Baxter equation on $B$; we can then
obtain solutions of the Yang-Baxter equation over any field by using $B$ as
the basis for a vector space (see [15, §2], for example).

The connection between skew braces and Hopf-Galois structures on Galois
extensions is noted by Bachiller [1]. One formulation is as follows: a finite
skew brace $(B,\star,\circ)$, yields two left regular representation maps
$\lambda_{\star},\lambda_{\circ}:B\hookrightarrow\operatorname{\mbox{Perm}}(B)$;
we find that $\lambda_{\star}(B)$ is a $(B,\circ)$-stable regular subgroup of
$\operatorname{\mbox{Perm}}(B)$, and so corresponds to a Hopf-Galois structure
on a Galois extension with Galois group $(B,\circ)$. Conversely, if
$G=(G,\circ)$ is a finite group and $N$ is a $G$-stable regular subgroup of
$\operatorname{\mbox{Perm}}(G)$ then the map $\eta\mapsto\eta[e_{G}]$ is a
bijection from $N$ to $G$; we use this to define a second binary operation
$\star$ on $G$ by

$\eta[e_{G}]\star\mu[e_{G}]=\eta\mu[e_{G}].$

Then $(G,\star)\cong N$, and the fact that $N$ is $G$-stable implies that
$(G,\star,\circ)$ is a skew brace. As noted in [18, Proposition 2.1] and [17,
Proposition 3.1], two $G$-stable regular subgroups
$N,N^{\prime}\leq\operatorname{\mbox{Perm}}(G)$ yield isomorphic skew braces
if and only if $N^{\prime}=\varphi^{-1}N\varphi$ for some
$\varphi\in\operatorname{\mbox{Aut}}(G,\circ)$, yield the same skew brace if
and only if $N^{\prime}=\varphi^{-1}N\varphi$ for some
$\varphi\in\operatorname{\mbox{Aut}}(G,\star,\circ)$ (the group of
automorphisms of $(G,\circ)$ that also respect $\star$).

In [17, Proposition 4.1] we find the following alternative formulation of
$\rho$-conjugate subgroups, which enables us to study the corresponding skew
braces.

######  Proposition 4.1.

Let $N$ be a $G$-stable regular subgroup of $\operatorname{\mbox{Perm}}(G)$,
and let $g\in G$. Then $N_{g}=\varphi_{g}N\varphi_{g}^{-1}$, where
$\varphi_{g}$ is the inner automorphism of $G$ corresponding to $g$.

###### Proof.

We may write $\varphi_{g}=\rho(g)\lambda(g)$, and we then have

$\displaystyle\varphi_{g}N\varphi_{g}^{-1}$ $\displaystyle=$
$\displaystyle\rho(g)\lambda(g)N\lambda(g)^{-1}\rho(g)^{-1}$ $\displaystyle=$
$\displaystyle\rho(g)N\rho(g)^{-1}\mbox{ since $N$ is $G$-stable }$
$\displaystyle=$ $\displaystyle N_{g}.$

∎

This observation immediately implies that $\rho$-conjugate Hopf-Galois
structures yield isomorphic skew braces under the correspondence described
above ([17, Corollary 4.1]).

The number of distinct $\rho$-conjugates of a given $G$-stable regular
subgroup $N$ of $\operatorname{\mbox{Perm}}(G)$ can studied via the
corresponding skew brace.

######  Theorem 4.2.

Let $N$ be a $G$-stable regular subgroup of $\operatorname{\mbox{Perm}}(G)$,
let $(G,\star,\circ)$ be the corresponding skew brace, and let $g\in G$. The
following are equivalent:

  1. i)

The $G$-stable regular subgroup $N$ is normalized by $\rho(g)$;

  2. ii)

The inner automorphism $\varphi_{g}$ of $\operatorname{\mbox{Aut}}(G,\circ)$
is an element of $\operatorname{\mbox{Aut}}(G,\star,\circ)$;

  3. iii)

We have $(y\star z)\circ g=(y\circ g)\star g^{-1}\star(z\circ g)$ for all
$y,z\in G$.

######  Corollary 4.3.

Let $G^{\prime}=\\{g\in
G\mid\varphi_{g}\in\operatorname{\mbox{Aut}}(G,\star,\circ)\\}$. Then the
$G$-stable regular subgroup $N$ has $|G|/|G^{\prime}|$ distinct
$\rho$-conjugates.

######  Corollary 4.4.

The $G$-stable regular subgroup $N$ is normalized by $\rho(G)$ if and only if
$(G,\star,\circ)$ is a two-sided skew brace.

###### Proof of Theorem 4.2..

By Proposition 4.1 we have $N_{g}=\varphi_{g}N\varphi_{g}^{-1}$, and the
discussion above then implies that $N_{g}=N$ if and only if
$\varphi_{g}\in\operatorname{\mbox{Aut}}(G,\star,\circ)$. Hence (i) and (ii)
are equivalent.

Now suppose that (ii) holds. By the left skew brace relation (3), for all
$y,z\in G$ we have

$g\circ(y\star z)=(g\circ y)\star g^{-1}\star(g\circ z).$

Since $\varphi_{g}\in\operatorname{\mbox{Aut}}(G,\star,\circ)$, so is
$\varphi_{g}^{-1}=\varphi_{\overline{g}}$. Applying this to the equation above
yields

$\displaystyle\varphi_{\overline{g}}(g\circ(y\star
z))=\varphi_{\overline{g}}((g\circ y)\star g^{-1}\star(g\circ z))$
$\displaystyle\Rightarrow$ $\displaystyle\varphi_{\overline{g}}(g\circ(y\star
z))=\varphi_{\overline{g}}(g\circ
y)\star\varphi_{\overline{g}}(g^{-1})\star\varphi_{\overline{g}}(g\circ z)$
$\displaystyle\Rightarrow$ $\displaystyle\overline{g}\circ g\circ(y\star
z)\circ g=(\overline{g}\circ g\circ y\circ
g)\star\varphi_{\overline{g}}(g)^{-1}\star(\overline{g}\circ g\circ z\circ g)$
$\displaystyle\Rightarrow$ $\displaystyle(y\star z)\circ g=(y\circ g)\star
g^{-1}\star(z\circ g);$

Hence (iii) holds.

Finally, suppose that (iii) holds. Then for all $y,z\in G$ we have

$\displaystyle\varphi_{\overline{g}}(y\star z)$ $\displaystyle=$
$\displaystyle\overline{g}\circ(y\star z)\circ g$ $\displaystyle=$
$\displaystyle\overline{g}\circ((y\circ g)\star g^{-1}\star(z\circ g))\mbox{
by (iii) }$ $\displaystyle=$ $\displaystyle(\overline{g}\circ y\circ
g)\star\overline{g}^{-1}\star(\overline{g}\circ
g^{-1})\star\overline{g}^{-1}\star(\overline{g}\circ z\circ g)\mbox{ by
Equation \ref{eqn_brace_relation}. }$

Now by a well known skew brace identity (see [12, Lemma 1.7, part (2)], for
example) we have

$\overline{g}^{-1}\star(\overline{g}\circ
g^{-1})\star\overline{g}^{-1}=(\overline{g}\circ g)^{-1}=e_{G};$

applying this to the final line above yields

$\varphi_{\overline{g}}(y\star z)=(\overline{g}\circ y\circ
g)\star(\overline{g}\circ z\circ
g)=\varphi_{\overline{g}}(y)\star\varphi_{\overline{g}}(z),$

and so $\varphi_{\overline{g}}\in\operatorname{\mbox{Aut}}(G,\star,\circ)$.
Therefore $\varphi_{g}\in\operatorname{\mbox{Aut}}(G,\star,\circ))$, so (ii)
holds. ∎

##  5\. Integral module structure

We now suppose that $L/K$ is a Galois extension of local or global fields (in
any characteristic) with Galois group $G$. As discussed briefly in Section 1,
Hopf-Galois structures can be used in this context to study the structure of
fractional ideals $\mathfrak{B}$ of $L$. More precisely: if $H$ is a Hopf
algebra giving a Hopf-Galois structure on the extension then $L$ is a free
$H$-module of rank $1$ (this is a Hopf-Galois analogue of the classical normal
basis theorem), and we seek integral analogues of this result. Each fractional
ideal $\mathfrak{B}$ of $L$ has an associated order in $H$:

${\mathfrak{A}}_{H}(\mathfrak{B})=\\{h\in H\mid h\cdot x\in\mathfrak{B}\mbox{
for all }x\in\mathfrak{B}\\},$

and we may study the structure of $\mathfrak{B}$ as an
${\mathfrak{A}}_{H}(\mathfrak{B})$-module, with particular emphasis on
determining criteria for it to be free (necessarily of rank $1$). By
construction, ${\mathfrak{A}}_{H}(\mathfrak{B})$ is the largest subring of $H$
for which $\mathfrak{B}$ is a module, and it is the only order in $H$ over
which $\mathfrak{B}$ can possibly be free.

