# Stronger arithmetic equivalence

Andrew V. Sutherland Supported by NSF grant DMS-1522526 and Simons Foundation
grant 550033.

###### Abstract

Motivated by a recent result of Prasad, we consider three stronger notions of
arithmetic equivalence: _local integral equivalence_ , _integral equivalence_
, and _solvable equivalence_. In addition to having the same Dedekind zeta
function (the usual notion of arithmetic equivalence), number fields that are
equivalent in any of these stronger senses must have the same class number,
and solvable equivalence forces an isomorphism of adele rings. Until recently
the only nontrivial example of integral and solvable equivalence arose from a
group-theoretic construction of Scott that was exploited by Prasad. Here we
provide infinitely many distinct examples of solvable equivalence, including a
family that contains Scott’s construction as well as an explicit example of
degree 96. We also construct examples that address questions of Scott, and of
Guralnick and Weiss, and shed some light on a question of Prasad.

title = Stronger arithmetic equivalence, author = Andrew V. Sutherland,
plaintextauthor = Andrew V. Sutherland, year=2021, number=23, received=28
April 2021, published=15 November 2021, doi=10.19086/da.29452,

[classification=text]

##  1 Introduction

Number fields that have the same Dedekind zeta function are said to be
_arithmetically equivalent_. Arithmetically equivalent number fields need not
be isomorphic, but they necessarily have the same normal closure and share
many arithmetic invariants. The first nontrivial example of arithmetically
equivalent number fields was given by Gassmann [15], who showed that all such
examples arise from a simple group-theoretic construction, a _Gassmann triple_
$(G,H_{1},H_{2})$ of finite groups in which $H_{1}$ and $H_{2}$ are subgroups
of $G$ that intersect every $G$-conjugacy class with the same cardinality; see
Proposition 2.6 for several equivalent definitions. Gassmann proved that
number fields $K_{1}$, $K_{2}$ with Galois closure $L$ are arithmetically
equivalent if and only if
$(\operatorname{Gal}(L/\mathbf{Q}),\operatorname{Gal}(L/K_{1}),\operatorname{Gal}(L/K_{2}))$
is a Gassmann triple. The number fields $K_{1}$ and $K_{2}$ are isomorphic if
and only if $\operatorname{Gal}(L/K_{1})$ and $\operatorname{Gal}(L/K_{2})$
are conjugate in $\operatorname{Gal}(L/\mathbf{Q})$; we are thus interested in
_nontrivial Gassmann triples_ $(G,H_{1},H_{2})$, those in which $H_{1}$ and
$H_{2}$ are nonconjugate subgroups of $G$.

Gassmann triples $(G,H_{1},H_{2})$ naturally arise in many other settings,
most notably in the construction of isospectral manifolds. As shown by Sunada
[47], if $\pi\colon M\to M_{0}$ is a normal finite Riemannian covering with
transformation group $G$, the quotient manifolds $M/H_{1}$ and $M/H_{2}$ are
_isospectral_ : they have the same sequence of Laplacian eigenvalues. Unlike
the number field case, $M_{1}$ and $M_{2}$ may be isometric even when $H_{1}$
and $H_{2}$ are nonconjugate, but if $H_{1}$ and $H_{2}$ are nonisomorphic and
$M$ is the universal covering of $M_{0}$, then $M/H_{1}$ and $M/H_{2}$ have
nonisomorphic fundamental groups $H_{1}$ and $H_{2}$ and cannot be isometric;
see [44, §4.2] and [47, Corollary 1]. A consequence of this result is that
there are infinitely many distinct ways in which one cannot “hear the shape of
a drum” [16, 23, 33]. A similar result holds in algebraic geometry: if $X$ is
a projective curve over a number field $K$ and $(G,H_{1},H_{2})$ is a Gassmann
triple with $G\subseteq\operatorname{Aut}(X)$, then the Jacobians of the
quotient curves $X/H_{1}$ and $X/H_{2}$ are isogenous over $K$, as proved by
Prasad and Rajan in [39]. As shown in [1], this result can be generalized to
étale Galois covers of $K$-varieties. There is also a discrete analog to
Sunada’s theorem in which one considers a finite graph $\Gamma$ with
automorphism group $G$: Gassmann triples $(G,H_{1},H_{2})$ can be used to
construct nonisomorphic isospectral graphs $\Gamma/H_{1}$ and $\Gamma/H_{2}$,
subject to conditions on $H_{1}$ and $H_{2}$; see [19]. An introduction to the
topics of arithmetic equivalence and isospectrality can be found in [49].