The most famous results in this area are due to Byott [3], who exhibits Galois
extensions $L/K$ of $p$-adic fields for which the valuation ring
${\mathfrak{O}}_{L}$ is not free over
${\mathfrak{A}}_{K[G]}({\mathfrak{O}}_{L})$, but is free over
${\mathfrak{A}}_{H}({\mathfrak{O}}_{L})$ for some other Hopf algebra $H$
giving a Hopf-Galois structure on the extension. On the other hand, in [19]
and [20] we show that a fractional ideal $\mathfrak{B}$ of $L$ is free over
its associated order in a particular Hopf-Galois structure if and only if it
is free over its associated order in the opposite Hopf-Galois structure. The
main result of this section is in a similar vein. Recall that an ambiguous
ideal of $L$ is a fractional idea of $L$ that is invariant under the action of
$G$. (In particular: the ring of algebraic integers ${\mathfrak{O}}_{L}$ is an
ambiguous ideal of $L$, and if $L/K$ is an extension of local fields then
every fractional ideal of $L$ is an ambiguous ideal.) We will prove:

######  Theorem 5.1.

Let $L[N]^{G}$ give a Hopf-Galois structure on $L/K$, and let $g\in G$. Then
an ambiguous ideal $\mathfrak{B}$ of $L$ is free over its associated order in
$L[N]^{G}$ if and only if it is free over its associated order in
$L[N_{g}]^{G}$.

######  Corollary 5.2.

An ambiguous ideal of $L$ is free over its associated order in $L[N]^{G}$ if
and only if it is free over its associated order in $L[N_{g}]^{G}$ for all
$g\in G$.

The proof of Theorem 5.1 does not depend upon the fact that $L$ is a field,
and so we have

######  Corollary 5.3.

If $L/K$ is an extension of global fields then an ambiguous ideal of $L$ is
locally free over its associated order in $L[N]^{G}$ if and only if it is
locally free over its associated order in $L[N_{g}]^{G}$ for all $g\in G$.

###### Proof of Theorem 5.1..

Let ${\mathfrak{A}}$ denote the associated order of $\mathfrak{B}$ in
$L[N]^{G}$, and let ${\mathfrak{A}}_{g}$ denote the associated order of
$\mathfrak{B}$ in $L[N_{g}]^{G}$. Suppose that $\mathfrak{B}$ is a free
${\mathfrak{A}}$-module, with generator $x\in\mathfrak{B}$. We shall show that
${\mathfrak{A}}_{g}=\phi({\mathfrak{A}})$ and that $\mathfrak{B}$ is a free
${\mathfrak{A}}_{g}$-module with generator $g(x)$ (note that
$g(x)\in\mathfrak{B}$ because $\mathfrak{B}$ is an ambiguous ideal of $L$).

Since $\phi$ is an isomorphism of $K$-Hopf algebras, it is immediate that
$\phi({\mathfrak{A}})$ is an order in $L[N_{g}]^{G}$. Now let
$y\in\mathfrak{B}$. Since $\mathfrak{B}$ is an ambiguous ideal of $L$ we have
$g^{-1}(y)\in\mathfrak{B}$, and since $\mathfrak{B}$ is a free
${\mathfrak{A}}$-module with generator $x$ there exists a unique element
$z\in{\mathfrak{A}}$ such that $z\cdot x=g^{-1}(y)$. Now consider the action
of the element $\phi(z)\in\phi({\mathfrak{A}})$ on the element
$g(x)\in\mathfrak{B}$. We have:

$\displaystyle\phi(z)\cdot g(x)$ $\displaystyle=$ $\displaystyle g(z\cdot
g^{-1}(g(x)))\mbox{ by Proposition \ref{prop_conjugation_action}) }$
$\displaystyle=$ $\displaystyle g(g^{-1}(y))$ $\displaystyle=$ $\displaystyle
y.$

Hence given $y\in\mathfrak{B}$ there exists a unique element $\phi(z)$ of the
order $\phi({\mathfrak{A}})$ in $L[N_{g}]^{G}$ such that $\phi(z)\cdot
g(x)=y$, and so $\mathfrak{B}$ is a free $\phi({\mathfrak{A}})$-module with
generator $g(x)$. Since ${\mathfrak{A}}_{g}$ is the only order in
$L[N_{g}]^{G}$ over which $\mathfrak{B}$ can be free, we conclude that
$\phi({\mathfrak{A}})={\mathfrak{A}}_{g}$.

For the converse statement we replace $N$ by $N_{g}$ and $g$ by $g^{-1}$ in
the argument above. ∎

The interaction between Theorem 5.1 and the previously mentioned results
concerning integral module structure with respect to opposite Hopf-Galois
structures can cause one instance of freeness to propagate to a whole family.

######  Example 5.4.

Let $p,q$ be odd primes with $p\equiv 1\pmod{q}$ and let $L/K$ be a Galois
extension of local or global fields whose Galois group $G$ is isomorphic to
the metacyclic group of order $pq$

$G=\langle s,t\mid s^{p}=t^{q}=e,\;tst^{-1}=s^{d}\rangle,$

as in Example 2.3. For $k=0,\ldots,p-1$ let

$N_{k}=\langle\lambda(s),\lambda(t)\rho(s^{k}t)\rangle.$

It is routine to verify that each $N_{k}$ is distinct, $G$-stable, regular,
and isomorphic to $G$. We find that $\rho(t)$ normalizes each $N_{k}$, and
that $\rho(s)N_{k}\rho(s^{-1})=N_{k+1-d}$. Thus the subgroups $N_{k}$ are
mutually $\rho$-conjugate, and by [17, Corollary 7.2] the same is true for
their opposites $N_{k}^{opp}$.

Now we write $H_{k}$ (repectively $H_{k}^{opp}$) for the Hopf algebra giving
the Hopf-Galois structure corresponding to $N_{k}$ (respectively
$H_{k}^{opp}$), and consider questions of integral module structure. By
Theorem 5.1, if the ring of integers ${\mathfrak{O}}_{L}$ is free over its
associated order in one of the $H_{k}$ then it is free over its associated
order in each of the $H_{k}$, and by [20, Theorem 1.2] if it is free over its
associated order in a particular $H_{k}$ then it is free over its associated
order in $H_{k}^{opp}$. Thus ${\mathfrak{O}}_{L}$ is either free over its
associated orders in all of these $2p$ Hopf-Galois structures or none of them.

##  6\. Interactions with existing constructions

In this section we study the interactions between $\rho$-conjugation and
various results concerning the construction and enumeration of Hopf-Galois
structures on a Galois extension $L/K$. (We no longer assume that $L/K$ is an
extension of local or global fields.)

###  6.1. Byott’s translation theorem and fixed-point free pairs of
homomorphisms

Many results concerning the construction and enumeration of Hopf-Galois
structures on separable field extensions employ Byott’s translation theorem
[2] to address some of the combinatorial difficulties inherent to Greither-
Pareigis theory. Specializing to the Galois case, Byott’s theorem may be
summarized as follows: let $M$ be an abstract group of the same order as $G$,
and consider regular embeddings
$\alpha:M\rightarrow\operatorname{\mbox{Perm}}(G)$ (that is: embeddings with
regular image). Such an embedding yields a bijection $a:M\rightarrow G$
defined by $a(\mu)=\alpha(\mu)[e_{G}]$ for all $\mu\in M$, and thence a
regular embedding $\beta:G\rightarrow\operatorname{\mbox{Perm}}(M)$ defined by
$\beta(g)[\mu]=a^{-1}\lambda(g)a$ for all $g\in G$. Byott’s theorem asserts
that this construction yields a bijection between regular embeddings of $M$
into $\operatorname{\mbox{Perm}}(G)$ and regular embeddings of $G$ into
$\operatorname{\mbox{Perm}}(M)$. Furthermore: $\alpha(M)$ is $G$-stable if and
only if
$\beta(G)\subseteq\operatorname{\mbox{Hol}}(M)=\lambda(M)\operatorname{\mbox{Aut}}(M)\subseteq\operatorname{\mbox{Perm}}(M)$,
and $\alpha(M)=\alpha^{\prime}(M)$ if and only if there exists
$\theta\in\operatorname{\mbox{Aut}}(M)$ such that
$\beta^{\prime}(g)=\theta\beta(g)\theta^{-1}$ for all $g\in G$.

We can study $\rho$-conjugate Hopf-Galois structures from this perspective,
using the reformulation expressed in Proposition 4.1. Given a regular
embedding $\alpha:M\rightarrow\operatorname{\mbox{Perm}}(G)$ and an element
$g\in G$, the map $\alpha_{g}:M\rightarrow\operatorname{\mbox{Perm}}(G)$
defined by $\alpha_{g}(\mu)=\varphi_{g}\alpha(\mu)\varphi_{g}^{-1}$ for all
$\mu\in M$ is a regular embedding with image $\alpha(M)_{g}$. We then find the
following:

######  Lemma 6.1.

Let $\beta:G\rightarrow\operatorname{\mbox{Hol}}(M)$ be the embedding
corresponding to $\alpha$. Then then embedding
$\beta_{g}:G\rightarrow\operatorname{\mbox{Hol}}(M)$ corresponding to
$\alpha_{g}$ is given by $\beta_{g}=\beta\varphi_{g^{-1}}$.

###### Proof.

The bijection $a_{g}:M\rightarrow G$ corresponding to $\alpha_{g}$ is given by

$a_{g}(\mu)=\varphi_{g}\alpha(\mu)\varphi_{g^{-1}}[e_{G}]=\varphi_{g}\alpha(\mu)[e_{G}]=\varphi_{g}a(\mu).$

Hence for $h\in G$ and $\mu\in M$ we have

$\displaystyle\beta_{g}(h)[\mu]$ $\displaystyle=$ $\displaystyle
a^{-1}\varphi_{g}^{-1}\lambda_{G}(h)\overline{\varphi}_{g}a[\mu]$
$\displaystyle=$ $\displaystyle a^{-1}\varphi_{g}^{-1}(hga[\mu]g^{-1})$
$\displaystyle=$ $\displaystyle a^{-1}(g^{-1}hga[\mu]g^{-1}g)$
$\displaystyle=$ $\displaystyle a^{-1}\lambda(\varphi_{g^{-1}}(h))a[\mu]$
$\displaystyle=$ $\displaystyle\beta(\varphi_{g^{-1}}(h))[\mu].$

Hence $\beta_{g}=\beta\varphi_{g^{-1}}$, as claimed. ∎

As an application of Lemma 6.1 we study the interaction between
$\rho$-conjugation and Hopf-Galois structures on Galois extensions arising
from fixed point free pairs of homomorphisms, due to Byott and Childs [5].
Given finite groups $G,M$ of the same order, a pair of homomorphisms
$f_{1},f_{2}:G\rightarrow M$ is called fixed point free if the only element
$h\in G$ for which $f_{1}(h)=f_{2}(h)$ is $h=e_{G}$. Given such a pair of
homomorphisms, the map
$\beta_{f_{1},f_{2}}:G\rightarrow\operatorname{\mbox{Hol}}(M)$ defined by

$\beta_{f_{1},f_{2}}(h)=\lambda(f_{1}(h))\rho(f_{2}(h))\mbox{ for all }h\in G$

is a regular embedding of $G$ into $\operatorname{\mbox{Hol}}(M)$ (where
$\lambda,\rho$ denote the left and right regular representations of $M$), and
therefore yields a Hopf-Galois structure of type $M$ on a Galois extension
with Galois group $G$. Applying Lemma 6.1 we obtain immediately

######  Proposition 6.2.

Let $f_{1},f_{2}:G\rightarrow\operatorname{\mbox{Hol}}(M)$ be a fixed point
free pair of homomorphisms, with corresponding regular embedding
$\beta:G\rightarrow\operatorname{\mbox{Hol}}(M)$, and let $g\in G$. For
$i=1,2$ let $f_{i,g}=f_{i}\varphi_{g}$. Then $f_{1,g},f_{2,g}$ is a fixed
point free pair of homomorphisms, and the corresponding embedding of $G$ into
$\operatorname{\mbox{Hol}}(M)$ is $\beta_{g}$.

######  Example 6.3.

Let $n$ be an even natural number and let $G=\langle r,s\rangle$ and
$M=\langle\mu,\pi\rangle$ both be isomorphic to the dihedral group of order
$2n$ (here $r,\mu$ have order $n$ and $s,\pi$ have order $2$). Fix $0\leq
k\leq n/2-1$ and define $f_{1}:G\rightarrow M$ by
$f_{1}(r^{i}s^{j})=\mu^{i}\pi^{j}$ and $f_{2}:G\rightarrow M$ by
$f_{2}(r^{i}s^{j})=(\mu^{2k}\pi)^{i}$. Then $f_{1},f_{2}$ form a fixed point
free pair of homomorphisms, and the corresponding embedding
$\beta:G\hookrightarrow\operatorname{\mbox{Hol}}(M)$ is given by

$\beta(r)=\lambda(\mu)\rho(\mu^{2k}\pi),\quad\beta(s)=\lambda(\pi).$

(Here $\lambda,\rho$ denote the left and right regular representations of
$M$.) We find that composing $\beta$ with $\varphi_{s}$ results in an
equivalent embedding, whereas composing with each $\varphi_{r^{\ell}}$ for
$0\leq\ell\leq n/2-1$ gives an inequivalent embedding. Thus we obtain $n/2$
mutually $\rho$-conjugate Hopf-Galois structures of dihedral type on a Galois
extension with Galois group isomorphic to $G$. In fact, these are the Hopf-
Galois structures described in Example 2.4.

###  6.2. Hopf-Galois structures arising from abelian maps

An abelian map is a group endomorphism with abelian image. In [7] Childs shows
that fixed point free abelian maps on a finite group $G$ yield a family of
Hopf-Galois structures on a Galois extension $L/K$ with Galois group $G$. This
approach is generalized by Koch [13], as follows: let $\psi:G\rightarrow G$ be
an abelian map, and for each $h\in G$ define
$\eta_{\psi}(h)\in\operatorname{\mbox{Perm}}(G)$ by

$\eta_{\psi}(h)=\lambda(h\psi(h)^{-1})\rho(\psi(h)^{-1}).$

Then $N_{\psi}=\\{\eta_{\psi}(h)\mid h\in G\\}$ is a $G$-stable regular
subgroup of $\operatorname{\mbox{Perm}}(G)$ [13, Theorem 3.1], which therefore
corresponds to a Hopf-Galois structure on $L/K$.

######  Proposition 6.4.

Let $\psi$ be an abelian map on $G$ and let $N_{\psi}$ be the corresponding
$G$-stable regular subgroup of $\operatorname{\mbox{Perm}}(G)$. Then for each
$g\in G$ the subgroup $N_{\psi,g}$ arises from an abelian map on $G$.

###### Proof.

By Proposition 4.1 we have $N_{\psi,g}=\varphi_{g}N_{\psi}\varphi_{g}^{-1}$.
It is shown in [17, Proposition 5.1] that if $\psi$ is an abelian map on $G$
and $\varphi\in\operatorname{\mbox{Aut}}(G)$ then the map
$\varphi\psi\varphi^{-1}$ is also an abelian map on $G$, and
$N_{\varphi\psi\varphi^{-1}}=\varphi N_{\psi}\varphi^{-1}$. Thus $N_{\psi,g}$
arises from the abelian map $\psi^{\prime}=\varphi_{g}\psi\varphi_{g}^{-1}$ on
$G$. ∎

######  Example 6.5.

Let $n\geq 5$ and suppose that $G$ is isomorphic to the symmetric group
$S_{n}$. In [13, Example 3.7] Koch shows that the abelian maps on $G$
correspond bijectively with elements $x\in S_{n}$ of order at most $2$, as
follows:

$\psi_{x}(t)=\left\\{\begin{array}[]{ll}e&\mbox{if }t\in A_{n}\\\ x&\mbox{if
}t\not\in A_{n}\end{array}\right.$

Let $N_{x}$ be the $G$-stable regular subgroup arising from the abelian map
$\psi_{x}$. Applying Proposition 6.4 we see that for each $g\in G$ the
subgroup $N_{\psi,g}$ arises from the abelian map
$\varphi_{g}\psi_{x}\varphi_{g}^{-1}$, which behaves as follows

$\varphi_{g}\psi_{x}\varphi_{g}^{-1}(t)=\left\\{\begin{array}[]{ll}e&\mbox{if
}t\in A_{n}\\\ gxg^{-1}&\mbox{if }t\not\in A_{n}\end{array}\right.$

Thus $\varphi_{g}\psi_{x}\varphi_{g}^{-1}=\psi_{gxg^{-1}}$, and so the
$\rho$-conjugate subgroups correspond to conjugacy classes of elements of
order at most $2$ in $G$. In particular, the subgroups arising from
transpositions are mutually $\rho$-conjugate.

###  6.3. Induced Hopf-Galois structures

The method of induced Hopf-Galois structures is due to Crespo, Rio, and Vela
[10]: if $T$ is a subgroup of $G$ having a normal complement $S$ in $G$ then,
given a Hopf-Galois structure on $L/L^{T}$ and a Hopf-Galois structure on
$L^{T}/K$, we can induce a Hopf-Galois structure on $L/K$. Note that $T$ need
not be a normal subgroup of $G$, and so the extension $L^{T}/K$ need not be a
Galois extension. In order to study Hopf-Galois structures on this extension,
we need the full strength of the Greither-Pareigis classification. In this
context, it implies that the Hopf-Galois structures on $L^{T}/K$ correspond
bijectively with regular subgroups of $\operatorname{\mbox{Perm}}(G/T)$ (the
permutation group of the left coset space) that are $G$-stable in the sense
that they are normalized by the image of $G$ under the left translation map
$\lambda:G\rightarrow\operatorname{\mbox{Perm}}(G/T)$ defined by
$\lambda(h)[xT]=hxT$ for all $h\in G$ and $xT\in G/T$.