Subgroups $H_{1},H_{2}$ of $G$ form a Gassmann triple $(G,H_{1},H_{2})$ if and
only if the permutation modules $\mathbf{Q}[H_{1}\backslash G]$ and
$\mathbf{Q}[H_{2}\backslash G]$ given by the $G$-action on right cosets are
isomorphic as $\mathbf{Q}[G]$-modules. We then say that $H_{1}$ and $H_{2}$
are _rationally equivalent_ , and if $\mathbf{Z}[H_{1}\backslash G]$ and
$\mathbf{Z}[H_{2}\backslash G]$ are isomorphic as $\mathbf{Z}[G]$-modules, we
say that $H_{1}$ and $H_{2}$ are _integrally equivalent_. Prasad calls
$(G,H_{1},H_{2})$ a _refined Gassmann triple_ when $H_{1},H_{2}\leq G$ are
integrally equivalent, and shows that if $G$ is the Galois group of a Galois
number field $L/\mathbf{Q}$ then the fixed fields $K_{1}:=L^{H_{1}}$ and
$L_{2}:=K_{2}^{H_{2}}$ not only have the same Dedekind zeta function, they
must also have isomorphic idele groups (and in particular, isomorphic class
groups); see [38, Theorem 2]. The first (and so far only) nontrivial example
of a refined Gassmann triple was constructed by Scott [42] more than thirty
years ago. Prasad notes that this example can be realized by number fields,
and that such number fields not only have the same Dedekind zeta function and
isomorphic idele groups, they have isomorphic rings of adeles [38, Theorem 3],
and are thus _locally isomorphic_ , meaning their local algebras are
isomorphic at every place (see Theorem 2.16). Thus even when taken in
aggregate these invariants are not enough to guarantee an isomorphism of
number fields.111One can attach auxiliary $L$-functions to a number field that
in combination with the Dedekind zeta function ensure an isomorphism of number
fields whenever all of these $L$-functions coincide [9, 46]; for function
fields see [4, 8, 45].

As noted by Prasad, Scott’s example is essentially the only nontrivial example
of integral equivalence currently known [38, Remark 1]. It is not clear
whether the particular feature of Scott’s refined Gassmann triple that allowed
Prasad to prove an isomorphism of adele rings is necessarily enjoyed by others
(assuming there are any). Whether Scott’s example is a singular special case
or just the most accessible example of a general phenomenon remains an open
question.

In this article we consider two alternative strengthenings of the notion of
arithmetic equivalence: _local integral equivalence_ and _solvable
equivalence_. The latter implies the former and is sufficient to prove
equality of the number field invariants considered by Prasad, notably
including local isomorphism, which is not obviously implied by integral
equivalence (indeed, we show that it is not implied by the similar but weaker
notion of local integral equivalence).

An attractive feature of both local integral equivalence and solvable
equivalence is that they are much easier conditions to check than integral
equivalence; see Propositions 2.2 and 3.1, and Definition 3.5. In this article
we provide infinitely many nontrivial examples of solvably equivalent triples
$(G,H_{1},H_{2})$ that can be realized as Galois groups of number fields (see
Theorem 3.9), and we prove that

  * •

locally integrally equivalent subgroups need not be isomorphic (see §4.1);

  * •

locally integrally equivalent number fields need not be locally isomorphic
(see §4.2);

  * •

locally integrally equivalent subgroups need not be integrally equivalent (see
§4.3);

  * •

solvably equivalent subgroups need not be integrally equivalent (see §4.4).

We construct an explicit example of locally integrally equivalent number
fields of degree 32 arising from a triple $(G,H_{1},H_{2})$ with
$H_{1}\not\simeq H_{2}$, and an explicit example of solvably equivalent number
fields of degree 96 that are not integrally equivalent. Thus solvable
equivalence does not imply integral equivalence; we leave open the question of
whether integral equivalence implies solvable equivalence.

The example in §4.1 negatively answers a question of Guralnick and Weiss [18,
Question 2.11] and is relevant to the question of Prasad [38, Question 1] as
to whether integrally equivalent subgroups must be isomorphic, since it shows
that locally integrally equivalent subgroups need not be. The solvably
equivalent subgroups we construct are all isomorphic, which leads to the
question of whether solvably equivalent subgroups are necessarily isomorphic.
This question is perhaps more accessible than Prasad’s question, since the
only example of integral equivalence currently known arises in a setting where
rational equivalence is already enough to force isomorphism (rationally
equivalent subgroups of ${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$,
${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$,
${\mathbf{PSL}}_{2}(\mathbf{F}_{\\!p})$ must be isomorphic, see [38, Question
1] and [48, Remark 3.7]). The example of solvable equivalence given in §4.4
does not arise in this setting, and one can find many others.

The example in §4.2 refines an answer to a question of Stuart and Perlis [37,
§4] given by Mantilla-Soler [30, Theorem 3.7] by showing that the sum of the
ramification indices above a given prime in arithmetically equivalent number
fields need not coincide even when their products do (as they must for number
fields that are locally integrally equivalent; see Proposition 3.1).

The examples in §4.3, §4.4 negatively answer a question of Guralnick and Weiss
[18, Question 2.10] as to whether local integral equivalence implies integral
equivalence (this question appears to have also been addressed in the thesis
of D. Hahn in the case of solvable groups [40]). The example in §4.4 also
addresses a question of Scott [42, Remark 4.3] regarding low rank permutation
modules.

##  2 Background and preparation

In this section we recall background material, set notation, and summarize
some of the results we will use. The material in this section is well known to
experts, but we provide short proofs in cases where we were unable to find a
suitable reference (a few of these results seem to be folklore).

Let $G$ be a finite group. For each subgroup $H\leq G$ we use $[H\backslash
G]$ to denote the transitive $G$-set given by the (right) action of $G$ on
(right) cosets of $H$; this action is faithful if and only if the intersection
of all the $G$-conjugates of $H$ (its _normal core_ in $G$) is the trivial
group. We use $\chi_{H}\colon G\to\mathbf{Z}$ to denote the permutation
character $g\mapsto\\#[H\backslash G]^{g}$ that sends $g$ to the number of
$H$-cosets it fixes (the induced character $1_{H}^{G}$), and note that
$\chi_{H}(g)\neq 0$ if and only if $g$ is conjugate to an element of $H$.

We extend $\chi_{H}$ to subgroups $K$ of $G$ by defining
$\chi_{H}(K)\coloneqq\\#[H\backslash G]^{K}$ (the _mark_ of $K$ on
$[H\backslash G]$), so that $\chi_{H}(\langle g\rangle)=\chi_{H}(g)$;
equivalently, $\chi_{H}(K)$ is the number of singleton fibers in the map
$[H\backslash G]\to[H\backslash G/K]$ defined by $Hg\mapsto HgK$. For $g\in G$
we have $HgK=Hg$ if and only if $gKg^{-1}\subseteq H$, and it follows that

$\chi_{H}(K)=\frac{\\#\bigl{\\{}g\in G:gKg^{-1}\leq
H\bigr{\\}}}{\\#H}=\frac{\\#N_{G}(K)}{\\#H}\\#\bigl{\\{}gKg^{-1}\leq H:g\in
G\bigr{\\}},$ (2.1)

where $N_{G}(K)$ denotes the normalizer of $K$ in $G$.

######  Lemma 2.1.

Let $H$ and $K$ be subgroups of a finite group $G$. The integer $\chi_{H}(K)$
depends only on the $G$-conjugacy classes of $H$ and $K$ and the function
$\chi_{H}$ depends only on the $G$-conjugacy class of $H$.

###### Proof.

Replacing either $H$ or $K$ with a $G$-conjugate does not change the RHS of
(2.1). ∎

If $\mathcal{P}$ is a class of groups (e.g. cyclic groups or solvable groups),
we call its elements _$\mathcal{P}$ -groups_, and refer to the subgroups of a
group $G$ that lie in $\mathcal{P}$ as its _$\mathcal{P}$ -subgroups_. We say
that $\mathcal{P}$ is _subgroup-closed_ if it contains all subgroups of its
elements. A function that maps subgroups of $G$ to subgroups of $G$ is _$G$
-class preserving_ if it maps subgroups to $G$-conjugates.

######  Proposition 2.2.

Let $G$ be a finite group and let $\mathcal{P}$ be a subgroup-closed class of
groups. For any two subgroups $H_{1},H_{2}\leq G$ the following are
equivalent:

  1. (i)

There is $G$-class preserving bijection between the sets of
$\mathcal{P}$-subgroups of $H_{1}$ and $H_{2}$;

  2. (ii)

$\\#\\{gKg^{-1}\leq H_{1}:g\in G\\}=\\#\\{gKg^{-1}\leq H_{2}:g\in G\\}$ for
every $\mathcal{P}$-subgroup $K$ of $G$;

  3. (iii)

$\chi_{H_{1}}(K)=\chi_{H_{2}}(K)$ for every $\mathcal{P}$-subgroup $K$ of $G$;

  4. (iv)

$\chi_{H_{1}}(K)=\chi_{H_{2}}(K)$ for every $\mathcal{P}$-subgroup $K$ of
$H_{1}$ or $H_{2}$;

  5. (v)

the $G$-sets $[H_{1}\backslash G]$ and $[H_{2}\backslash G]$ are isomorphic as
$K$-sets for every $\mathcal{P}$-subgroup $K$ of $G$.

  6. (vi)

the $G$-sets $[H_{1}\backslash G]$ and $[H_{2}\backslash G]$ are isomorphic as
$K$-sets for every $\mathcal{P}$-subgroup $K$ of $H_{1}$ or $H_{1}$.

###### Proof.

(i) $\Leftrightarrow$ (ii) is immediate, (ii) $\Leftrightarrow$ (iii) follows
from (2.1), (iii) $\Leftrightarrow$ (iv) follows from the fact that
$\chi_{H_{i}}(K)=0$ if $K$ is not conjugate to a subgroup of $H_{i}$, and the
implications (v) $\Rightarrow$ (vi) $\Rightarrow$ (iv) are clear; it thus
suffices to show (iii) $\Rightarrow$ (v).