Given a $G$-stable regular subgroup $A\subset\operatorname{\mbox{Perm}}(G/T)$
and a $T$-stable regular subgroup $B\subset\operatorname{\mbox{Perm}}(T)$, we
use the fact that $G$ is the semidirect product of $S$ and $T$ to identify $A$
with a regular subgroup of $\operatorname{\mbox{Perm}}(S)$, and then to define
a map $\eta:A\times B\rightarrow\operatorname{\mbox{Perm}}(G)$ by

$\eta(a,b)[st]=a[s]b[t]\mbox{ for all }s\in S\mbox{ and }t\in T.$

It is easy to see that $\eta$ is an embedding whose image $N$ is a regular
subgroup of $\operatorname{\mbox{Perm}}(G)$ isomorphic to $A\times B$; [10,
Theorem 3] shows that $N$ is $G$-stable, and hence corresponds to a Hopf-
Galois structure on $L/K$. We say that the subgroup $N$ is induced from the
subgroups $A$ and $B$, and that the Hopf-Galois structure on $L/K$
corresponding to $N$ is induced from those corresponding to $A$ on $L^{T}/K$
and $B$ on $L/L^{T}$.

Now let $g\in G$ and let $\varphi=\varphi_{g}$ be the inner automorphism
corresponding to $g$. Since $S$ is normal in $G$ we have $\varphi(S)=S$, and
it is easily shown that $G$ is the semidirect product of $S$ and $\varphi(T)$.
We shall show that the Hopf-Galois structure corresponding to $N_{g}$ is
induced from certain Hopf-Galois structures on $L^{\varphi(T)}/K$ and
$L/L^{\varphi(T)}$.

######  Lemma 6.6.

Let $g\in G$ and $\varphi=\varphi_{g}$. Then:

  1. i)

$\varphi A\varphi^{-1}$ identifies naturally with a $G$-stable regular
subgroup of $\operatorname{\mbox{Perm}}(G/\varphi(T))$;

  2. ii)

$\varphi B\varphi^{-1}$ identifies naturally with a $\varphi(T)$-stable
regular subgroup of $\operatorname{\mbox{Perm}}(\varphi(T))$.

###### Proof.

To show (i), let $a\in A$ and $s\varphi(T)\in G/\varphi(T)$ (with $s\in S$).
Then

$\varphi a\varphi^{-1}[s\varphi(T)]=\varphi
a(\varphi^{-1}(s)T)=s^{\prime}\varphi(T)$

for some element $s^{\prime}\in S$. Hence $\varphi A\varphi^{-1}$ permutes
$G/\varphi(T)$, and this action is regular because $A$ acts regularly on
$G/T$. To show $G$-stability, let $h\in G$ and consider

$\displaystyle\lambda(h)\varphi a\varphi^{-1}\lambda(h^{-1})[s\varphi(T)]$
$\displaystyle=$ $\displaystyle h\varphi
a[\varphi^{-1}(h^{-1})\varphi^{-1}(s)T]$ $\displaystyle=$
$\displaystyle\varphi(\varphi^{-1}(h)a[\varphi^{-1}(h^{-1})\varphi^{-1}(s)T]$
$\displaystyle=$
$\displaystyle\varphi(\varphi^{-1}(h)a\varphi^{-1}(h^{-1})\varphi^{-1}[s\varphi(T)]$
$\displaystyle=$ $\displaystyle\varphi a^{\prime}\varphi^{-1}[s\varphi(T)]$

for some $a^{\prime}\in A$, since $A$ is $G$-stable. Hence $\varphi
A\varphi^{-1}$ is $G$-stable. The proof of (ii) is similar. ∎

Note that if $g\in T$ then the regular subgroup $\varphi B\varphi^{-1}$ of
$\operatorname{\mbox{Perm}}(T)$ is $\rho$-conjugate to $B$.

######  Proposition 6.7.

For $A,B,N,$ and $g$ as above, the subgroup
$N_{g}\leq\operatorname{\mbox{Perm}}(G)$ is induced from $\varphi
A\varphi^{-1}\leq\operatorname{\mbox{Perm}}(G/\varphi(T))$ and $\varphi
B\varphi^{-1}\leq\operatorname{\mbox{Perm}}(\varphi(T))$.

###### Proof.

Let $h\in G$, and write $h=s\varphi(t)$ with $s\in S$ and $t\in T$. Now let
$\eta=\eta(a,b)\in N$ and consider

$\displaystyle\varphi\eta\varphi^{-1}[s\varphi(t)]$ $\displaystyle=$
$\displaystyle\varphi\eta[\varphi^{-1}(s)t]$ $\displaystyle=$
$\displaystyle\varphi(a[\varphi^{-1}(s)]b[t])\mbox{ (note that
$\varphi^{-1}(s)\in S$)}$ $\displaystyle=$ $\displaystyle(\varphi
a[\varphi^{-1}(s)])(\varphi b[t])$ $\displaystyle=$ $\displaystyle(\varphi
a[\varphi^{-1}(s)])(\varphi b[\varphi^{-1}(\varphi(t))])$ $\displaystyle=$
$\displaystyle\eta(\varphi a\varphi^{-1},\varphi b\varphi^{-1})[s\varphi(t)].$

Hence $N_{g}\leq\operatorname{\mbox{Perm}}(G)$ is induced from $\varphi
A\varphi^{-1}\leq\operatorname{\mbox{Perm}}(G/\varphi(T))$ and $\varphi
B\varphi^{-1}\leq\operatorname{\mbox{Perm}}(\varphi(T))$. ∎

We remark that the argument of Proposition 6.7 remains valid if $\varphi$ is
any automorphism of $G$ that preserves $S$.

######  Example 6.8.

Let $p,q$ be odd primes with $p\equiv 1\pmod{q}$ and let $L/K$ be a Galois
extension of fields whose Galois group $G$ is isomorphic to the metacyclic
group of order $pq$

$G=\langle s,t\mid s^{p}=t^{q}=e,\;tst^{-1}=s^{d}\rangle,$

as in Example 2.3. Let $T=\langle t\rangle$ and $S=\langle s\rangle$. Then the
conjugates of $T$ are precisely the subgroups $T_{i}=s^{i}Ts^{-i}$, and $S$ is
a normal complement to each of these subgroups in $G$. For each $i$ the
extensions $L/L^{T_{i}}$ and $L^{T_{i}}/K$ have prime degree and each admits a
unique Hopf-Galois structure (note that each $L^{T_{i}}/K$ is non-normal). In
each case, we can induce a Hopf-Galois structure on $L/K$ from these, and by
Proposition 6.7 we obtain a family of $p$ mutually $\rho$-conjugate Hopf-
Galois on $L/K$. Referring to Example 2.3 we see that these are all of the
Hopf-Galois structures of cyclic type admitted by $L/K$.

## References

  * [1] D. Bachiller.  Counterexample to a conjecture about braces.  J. Algebra, 453:160–176, 2016.
  * [2] N. P. Byott.  Uniqueness of Hopf Galois structure of separable field extensions.  Comm. Algebra, 24:3217–3228, 3705, 1996.
  * [3] N. P. Byott.  Galois structure of ideals in wildly ramified abelian $p$-extensions of a $p$-adic field, and some applications.  J. Théor. Nombres Bordeaux, 9:201–219, 1997.
  * [4] N. P. Byott.  Hopf-Galois structures on Galois field extensions of degree $pq$.  J. Pure Appl. Algebra, 188(1-3,2.2):45–57, 2004.
  * [5] N. P. Byott and L. N. Childs.  Fixed point free pairs of homomorphisms and Hopf Galois structures.  New York J. Math., 18:707–731, 2012.
  * [6] L. N. Childs.  Taming Wild Extensions: Hopf algebras and local Galois module theory, volume 80 of Mathematical Surveys and Monographs.  American Mathematical Society, 2000.
  * [7] L. N. Childs.  Fixed-point free endomorphisms and Hopf Galois structures.  Proc. Amer. Math. Soc., 141:1255–1265, 2013.
  * [8] L. N. Childs, C. Greither, K. P. Keating, A. Koch, T. Kohl, P. J. Truman, and R. Underwood.  Hopf algebras and Galois module theory, volume 260 of Mathematical Surveys and Monographs.  American Mathematical Society, 2021.
  * [9] T. Crespo, A. Rio, and M. Vela.  The Hopf Galois property in subfield lattices.  Comm. Algebra, 44:336–353, 2016.
  * [10] T. Crespo, A. Rio, and M. Vela.  On the Galois correspondence theorem in separable Hopf Galois theory.  Publ. Mat. (Barcelona), 60(1):221–234, 2016.
  * [11] C. Greither and B. Pareigis.  Hopf Galois theory for separable field extensions.  J. Algebra, 106:239–258, 1987.
  * [12] L. Guarneri and L. Vendramin.  Skew braces and the Yang-Baxter equation.  Math. Comp., 86(307):2519–2534, 2017.
  * [13] A. Koch.  Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures.  Proc. Amer. Math. Soc., 8(16):189–203, 2021.
  * [14] A. Koch, T. Kohl, P. J. Truman, and R. Underwood.  Isomorphism problems for Hopf-Galois structures on separable field extensions.  J. Pure Appl. Algebra, 223:2230–2245, 2019.
  * [15] A. Koch, L. Stordy, and P. J. Truman.  Abelian fixed point free endomorphisms and the yang-baxter equation.  New York J. Math., 26:1473–1492, 2020.
  * [16] A. Koch and P. J. Truman.  Opposite skew left braces and applications.  J. Algebra, 546:218–235, 2020.
  * [17] A. Koch and P. J. Truman.  Skew left braces and isomorphism problems for Hopf-Galois structures on Galois extensions.  J. Algebra Appl., 2022.
  * [18] K. Nejabati Zenouz.  Skew braces and Hopf-Galois structures of Heisenberg type.  J. Algebra, 524:187–225, 2019.
  * [19] P. J. Truman.  Canonical nonclassical Hopf–Galois module structure of nonabelian Galois extensions.  Comm. Algebra, 44(3):1119–1130, 2016.
  * [20] P. J. Truman.  Commuting Hopf-Galois structures on a separable extension.  Comm. Algebra, 46(4):1420–1427, 2018.


# A Comparison of Inquiry-Based Conceptual Feedback vs. Traditional Detailed
Feedback Mechanisms in Software Testing Education: An Empirical Investigation

Lucas Cordova  Western Oregon UniversityMonmouthORUSA<EMAIL_ADDRESS>,
Jeffrey Carver, Noah Gershmel  University of AlabamaTuscaloosaALUSA
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Gursimran Walia  Georgia
Southern UniversityStatesboroGAUSA<EMAIL_ADDRESS>
(2021)

###### Abstract.

The feedback provided by current testing education tools about the
deficiencies in a student’s test suite either mimics industry code coverage
tools or lists specific instructor test cases that are missing from the
student’s test suite. While useful in some sense, these types of feedback are
akin to revealing the solution to the problem, which can inadvertently
encourage students to pursue a trial-and-error approach to testing, rather
than using a more systematic approach that encourages learning. In addition to
not teaching students why their test suite is inadequate, this type of
feedback may motivate students to become dependent on the feedback rather than
thinking for themselves. To address this deficiency, there is an opportunity
to investigate alternative feedback mechanisms that include a positive
reinforcement of testing concepts. We argue that using an inquiry-based
learning approach is better than simply providing the answers. To facilitate
this type of learning, we present Testing Tutor, a web-based assignment
submission platform that supports different levels of testing pedagogy via a
customizable feedback engine. We evaluated the impact of the different types
of feedback through an empirical study in two sophomore-level courses. We use
Testing Tutor to provide students with different types of feedback, either
traditional detailed code coverage feedback or inquiry-based learning
conceptual feedback, and compare the effects. The results show that students
that receive conceptual feedback had higher code coverage (by different
measures), fewer redundant test cases, and higher programming grades than the
students who receive traditional code coverage feedback.

Testing, Education, Pedagogy, Tools

††journalyear: 2021††copyright: acmcopyright††conference: Proceedings of the
52nd ACM Technical Symposium on Computer Science Education; March 13–20, 2021;
Virtual Event, USA††booktitle: Proceedings of the 52nd ACM Technical Symposium
on Computer Science Education (SIGCSE ’21), March 13–20, 2021, Virtual Event,
USA††price: 15.00††doi: 10.1145/3408877.3432417††isbn:
978-1-4503-8062-1/21/03††ccs: Software and its engineering††ccs: Software and
its engineering Software creation and management††ccs: Software and its
engineering Software verification and validation

##  1\. Introduction

In programming courses, students often write and debug code by trial-and-
error, running it on sample input (often provided by an instructor), or just
using the compiler (i.e. believing that if it compiles, it must be correct)
(Edwards, 2004). Students’ lack of testing knowledge and ability mistakenly
leads them to believe that they can determine the correctness of their code
from a small number of test cases. They do not foresee their programs failing
other test cases that may be used by their instructor and thus do not
understand why they receive a low grade. Moreover, after their formal
education, students are not properly equipped to enter the workplace because
many graduate with a knowledge gap about software testing (Begel and Simon,
2008; Haddad, 2002; Kitchenham et al., 2005; Lethbridge, 1998, 2000a, 2000b;
Wang et al., 2008)

Researchers have developed pedagogical tools (Collofello and Vehathiri, 2005;
Spacco et al., 2006; Edwards, 2003a) to address the shortcomings in testing
education. These tools (e.g., Marmoset, WebCAT) each provide students with
different types of information about their code and tests including
information like: whether the instructor’s tests have passed, code coverage,
and even pointing out exactly which portion of code is problematic. While
these tools do provide useful information, one of the main drawbacks is that
these tools tend to provide the ‘answers’ (usually after the students meet
some criteria). For example, a tool might tell a student exactly which of the
instructor’s test cases are missing from his or her test suite. In addition,
code coverage tools indicate exactly which portions of code are not fully
tested. While this information may be useful for improving the current test
suite, beginning students lack the metacognitive ability to use that
information to identify their knowledge gaps about fundamental testing
concepts and determine why their test suite was inadequate. Tools that provide
this type of automated feedback may actually discourage students from thinking
on their own and instead encourage them to rely upon the automated feedback
(Buffardi and Edwards, 2015).

We approach this limitation by focusing on an inquiry-based learning (IBL)
approach (Prince and Felder, 2006). IBL is based on the Cognitive
Constructivist learning theory (Vygotskij and Cole, 1981) that students are
more engaged when they actively construct and validate their own knowledge
(through experience) rather than simply receiving the answers. IBL has
improved student learning in many disciplines including: Social Science
(Justice et al., 2002), Geography (Spronken-Smith et al., 2007), Psychology
(Muukkonen et al., 2005), Medicine (Houlden et al., 2004), Physics (Abell,
2005), Meteorology (Yarger et al., 2000), Chemistry (Zoller, 1999), and
Forestry (Yin, 2006). Meta-analyses of IBL experiments showed that, compared
with a traditional classroom approach, IBL results in an improved academic
achievement, deeper understanding of content, critical thinking skills,
motivation, engagement, and creativity (Shymansky et al., 1990; Smith, 1996).
Previous work about IBL in CS (when used to teach the LOGO programming
language) has shown that learning should emphasize discovery and that there
needs to be some tutor facilitation to guide the learning process rather than
leaving it open-ended (Pettit et al., 2015; Hu et al., 2016).

To implement an IBL-based approach for improving software testing education,
we built Testing Tutor, a web-based assignment submission platform that
supports testing pedagogy via a customizable feedback engine that can be
integrated into any level of the CS curriculum. Testing Tutor’s innovative
pedagogical contribution is in the type of feedback it provides. Rather than
providing a student with the ’answer’ (e.g. exactly which tests are missing)
or the level of test coverage, Testing Tutor provides students with conceptual
feedback. Conceptual feedback informs the student which underlying fundamental
testing concepts their test suites does not adequately cover and provides
suggestions for the student to initiate their own learning process about those
concepts. This type of feedback will allow the student to determine on his or
her own how to improve their test suite, rather than being told by the system
exactly what is missing. Examples of conceptual feedback for a CS2-level
assignment are “the test suite has not fully tested all boundary conditions”
and “the test suite misses part of a compound Boolean expression” along with
resources provided (examples, videos).

The primary goal of this paper is to assess whether conceptual feedback helps
students produce better, more concise and comprehensive test suites.

##  2\. Previous Approaches

Some educators have developed approaches to improve software testing pedagogy.
Testing Tutor builds upon the ideas and shortcomings in these existing
approaches.

Collofello and Vehathiri (Collofello and Vehathiri, 2005) developed a testing
simulator that executes student test cases against built-in buggy programs and
displays missing test cases. The tool reports the following metrics: 1) test
completeness – a measure of input coverage, 2) flow coverage – a measure of
statement/path coverage, 3) correctness – a measure of student test outputs
correspondence to expected test outputs, and 4) a fault detection metric – a
measure of fault detection effectiveness. At the conclusion of the testing
exercise, the tool gives the students the ’answers’ (e.g. the correct set of
tests). While this approach does encourage students to improve the test suite
for their current project, it does not provide feedback about why the test
suite is incomplete or help students carry the knowledge over to future
assignments. Testing Tutor borrows the idea of learning how to test by testing
someone else’s code. But different from this approach, which allows students
to succeed through trial-and-error, Testing Tutor provides conceptual feedback
about why the test suite is incomplete and provides support for learning
fundamental testing concepts that should carry over to future assignments.

Marmoset (Spacco et al., 2006) and WebCAT (Edwards, 2004, 2003a, 2003b) use
test coverage to provide students with automated feedback on their code.
Marmoset provides public tests (i.e. visible to students) and private tests
(i.e. not visible to students). When students’ code passes the public tests,
Marmoset executes the private tests and informs the student of how many
private tests failed along with the names of two of those failed tests.
Similarly, WebCAT uses a concept of public and private tests to perform
automated grading and provides students with feedback during the development
process. WebCAT provides students with detailed feedback on failed tests and
can provide code annotations with suggestions for improvement. A major
drawback to these tools is the extra grading of tests cases required of the
instructors. Testing Tutor borrows the idea of encouraging students to write
better test cases but does not add additional work for instructors to grade
test cases. In addition, rather than providing feedback about specific test
cases, Testing Tutor provides guidance on how students can improve their own
tests, which should encourage students to better learn the underlying testing
concepts.