Let $K$ be a $\mathcal{P}$-group and consider the $K$-sets
$X_{1}:=[H_{1}\backslash G]$ and $X_{2}=[H_{2}\backslash G]$. We have
$\chi_{H_{1}}(k)=\chi_{H_{2}}(k)$ for all $k\in K$, thus
$\\#X_{1}=\chi_{H_{i}}(1)=\\#X_{2}$, and $X_{1}$ and $X_{2}$ both have
$\frac{1}{\\#K}\sum_{k}\chi_{H_{i}}(k)$ orbits. The stabilizer $H\leq K$ of
any element $x$ in a $K$-orbit $X$ of $X_{1}$ or $X_{2}$ is a
$\mathcal{P}$-group, and $\chi_{H_{1}}(H)=\chi_{H_{2}}(H)$ implies that the
$K$-orbits of $X_{1}$ and $X_{2}$ can be put in a bijection that preserves
conjugacy classes of stabilizers (and thus preserves cardinalities). If $H$ is
the stabilizer of an element of a $K$-orbit $X$ in $X_{1}$ or $X_{2}$, then
$X$ is isomorphic to the $K$-set $[H\backslash K]$. It follows that $X_{1}$
and $X_{2}$ are isomorphic $K$-sets. ∎

We call subgroups $H_{1},H_{2}\leq G$ that satisfy the equivalent properties
of Proposition 2.2 _$\mathcal{P}$ -equivalent_, and this defines an
equivalence relation on the subgroups of $G$. A necessary condition for
$\mathcal{P}$-equivalence is that $H_{1}$ and $H_{2}$ must have the same
_$\mathcal{P}$ -statistics_, meaning that they contain the same number of
$\mathcal{P}$-subgroups in every isomorphism class of groups. When
$\mathcal{P}$ is the class of cyclic groups this amounts to having the same
_order statistics_ (numbers of elements of each order).

######  Remark 2.3.

Condition (iv) of Proposition 2.2 provides an efficient method for testing
whether two subgroups of $G$ are $\mathcal{P}$-equivalent. Conditions (ii) and
(iii) can both be used to efficiently partition the set of subgroups of $G$
into $\mathcal{P}$-equivalence classes without the need for pairwise testing;
conjugate subgroups lie in the same equivalence class, so it suffices to work
with a set of conjugacy class representatives.

######  Lemma 2.4.

Let $K,H_{1},H_{2}$ be finite groups with $|H_{1}|=|H_{2}|=n$, let
$\phi_{1}\colon K\to H_{1}$ and $\phi_{2}\colon K\to H_{2}$ be injective group
homomorphisms, and let $\rho_{1}\colon H_{1}\to S_{n}$ and $\rho_{2}\colon
H_{2}\to S_{n}$ be left regular representations of $H_{1}$ and $H_{2}$. Then
$K_{1}\coloneqq\rho_{1}(\phi_{1}(H_{0}))$ and
$K_{2}\coloneqq\rho_{2}(\phi_{2}(H_{0}))$ are conjugate subgroups of $S_{n}$.

###### Proof.

The $K$-sets $X_{1}=X_{2}=\\{1,\ldots,n\\}$ given by the actions of
$K_{1}\coloneqq\rho_{1}(\phi_{1}(H_{0}))$ and
$K_{2}\coloneqq\rho_{2}(\phi_{2}(H_{0}))$ are free $K$-sets with the same
number of orbits, hence isomorphic. There is thus a $K$-equivariant map
$\sigma\colon X_{1}\to X_{2}$ with $\sigma\in S_{n}$ satisfying
$\rho_{1}(\phi_{1}(k))\sigma=\sigma\rho_{2}(\phi_{2}(k))$ for $k\in K$, and
$\sigma^{-1}K_{1}\sigma=K_{2}$. ∎

######  Corollary 2.5.

Let $\mathcal{P}$ be a subgroup-closed class of groups. Finite groups $H_{1}$
and $H_{2}$ of the same order can be embedded as $\mathcal{P}$-equivalent
subgroups of some group $G$ if and only if they have the same
$\mathcal{P}$-statistics.

###### Proof.

The necessity of having the same $\mathcal{P}$-statistics is obvious, and
Lemma 2.4 proves sufficiency. ∎

For any integral domain $R$, we use $R[H\backslash G]$ to denote the
corresponding permutation module; this is the free $R$-module with basis
$[H\backslash G]$ equipped with the $R$-linear extension of the $G$-action on
$[H\backslash G]$; we thus view $R[H\backslash G]$ as a (right) $R[G]$-module.