ProgTest (de Souza et al., 2014) uses two sets of code and test cases, one
provided by the student and one provided by the tool or the instructor.
ProgTest runs the student’s tests against both their own code and the code
provided by the tool and runs the tests provided by the tool against the
student’s code. By comparing the coverage between the student’s code and the
instructor’s code, the student can continuously improve their code and tests.
This approach produces higher quality outcomes in terms of semantic
correctness compared with developing tests only after developing a complete
code solution (Kazerouni et al., 2019). Testing Tutor expands upon the type of
feedback provided by ProgTest to include conceptual information about the
tests that are missing.

##  3\. Testing Tutor

Testing Tutor is a web-based tool that supports testing pedagogy by helping
students learn to develop higher-quality test suites and become more effective
testers. Testing Tutor uses a reference implementation and its corresponding
test suite to identify which tests are missing from a student’s test suite.
Each missing test has one or more fundamental testing concept (e.g. testing
boundary conditions or testing for data integrity) attributed to it. Based on
the missing tests, Testing Tutor helps the students understand which
fundamental testing concept knowledge they lack so they can improve their own
test suite. Testing Tutor can be used in either Learning Mode or Development
Mode. In Learning Mode, Testing Tutor teaches a student how to develop a
complete test suite for a reference implementation of a program. In
Development Mode, Testing Tutor helps a student completely test their own
newly written code. Learning Mode can be used by itself as a testing exercise
or to prepare students to write their own code for later use with Development
Mode. If instructors use Learning Mode in a stand-alone fashion (i.e. students
will not later implement the assignment), then instructors can decide whether
students see the reference code (white-box testing) or only see the
specification (black-box testing). If instructors use Learning Mode to prepare
students for the Development Mode, then the students will not see the
reference code (black-box testing).

Testing Tutor is different from existing software testing education pedagogy
because of its’ customizable feedback mechanism. Testing Tutor allows an
instructor to tailor the level and type of feedback provided to the students.
By annotating the test cases in the reference implementation and configuring
the type of feedback Testing Tutor provides, an instructor can choose which
learning concepts she or he wants to be the focus. Currently, Testing Tutor
supports three types of feedback mechanisms, as discussed in this paper
(detailed feedback, conceptual feedback, and no feedback). This design creates
an opportunity for instructors and researchers to investigate which feedback
mechanisms best promote learning and improvement while teaching software
testing concepts.

The features of Testing Tutor include:

  * •

Web-based interface \- access to the tool via any web browser;

  * •

Authorization and authentication management \- supports institution
hierarchies for courses, students, faculty, administrators, and assignments;

  * •

Multi-institution support \- each institution can configure Testing Tutor so
that users, assignments, and reports remain separate from other institutions;

  * •

Assignment repository \- assignments can be private (to the instructor),
shared only within an institution, or public;

  * •

Tailored feedback \- instructors select the type of feedback: no feedback,
detailed feedback, or conceptual feedback;

  * •

Course management \- generate reports for courses, instructors, and individual
students, and analysis for assignments or groups of assignments; and

  * •

Plug-in platform architecture \- facilitates the addition of more programming
languages and is built to scale.

##  4\. Experiment

To investigate the impact of conceptual feedback compared with traditional
detailed code coverage feedback using Testing Tutor’s Learning Mode, we pose
the following high-level research questions:

RQ1: How do different types of feedback (conceptual, detailed, none) affect
the quality of student test suites?

Then, to specifically evaluate the usefulness of Testing Tutor, we pose a
second research question:

RQ2: What are the students’ perception of the usefulness of Testing Tutor in
terms of its usability and the feedback provided?

To answer these research questions, we conducted a series of two quasi-
experiments (an initial study followed by a replication). The remainder of
this section provides details on these studies.

###  4.1. Participating Subjects and Artifacts

We performed the studies in a sophomore-level software testing course at
Oregon Institute of Technology in the Spring and Summer 2019 semesters. We
chose this course because it has the goal of helping students produce and
improve the quality of code. Prior to this course, students completed CS1, CS2
and CS3 (Data Structures).

For each study (the original and the replication), we split the students into
two groups based on their course section. The students in Group A received the
traditional detailed coverage feedback. The students in Group B received the
conceptual feedback. Table 1 illustrates the number of participants in each
group.

Table 1. Study group compositions Study | Group A Participants | Group B Participants
---|---|---
Spring 2019 | 15 | 16
Summer 2019 | 13 | 15

Over the course of the study, the students in each group received the same
five assignments, all written in Java 1.11. We instructed the students to
produce the most comprehensive, yet smallest, test suite possible for each
assignment. The five assignments were:

  * •

Assignment 1 \- An I/O program (a calendar program taking a date as input and
returning the date of the day before, the day after, one week before, or one
week ahead).

  * •

Assignment 2 \- A state-based data structure (a queue) supporting all queue
operations and exception-handling.

  * •

Assignment 3 \- An object-oriented calculator containing multiple interfaces
and inheritance.

  * •

Assignment 4 \- A comma-separated value (CSV) parser built using the Visitor
design pattern.

  * •

Assignment 5 \- A banking application built using the Observer pattern for
support of multiple clients.

###  4.2. Independent Variable – Type of Feedback

To measure the impact the type of feedback had students’ testing performance,
we defined the following levels for the independent variable, and configured
Testing Tutor accordingly:

  * •

Treatment A \- Traditional detailed feedback similar to code coverage output
from tools like JaCoCo and CodeCover.

  * •

Treatment B \- Conceptual feedback which provides the student with the testing
concepts that are not adequately tested from the perspective of the instructor
and includes resources to review (textual and video).

Figure 1 provides an example of the detailed coverage type of feedback
(Treatment A). Figure 2 provides an annotated example of conceptual feedback
(Treatment B).

Figure 1. Detailed feedback example Figure 2. Conceptual feedback example

###  4.3. Study Design

We received IRB approval for the studies. We employed the same seven phases in
both instances of the course. When using Testing Tutor, we instructed the
students to create the smallest, yet most complete test suite possible. We
allowed the students to submit their code and tests as often as they would
like to receive feedback.

  * •

Phase 1 - Training : Training on using Testing Tutor.

  * •

Phase 2 - Pre-test : Students completed Assignment 1 using Testing Tutor,
configured to provide no feedback (baseline).

  * •

Phases 3-5 - Data Collection : Students completed assignments 2-4 using
Testing Tutor. Students in Group A received the detailed feedback (Treatment
A). Students in Group B received the conceptual feedback (Treatment B).

  * •

Session 6 - Post-test : Students completed assignment 5 using Testing Tutor,
configured to provide no feedback. This data helps us understand the students’
retention of information learned from the feedback provided by Treatments A
and B.

  * •

Session 7 - Survey : Students complete a survey about their experience using
Testing Tutor and the likelihood they would use it in future courses.

###  4.4. Dependent Variables

To help answer our research questions, we gathered data to support multiple
dependent variables. First, for each assignment submission, Testing Tutor
collected the following data:

  * •

Line coverage \- The percentage of line coverage obtained.

  * •

Branch coverage \- The percentage of branch coverage obtained.

  * •

Conditional coverage \- The percentage of conditional coverage obtained.

  * •

Redundant tests \- The number of tests in the test suite that are redundant
(i.e. test code that is tested by another test).

Second, for each assignment we gathered the following data outside of Testing
Tutor:

  * •

Assignment grade \- The instructor assigns a grade for the quality of the test
suite based on a
rubric111https://github.com/TestingTutor/Data/blob/master/SIGCSE21/Rubric.pdf
that includes code coverage achieved, the number of redundant tests, and a
visual inspection of test quality.

Lastly, we collected the following data at the end of the study:

  * •

Perception of student understanding of the feedback \- An end-of-study
optional and anonymous survey gathered the students’ perception of the
feedback provided by Testing Tutor as well as the usability of Testing Tutor.

##  5\. Results

We organize this section around the study phases (Section 4). We do not have
any a priori reason to believe that the students in the two semesters are
significantly different from each other. Therefore, in the analyses that
follow, we combine the data from both semesters. In our ANOVA tests, we
include the semester as one of the factors. For the sake of space, we do not
include all the details of the ANOVA results in the paper. These are available
in the online
appendix222https://github.com/TestingTutor/Data/tree/master/SIGCSE21.

###  5.1. Pre-test

First, we analyzed the performance of the students on the pre-test (Assignment
1) to serve as a baseline and ensure no systematic differences between the
groups. Table 2 provides an overview of the data. The results of the 2-way
ANOVA tests for each of the five dependent variables collected on Assignment 1
shows:

  * •

No significant differences between the groups for any of the dependent
variables

  * •

No main or interaction effects from the semester

Therefore, we can conclude based on the pre-test, that there was no built-in
bias in the grouping.

Table 2. Pre-test Results Dependent Variable |  Treatment A (Detailed) |  Treatment B (Conceptual)
---|---|---
Line Coverage |  35% |  35.7%
Branch Coverage |  35.3% |  34.9%
Conditional Coverage |  35.1% |  36.6%
Redundant Tests |  4.86 |  4.90
Assignment Grade |  57.95% |  58.42%

###  5.2. Comparison of Approaches

Next, we analyzed whether there were any differences in the dependent
variables between the two groups for Assignments 2-4. In this case, the ANOVA
tests were slightly more complicated. Because we had no reason to believe the
specific assignments would impact the dependent variables, we analyzed
Assignments 2-4 together. As a result, we added a factor to the ANOVA. For
these five ANOVAs (one for each dependent variable), we have the following
factors:

  * •

Treatment (Detailed/Conceptual)

  * •

Assignment (2/3/4)

  * •

Semester (Spring/Summer)

Table 3 overviews the results which show:.

  * •

Significant differences (p $<$ .001) for all dependent variables;

  * •

One case (Branch Coverage) in which the Assignment factor had showed a main
effect;

  * •

One case (Conditional Coverage) in which the Semester showed a main effect;
and

  * •

Two cases (Branch Coverage and Grade) where the Assignment showed an
interaction with the Treatment

Overall, the results show students who received the conceptual feedback
performed significantly better than those who received the detailed feedback.
At this point, we do not have a good explanation for few cases where other
factors showed main or interaction effects, but will continue to explore these
in future studies.

Table 3. Main Results Dependent Variable |  Treatment A (Detailed) |  Treatment B (Conceptual)
---|---|---
Line Coverage |  43.4% |  55.1%
Branch Coverage |  43.1% |  52.7%
Conditional Coverage |  45.4% |  57.5%
Redundant Tests |  4.86 |  3.33
Assignment Grade |  60.37% |  68.27%
*[all differences significant p $<$ .05]

###  5.3. Post-test

The results of the post-test (Assignment 5) provide insight into whether the
type of feedback received on Assignments 2-4 affected the students’ ability to
test when that feedback was removed. In Assignment 5, the students received no
Testing Tutor feedback. In this case, we ran five 2-way ANOVAs, similar to the
pre-test. The results of these tests (summarized in Table 4) showed:

  * •

Significant differences for all five dependent variables; and

  * •

No main or interaction effects from the Semester

Therefore, we can conclude that the type of feedback received did have an
effect on learning. The students who received the conceptual feedback were
able to carry over what they learned more effectively than those received the
detailed feedback.

Table 4. Post-test Results Dependent Variable |  Treatment A (Detailed) |  Treatment B (Conceptual)
---|---|---
Line Coverage |  37.9% |  68.8%
Branch Coverage |  38.6% |  69.4%
Conditional Coverage |  44.8% |  72.6%
Redundant Tests |  4.29 |  2.29
Assignment Grade |  60.31% |  78.95%
*[all differences significant p $<$ .05]

###  5.4. Survey

The end-of-study optional survey contained nine questions each using a 7-point
rating scale and three open-ended questions to gather information about
usability. To examine whether there was any significant difference between the
students in the two groups, we conducted a t-test for each question. Table 5
shows the averages for each group across both studies. In all cases, those
receiving conceptual feedback viewed Testing Tutor significantly more
favorably. Due to space, detailed results are available in the online
appendix.

Table 5. Survey Results (all questions on a 7-point scale) |  Question Text |  Treatment A (Detailed) |  Treatment B (Conceptual)
---|---|---|---
1 |  The information that Testing Tutor provided helped me discover deficiencies in the code coverage. |  3.57 |  5.82
2 |  The information Testing Tutor provided helped me discover redundant tests. |  4.25 |  5.21
3 |  The information Testing Tutor provided regarding code coverage deficiencies made a lasting impression on how I approach software testing in the future. |  3.75 |  5.52
4 |  The information Testing Tutor provided regarding redundant tests made a lasting impression on how I approach software testing in the future. |  3.18 |  4.85
5 |  Testing Tutor helped me become more EFFECTIVE at testing (achieving higher code coverage and reducing redundant tests) |  3.43 |  5.61
6 |  Testing Tutor helped me become more PRODUCTIVE at testing (achieving higher code coverage and reducing redundant tests during the amount of time spent). |  3.82 |  5.97
7 |  Testing Tutor is easy to use. |  3.89 |  5.79
8 |  I learned to use Testing Tutor quickly. |  5.14 |  6.21
9 |  I would recommend Testing Tutor to someone learning software testing. |  3.40 |  6.21
*[all differences significant p $<$ .05]

##  6\. Discussion

The objective of these experiments were to compare the effects of inquiry-
based conceptual feedback with those of more traditional feedback mechanisms
for software testing education. We now discuss insights and possible
implications for software testing education as well as the limitations to
these studies.

###  6.1. Answers to Research Questions

To summarize the results in the previous section, we provide an answer for
each research question.

RQ1 - Effects of Feedback: Perhaps of greatest practical significance, our
results show that students, who began on an equal footing relative to testing
knowledge and skill (based on our pre-test), achieved significantly different
levels of code coverage, test redundancies, and programming grades based on
the type of feedback they received. On average, students who received
conceptual feedback had higher code coverage (line, branch, and conditions),
fewer redundant test cases, and higher programming grades compared with the
students who received detailed feedback. This increased performance occurred
both while the students received the feedback (during Assignments 2-4) and
once the feedback was removed (during Assignment 5).

The fact that the effect carried over to Assignment 5 indicates students who
received conceptual feedback were able to learn how to be better testers,
which resulted in their better performance on Assignment 5. Therefore, we can
conclude that student obtained more long-term benefits from the conceptual
feedback than from the detailed feedback.

RQ2 - Student Perceptions: The end-of-study survey results illustrated the
students’ preference for conceptual feedback. Students who received conceptual
feedback indicated that Testing Tutor helped them meet the objectives of the
assignments (achieving higher code coverage and reducing redundant tests) in a
more productive and effective way than students who received detailed
feedback. The survey results also indicated that conceptual feedback had an
effect on the students’ perception of Testing Tutor’s usability, ease of use,
and whether they would recommend Testing Tutor to someone learning software
testing. All of these results were statistically significant in favor of the
conceptual feedback.

###  6.2. Threats to Validity

Internal validity: Sample size is the primary threat to internal validity. The
sample sizes were constrained by the size of the class sections at the
university during the time the studies were conducted. Although our pre-test
showed no built-in bias in the grouping, we used ANOVAs and t-tests, which
assume a normal distribution.

Construct validity: Our definition of a high-quality test suite is the primary
threat to construct validity. We used code coverage metrics and redundancies
in the test suite as our measure of high-quality. It is possible that other
measures of test-suite quality could produce different results. However, we
believe that our definition is valid and represents an important way to
measure test suite quality. Another potential threat would be how the grades
were assigned to programs. To reduce the chances of bias, the person who
graded the assignments was not aware of the student’s treatment group.

External validity: Because the student population in this academic program
tend to have some professional programming experience, they may not be
representative of other student populations. To increase external validity, we
chose a series of assignments that would be common for a course of this level.
Another potential threat is that the length of the semesters in which the
studies were conducted was different due to a shortened summer term. To
mitigate this threat, we kept the length of the time given for the assignments
consistent between the two studies. Furthermore, the populations in the two
studies were similar in terms of prior knowledge as validated by the pretests.

##  7\. Conclusion and Future Work

These outcomes of this study can be explained by the power of IBL-based
conceptual feedback. IBL-based conceptual feedback informs the student about
the underlying fundamental testing concepts that she or he did not test in her
or his test suite, rather than providing the student with code coverage
analytical feedback. This IBL type of feedback allows the student to determine
on her or his own how to improve the test suite. The feedback also has the
positive side-effects of helping the student gain knowledge, experience, and
reinforcement of fundamental testing concepts, which. ultimately, will make
the student better tester in the long term.

Testing Tutor’s approach falls well within the purview of best pedagogy
practices regarding providing students with the information to reach their
learning objectives, rather than simply following the traditional right/wrong
dichotomy of traditional testing coverage feedback. Testing Tutor trains
students to think about testing in a specific and logical manner while still
allowing them the opportunity to use their critical thinking skills to solve
complex problems. From a pedagogical perspective, the results indicate that
Testing Tutor can be used as an efficient modality to both analyze and
reinforce testing concepts taught as part of the CS curriculum.

We plan additional development work for Testing Tutor including additional
student and class analysis for the instructor, developing a plug-in that
allows a student to submit their tests through an Integrated Development
Environment (IDE), and additional user experience improvements. In addition,
we also plan to perform additional empirical studies with the following
objectives: 1) improve the feedback mechanisms 2) understand the effectiveness
of Testing Tutor’s feedback mechanisms at different levels of the curriculum
3) understand how Testing Tutor can be used as a tool for instructors to gauge
learning and determine whether intervention is necessary to improve students’
learning.