If $H_{1},H_{2}\leq G$ have the same index $n$, after ordering the $G$-sets
$[H_{1}\backslash G]$ and $[H_{2}\backslash G]$, we may uniquely identify each
$R[G]$-module homomorphism $R[H_{1}\backslash G]\to R[H_{2}\backslash G]$ with
a matrix $M\in R^{n\times n}$ whose determinant $\operatorname{det}M$ does not
depend on our choices. If $\rho_{1},\rho_{2}\colon G\to S_{n}$ are the
permutation representations of $G$ acting on $\\{1,\ldots,n\\}$ via our chosen
orderings of $[H_{1}\backslash G]$ and $[H_{2}\backslash G]$, respectively,
then the matrices $M\in R^{n\times n}$ that correspond to elements of
$\operatorname{Hom}_{R[G]}(R[H_{1}\backslash G],R[H_{2}\backslash G])$ are
precisely those that are fixed by the diagonal action of
$\rho_{1}\times\rho_{2}$ on matrix entries; in other words, the entries of $M$
must satisfy

$M_{ij}=M_{\rho_{1}(g)(i),\rho_{2}(g)(j)}\qquad(\text{for all }g\in G).$

We define

$d(H_{1},H_{2}):=\gcd\left\\{\operatorname{det}M:M\in\operatorname{Hom}_{\mathbf{Z}[G]}(\mathbf{Z}[H_{1}\backslash
G],\mathbf{Z}[H_{2}\backslash G])\right\\},$

and extend this definition to all subgroups of $G$ by defining
$d(H_{1},H_{2})=0$ whenever $\\#H_{1}\neq\\#H_{2}$.

We now give several equivalent conditions for subgroups to be
$\mathcal{P}$-equivalent when $\mathcal{P}$ is the class of cyclic groups.

######  Proposition 2.6.

Let $G$ be a finite group. For all subgroups $H_{1}$ and $H_{2}$ of $G$ the
following are equivalent:

  1. (i)

There is a bijection of sets $H_{1}\leftrightarrow H_{2}$ that preserves
$G$-conjugacy;

  2. (ii)

$\\#(H_{1}\cap C)=\\#(H_{2}\cap C)$ for every conjugacy class $C$ of $G$;

  3. (iii)

$\chi_{H_{1}}(K)=\chi_{H_{2}}(K)$ for every cyclic $K\leq G$;

  4. (iv)

The $G$-sets $[H_{1}\backslash G]$ and $[H_{2}\backslash G]$ are isomorphic as
$K$-sets for every cyclic $K\leq G$;

  5. (v)

$\mathbf{Q}[H_{1}\backslash G]\simeq\mathbf{Q}[H_{2}\backslash G]$;

  6. (vi)

$d(H_{1},H_{2})\neq 0$.

###### Proof.

The equivalence of (i) and (ii) is immediate. The equivalence of (ii) and
(iii) follows from the formula $\chi_{H_{i}}(g)\\#H_{i}=\\#(H_{i}\cap
C(g))\\#Z(g)$, where $C(g)$ is the conjugacy class of $g$ and $Z(g)$ is its
centralizer (in $G$); see [36, Eq. 8]. The equivalence of (iii) and (iv)
follows from applying Proposition 2.2 to the class of cyclic groups. For the
equivalence of (iii) and (v), note that
$\dim_{\mathbf{Q}}(\mathbf{Q}[H_{i}\backslash G]^{K})=\chi_{H_{i}}(K)$ for
cyclic $K\leq G$ and then apply the corollary to [43, Theorem 30]. Clearing
the denominators in
$M\in\operatorname{Hom}_{\mathbf{Q}[G]}(\mathbf{Q}[H_{1}\backslash
G],\mathbf{Q}[H_{2}\backslash G])$ shows the equivalence of (v) and (vi). ∎

######  Remark 2.7.

The condition $K\leq G$ in (iii), (iv) can be replaced by “$K\leq H_{1}$ or
$K\leq H_{2}$” via Lemma 2.2.

######  Definition 2.8.

Subgroups $H_{1}$ and $H_{2}$ of a finite group $G$ that satisfy the
equivalent conditions of Proposition 2.6 are said to be _rationally
equivalent_ (or _Gassmann equivalent_).

A triple of groups $(G,H_{1},H_{2})$ with $H_{1},H_{2}\leq G$ rationally
equivalent is called a _Gassmann triple_ [15]. By Proposition 2.6, rational
equivalence defines an equivalence relation on the subgroups of $G$. Conjugate
subgroups of $G$ are necessarily rational equivalent, so we may view this as
an equivalence relation on conjugacy classes of subgroups. Rational
equivalence classes may be arbitrarily large [27].

We are interested in the nontrivial rational equivalence classes, those which
contain nonconjugate but rationally equivalent subgroups $H_{1},H_{2}\leq G$.
Equivalently, we are interested in the cases where $[H_{1}\backslash
G]\not\simeq[H_{2}\backslash G]$ as $G$-sets, but $\mathbf{Q}[H_{1}\backslash
G]\simeq\mathbf{Q}[H_{2}\backslash G]$ as $G[\mathbf{Q}]$-modules. Standard
examples include the subgroups
$H_{1}\coloneqq\bigl{\\{}\bigl{[}\begin{smallmatrix}1&*\\\
0&*\end{smallmatrix}\bigr{]}\in{\mathbf{GL}}_{2}(\mathbf{F}_{p})\bigr{\\}}$
and $H_{2}\coloneqq\bigl{\\{}\bigl{[}\begin{smallmatrix}1&0\\\
*&*\end{smallmatrix}\bigr{]}\in{\mathbf{GL}}_{2}(\mathbf{F}_{p})\bigr{\\}}$ of
$G\coloneqq{\mathbf{GL}}_{2}(\mathbf{F}_{p})$, where $p$ is an odd prime [11],
and similar examples in ${\mathbf{GL}}_{n}(\mathbf{F}_{p})$ for $n>2$ and any
prime $p$. In these examples the subgroups $H_{1}$ and $H_{2}$ are not
$G$-conjugate, but transposition gives a bijection $H_{1}\leftrightarrow
H_{2}$ that preserves $G$-conjugacy. The smallest example occurs for the group
$G$ with GAP identifier $\langle 32,43\rangle$, which contains two
nonconjugate rationally equivalent subgroups $H_{1}$ and $H_{2}$ isomorphic to
the Klein $4$-group.222A GAP identifier $\langle m,n\rangle$ denotes the
isomorphism class of an abstract group of order $m$; the positive integer $n$
is an ordinal that distinguishes distinct isomorphism classes of groups of
order $m$. For $m\leq 2000$ not equal to 1024 explicit presentations of these
groups can be found in the small groups database [3], which is available in
both GAP [14] and Magma [5].

######  Remark 2.9.

Rationally equivalent subgroups necessarily have the same order but need not
be isomorphic. The smallest example of a Gassmann triple $(G,H_{1},H_{2})$
with $H_{1}\not\simeq H_{2}$ arises for $G\simeq\langle 384,5755\rangle$ with
subgroups $H_{1}\simeq\langle 16,3\rangle$ and $H_{2}\simeq\langle
16,10\rangle$. The groups $H_{1}$ and $H_{2}$ are the first of infinitely many
pairs of nonisomorphic groups with the same order statistics (one can take
$(\mathbf{Z}/p\mathbf{Z})^{3}$ and the Heisenberg group
$H_{3}(\mathbf{F}_{\\!p})$ for any prime $p$, for example). Corollary 2.5
implies that all such pairs $H_{1}$ and $H_{2}$ can be realized as part of a
Gassmann triple $(G,H_{1},H_{2})$.

The original motivation for studying rational equivalence stems from its
relationship to zeta functions of number fields. Recall that the _Dedekind
zeta function_ of a number field $K$ is defined by

$\zeta_{K}(z)\coloneqq\prod_{\mathfrak{p}}(1-N(\mathfrak{p})^{-z})^{-1},$

where $\mathfrak{p}$ varies over primes of $K$ (nonzero prime ideals of its
ring of integers $\mathcal{O}_{K}$) and
$N(\mathfrak{p})\coloneqq[\mathcal{O}_{K}:\mathfrak{p}]$ is the cardinality of
the residue field at $\mathfrak{p}$ (its absolute norm). The Euler product for
$\zeta_{K}(z)$ defines a holomorphic function on $\operatorname{Re}(z)>1$ that
extends to a meromorphic function on $\mathbf{C}$ with a simple pole at $z=1$
whose residue is given by the _analytic class number formula_ :

$\lim_{z\to
1^{+}}(z-1)\zeta_{K}(z)=\frac{2^{r}(2\pi)^{s}h_{K}R_{K}}{\\#\mu(K)|D_{K}|^{1/2}}.$
(2.2)

Here $r$ and $s$ are the number of real and complex places of $K$ (its
_signature_), $h_{K}$ is the class number, $R_{K}$ is the regulator, $\mu(K)$
is the group of roots of unity in $K^{\times}$, and $D_{K}$ is the
discriminant of $K$.

######  Theorem 2.10.

For number fields $K_{1}$ and $K_{2}$ the following are equivalent:

  1. (i)

$\zeta_{K_{1}}(s)=\zeta_{K_{2}}(s)$;

  2. (ii)

$K_{1}$ and $K_{2}$ have Galois closure $L$ with
$\operatorname{Gal}(L/K_{1}),\operatorname{Gal}(L/K_{2})\leq\operatorname{Gal}(L/\mathbf{Q})$
rationally equivalent;

  3. (iii)

There is a bijection between the primes of $K_{1}$ and $K_{2}$ that preserves
residue fields.

###### Proof.

These equivalences all follow from [35, Theorem 1]. ∎

######  Definition 2.11.

Number fields $K_{1}$ and $K_{2}$ that satisfy the equivalent conditions of
Theorem 2.10 are said to be _arithmetically equivalent_.