## References

  * (1)
  * Abell (2005) Sandra K. Abell. 2005\.  University Science Teachers as Researchers: Blurring the Scholarship Boundaries.  _Research in Science Education_ 35, 2-3 (2005), 281–298.  https://doi.org/10.1007/s11165-004-5600-x
  * Begel and Simon (2008) Andrew Begel and Beth Simon. 2008.  Struggles of new college graduates in their first software development job.  _ACM SIGCSE Bulletin_ 40, 1 (2008), 226–230.  https://doi.org/10.1145/1352322.1352218
  * Buffardi and Edwards (2015) Kevin Buffardi and Stephen H. Edwards. 2015.  Reconsidering Automated Feedback: A Test-Driven Approach. In _Proceedings of the 46th ACM Technical Symposium on Computer Science Education_ (Kansas City, Missouri, USA) _(SIGCSE ’15)_. Association for Computing Machinery, New York, NY, USA, 416–420.  https://doi.org/10.1145/2676723.2677313
  * Collofello and Vehathiri (2005) J. Collofello and K. Vehathiri. 2005.  An Environment for Training Computer Science Students on Software Testing.  _Proceedings Frontiers in Education 35th Annual Conference_.  https://doi.org/10.1109/fie.2005.1611937
  * de Souza et al. (2014) D. M. de Souza, B. H. Oliveira, J. C. Maldonado, S. R. S. Souza, and E. F. Barbosa. 2014\.  Towards the use of an automatic assessment system in the teaching of software testing. In _2014 IEEE Frontiers in Education Conference (FIE) Proceedings_. 1–8.
  * Edwards (2003a) Stephen H. Edwards. 2003a.  Improving student performance by evaluating how well students test their own programs.  _Journal on Educational Resources in Computing_ 3, 3 (2003), 1–24.  https://doi.org/10.1145/1029994.1029995
  * Edwards (2003b) Stephen H. Edwards. 2003b.  Teaching Software Testing: Automatic Grading Meets Test-First Coding. In _Companion of the 18th Annual ACM SIGPLAN Conference on Object-Oriented Programming, Systems, Languages, and Applications_ (Anaheim, CA, USA) _(OOPSLA ’03)_. Association for Computing Machinery, New York, NY, USA, 318–319.  https://doi.org/10.1145/949344.949431
  * Edwards (2004) Stephen H. Edwards. 2004\.  Using software testing to move students from trial-and-error to reflection-in-action.  _Proceedings of the 35th SIGCSE technical symposium on Computer science education - SIGCSE 04_ (May 2004), 26.  https://doi.org/10.1145/971300.971312
  * Haddad (2002) Hisham Haddad. 2002\.  Post-graduate assessment of CS students: experience and position paper.  _Journal of Computing Sciences in Colleges_ 18, 2 (2002), 189–197.
  * Houlden et al. (2004) Robyn L. Houlden, Jamila B. Raja, Christine P. Collier, Albert F. Clark, and Jennifer M. Waugh. 2004.  Medical students’ perceptions of an undergraduate research elective.  _Medical Teacher_ 26, 7 (2004), 659–661.  https://doi.org/10.1080/01421590400019542
  * Hu et al. (2016) Helen H. Hu, Clifton Kussmaul, Brian Knaeble, Chris Mayfield, and Aman Yadav. 2016.  Results from a Survey of Faculty Adoption of Process Oriented Guided Inquiry Learning (POGIL) in Computer Science. In _Proceedings of the 2016 ACM Conference on Innovation and Technology in Computer Science Education_ (Arequipa, Peru) _(ITiCSE ’16)_. Association for Computing Machinery, New York, NY, USA, 186–191.  https://doi.org/10.1145/2899415.2899471
  * Justice et al. (2002) Christopher Justice, Wayne Warry, and Carl Cuneo. 2002.  A grammar for inquiry: linking goals and methods in a collaboratively taught social sciences inquiry course.  _Alan Blizzard Award_ (2002), 15–27.
  * Kazerouni et al. (2019) Ayaan M. Kazerouni, Clifford A. Shaffer, Stephen H. Edwards, and Francisco Servant. 2019.  Assessing Incremental Testing Practices and Their Impact on Project Outcomes. In _Proceedings of the 50th ACM Technical Symposium on Computer Science Education_ (Minneapolis, MN, USA) _(SIGCSE ’19)_. Association for Computing Machinery, New York, NY, USA, 407–413.  https://doi.org/10.1145/3287324.3287366
  * Kitchenham et al. (2005) Barbara Kitchenham, David Budgen, Pearl Brereton, and Philip Woodall. 2005.  An investigation of software engineering curricula.  _Journal of Systems and Software_ 74, 3 (2005), 325–335.
  * Lethbridge (1998) Timothy C. Lethbridge. 1998\.  A survey of the relevance of computer science and software engineering education.  _Proceedings 11th Conference on Software Engineering Education_ , 56–66.  https://doi.org/10.1109/csee.1998.658300
  * Lethbridge (2000a) Timothy C. Lethbridge. 2000a.  Priorities for the education and training of software engineers.  _Journal of Systems and Software_ 53, 1 (2000), 53–71.
  * Lethbridge (2000b) Timothy C. Lethbridge. 2000b.  What knowledge is important to a software professional?  _Computer (Long. Beach. Calif)_ 33, 5 (2000), 44–50.  https://doi.org/10.1109/2.841783
  * Muukkonen et al. (2005) Hanni Muukkonen, Minna Lakkala, and Kai Hakkarainen. 2005\.  Technology-Mediation and Tutoring: How Do They Shape Progressive Inquiry Discourse?  _Journal of the Learning Sciences_ 14, 4 (2005), 527–565.  https://doi.org/10.1207/s15327809jls1404_3
  * Pettit et al. (2015) Raymond Pettit, John Homer, Roger Gee, Susan Mengel, and Adam Starbuck. 2015.  An Empirical Study of Iterative Improvement in Programming Assignments. In _Proceedings of the 46th ACM Technical Symposium on Computer Science Education_ (Kansas City, Missouri, USA) _(SIGCSE ’15)_. Association for Computing Machinery, New York, NY, USA, 410–415.  https://doi.org/10.1145/2676723.2677279
  * Prince and Felder (2006) Michael J. Prince and Richard M. Felder. 2006.  Inductive Teaching and Learning Methods: Definitions, Comparisons, and Research Bases.  _Journal of Engineering Education_ 95, 2 (2006), 123–138.  https://doi.org/10.1002/j.2168-9830.2006.tb00884.x
  * Shymansky et al. (1990) James A. Shymansky, Larry V. Hedges, and George Woodworth. 1990\.  A reassessment of the effects of inquiry-based science curricula of the 60s on student performance.  _Journal of Research in Science Teaching_ 27, 2 (1990), 127–144.  https://doi.org/10.1002/tea.3660270205
  * Smith (1996) Deborah A. Smith. 1996\.  _A Meta-Analysis of Student Outcomes Attributable to the Teaching of Science as Inquiry as Compared to Traditional Methodology_.  Temple University.
  * Spacco et al. (2006) Jaime Spacco, David Hovemeyer, William Pugh, Fawzi Emad, Jeffrey K. Hollingsworth, and Nelson Padua-Perez. 2006.  Experiences with marmoset.  _Proceedings of the 11th annual SIGCSE conference on Innovation and technology in computer science education - ITICSE 06_ (2006).  https://doi.org/10.1145/1140124.1140131
  * Spronken-Smith et al. (2007) Rachel Spronken-Smith, Tom Angelo, Helen Matthews, Billy O’Steen, and Jane Robertson. 2007.  How Effective is Inquiry-Based Learning in Linking Teaching and Research?  _An International Colloquium on International Policies and Practices for Academic Enquiry_ 7, 4 (2007), 1–7.
  * Vygotskij and Cole (1981) Lev S. Vygotskij and Michael Cole. 1981.  _Mind in society: the development of higher psychological processes_.  Cambridge: Harvard University Press.
  * Wang et al. (2008) Taehyung (George) Wang, Diane Schwartz, and Robert Lingard. 2008\.  Assessing Student Learning in Software Engineering.  _J. Comput. Sci. Coll._ 23, 6 (June 2008), 239–248.  https://doi.org/10.5555/1352383.1352424
  * Yarger et al. (2000) Douglas N. Yarger, William A. Gallus, Michael Taber, J. Peter Boysen, and Paul Castleberry. 2000\.  A Forecasting Activity for a Large Introductory Meteorology Course.  _Bulletin of the American Meteorological Society_ 81, 1 (2000), 31–39.  https://doi.org/10.1175/1520-0477(2000)081<0031:afafal>2.3.co;2
  * Yin (2006) R. Yin. 2006.  Preparing Resource and Environmental Managers with International Understanding and Merits (PREMIUM): Introducing a research experience for undergraduates program.  _Journal of Forestry_ 104 (2006), 320–323.
  * Zoller (1999) Uri Zoller. 1999\.  Scaling-up of higher-order cognitive skills-oriented college chemistry teaching: An action-oriented research.  _Journal of Research in Science Teaching_ 36, 5 (1999), 583–596.  https://doi.org/10.1002/(SICI)1098-2736(199905)36:5<583::AID-TEA5>3.0.CO;2-M