If $K_{1}$ and $K_{2}$ are arithmetically equivalent number fields with common
Galois closure $L$ and we put $G:=\operatorname{Gal}(L/\mathbf{Q})$,
$H_{1}:=\operatorname{Gal}(L/K_{1})$, $H_{2}:=\operatorname{Gal}(L/K_{2})$,
then $(G,H_{1},H_{2})$ is a _faithful_ Gassmann triple, meaning that
$\mathbf{Q}[H_{1}\backslash G]\simeq\mathbf{Q}[H_{2}\backslash G]$ is a
faithful representation of $G$. Equivalently, $H_{1}$ and $H_{2}$ have trivial
normal core in $G$. There is no loss of generality in restricting our
attention to faithful Gassmann triples: if $H_{1},H_{2}\leq G$ are rationally
equivalent then they necessarily have the same normal core $N$, the quotients
$H_{1}/N,H_{2}/N\leq G/N$ are rationally equivalent, and $H_{1}/N$ and
$H_{2}/N$ are conjugate in $G/N$ if and only if $H_{1}$ and $H_{2}$ are
conjugate in $G$.

Arithmetically equivalent number fields share many (but not all) arithmetic
invariants.

######  Theorem 2.12.

Arithmetically equivalent number fields have the same degree, discriminant,
signature, and roots of unity.

###### Proof.

See [35, Theorem 1]. ∎

The analytic class number formula (2.2) implies that if $K_{1}$ and $K_{2}$
are arithmetically equivalent number fields then we must have

$h_{K_{1}}R_{K_{1}}=h_{K_{2}}R_{K_{2}},$

but it may happen that $h_{K_{1}}\neq h_{K_{2}}$ (in which case $R_{K_{1}}\neq
R_{K_{2}}$), and even when $h_{K_{1}}=h_{K_{2}}$ the class groups need not be
isomorphic.333The fields
$\mathbf{Q}[x]/(x^{7}-3x^{6}+10x^{5}-21x^{4}-6x^{3}+58x^{2}-41x-6)$ and
$\mathbf{Q}[x]/(x^{7}-x^{6}+x^{5}+5x^{4}+9x^{3}+5x^{2}-7x-4)$ with LMFDB [29]
labels 7.3.1427382162361.1 and 7.3.1427382162361.2 are an example; see [2] for
analogous exceptions in the context of isospectral Riemannian manifolds. It
follows from Theorem 2.12 that if $K_{1}$ and $K_{2}$ are arithmetically
equivalent then a prime $p$ of $\mathbf{Q}$ ramifies in $K_{1}$ if and only if
it ramifies in $K_{2}$.

######  Remark 2.13.

The ramified rational primes in arithmetically equivalent number fields
necessarily coincide, but they may have different factorization patterns. This
was shown by Perlis in [35, page 351] for the arithmetically equivalent number
fields $K_{1}\coloneqq\mathbf{Q}(\sqrt[8]{97})$ and
$K_{2}\coloneqq\mathbf{Q}(\sqrt[8]{1552})$ where we have

$2\mathcal{O}_{K_{1}}=\mathfrak{p}_{1}\mathfrak{p}_{2}\mathfrak{p}_{3}^{2}\mathfrak{p}_{4}^{4}\qquad\text{versus}\qquad
2\mathcal{O}_{K_{2}}=\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}^{2}\mathfrak{q}_{4}^{2}.$

In this example the products of the ramification indices differ, but the sums
are the same. As shown by Mantilla-Soler [30, Thm. 3.7], there are cases where
the sums also differ. Indeed, the number fields
$K_{1}:=\mathbf{Q}[x]/(x^{7}-3x^{6}+4x^{5}-5x^{4}+3x^{3}-x^{2}-2x+1)$ and
$K_{2}:=\mathbf{Q}[x]/(x^{7}-x^{5}-2x^{4}-2x^{3}+2x^{2}-x+4)$ with LMFDB
labels 7.3.30558784.1 and 7.3.30558784.2 are arithmetically equivalent with

$2\mathcal{O}_{K_{1}}=\mathfrak{p}_{1}\mathfrak{p}_{2}^{4}\qquad\text{versus}\qquad
2\mathcal{O}_{K_{2}}=\mathfrak{q}_{1}\mathfrak{q}_{2}^{2}.$

This example settled a question of Stuart and Perlis [37, §4].

For a number field $K$ with Galois closure $L$ that is the fixed field of
$H\leq G=\operatorname{Gal}(L/\mathbf{Q})$, the decomposition of rational
primes in $K$, can be computed using the $G$-set $[H\backslash G]$. The lemma
below can be used to explain the examples in Remark 2.13 and to prove part
(iii) of Theorem 2.10.

######  Lemma 2.14.

Let $L$ be a Galois extension of $\mathbf{Q}$ with Galois group $G$, let
$\mathfrak{p}$ be a prime of $L$ above $p:=\mathfrak{p}\cap\mathbf{Q}$ with
decomposition group $D_{\mathfrak{p}}$ and inertia group $I_{\mathfrak{p}}$,
and let $K$ be the fixed field of $H\leq G$.

  1. (i)

There is a bijection $[H\backslash G/D_{\mathfrak{p}}]\to\\{\text{primes of
$K$ above $p$}\\}$ defined by $H\sigma
D_{\mathfrak{p}}\mapsto\sigma(\mathfrak{p})\cap K$.

  2. (ii)

The prime $\sigma(\mathfrak{p})\cap K$ has ramification index $[H\sigma
I_{\mathfrak{p}}:H\sigma]$ and residue field degree $[H\sigma
D_{\mathfrak{p}}:H\sigma I_{\mathfrak{p}}]$.

###### Proof.

This is well known; see [34, §9] and [51], for example. ∎

Theorem 2.10 implies that if $K_{1}$ and $K_{2}$ are arithmetically equivalent
number fields then for every unramified rational prime $p$ there is a
bijection between the primes of $K_{1}$ above $p$ and the primes of $K_{2}$
above $p$ such that the completions of $K_{1}$ and $K_{2}$ at corresponding
primes above $p$ are isomorphic extensions of $\mathbf{Q}_{p}$, since (up to
isomorphism) there is a unique unramified extension of $\mathbf{Q}_{p}$ of
each degree. By Theorem 2.12, this also holds for the archimedean place
$\infty$ of $\mathbf{Q}$, since the signatures of $K_{1}$ and $K_{2}$
coincide, but as shown by the examples of Remark 2.13, this need not hold at
ramified primes.

######  Definition 2.15.

Two number fields $K_{1}$ and $K_{2}$ are said to be _locally isomorphic_ if
there is a bijection between the places of $K_{1}$ and the places of $K_{2}$
such that the completions at corresponding places are isomorphic (both as
topological rings and as $\mathbf{Q}_{p}$-algebras); equivalently,
$K_{1}\otimes_{\mathbf{Q}}\mathbf{Q}_{p}\simeq
K_{2}\otimes_{\mathbf{Q}}\mathbf{Q}_{p}$ for all $p\leq\infty$. If this holds
for all but finitely many places then $K_{1}$ and $K_{2}$ are said to be
_locally isomorphic almost everywhere_.

For a number field $K$ we use $\mathbf{A}_{K}$ to denote its ring of adeles,
which we may regard both as a topological ring and as an
$\mathbf{A}_{\mathbf{Q}}$-algebra.

######  Theorem 2.16.

Let $K_{1}$ and $K_{2}$ be number fields. The following hold:

  1. (i)

$K_{1}$ and $K_{2}$ are locally isomorphic almost everywhere if and only if
they are arithmetically equivalent, and if and only if almost every prime of
$\mathbf{Q}$ has the same number of primes above it in $K_{1}$ and $K_{2}$;

  2. (ii)

$K_{1}$ and $K_{2}$ are locally isomorphic if and only if
$\mathbf{A}_{K_{1}}\simeq\mathbf{A}_{K_{2}}$ (as topological rings and
$\mathbf{A}_{\mathbf{Q}}$-algebras);

  3. (iii)

if $K_{1}$ and $K_{2}$ are locally isomorphic then there is a natural
isomorphism of their Brauer groups that commutes with all restriction maps
induced by common inclusions of number fields.

###### Proof.

The first equivalence in (i) follows from [35, Theorem 1] and the second was
proved in [37], the forward implication in (ii) is immediate and the reverse
implication is due to Iwasawa [22, Lemma 7] (also see [26, Lemma 3]), and the
implication in (iii) is proved in [28]. ∎

######  Remark 2.17.

The converse of part (iii) of Theorem 2.16 is false. Arithmetically equivalent
number fields with naturally isomorphic Brauer groups need not be locally
isomorphic, as shown in [31].

Theorem 2.16 implies that locally isomorphic number fields are necessarily
arithmetically equivalent, but the converse need not hold. As observed in
Remark 2.13, arithmetically equivalent number fields may have incompatible
ramification indices, which precludes local isomorphism.

######  Remark 2.18.

Locally isomorphic number fields need not have the same class number; the
fields $\mathbf{Q}(\sqrt[8]{-33})$ and $\mathbf{Q}(\sqrt[8]{-528})$ with class
numbers 256 and 128 are an example [12, p. 214].

The following proposition provides an effective way to test for local
isomorphism.

######  Proposition 2.19.

Let $L,K_{1},K_{2}$ be number fields corresponding to a Gassmann triple
$(G,H_{1},H_{2})$, and let $D_{\mathfrak{p}}\subseteq G$ be the decomposition
group of a place $\mathfrak{p}$ of $L$ above a place $p$ of $\mathbf{Q}$. Then
$K_{1}\otimes_{\mathbf{Q}}\mathbf{Q}_{p}\simeq
K_{2}\otimes_{\mathbf{Q}}\mathbf{Q}_{p}$ if and only if $[H_{1}\backslash G]$
and $[H_{2}\backslash G]$ are isomorphic as $D_{\mathfrak{p}}$-sets. These
equivalent conditions necessarily hold for every unramified place $p$ of
$\mathbf{Q}$.

###### Proof.

Recall that for any field $F$ with separable closure $\Omega$ there is a
functorial equivalence between the category of étale $F$-algebras $A$ and the
category of finite $\operatorname{Gal}(\Omega/F)$-sets $S$; see [32, Theorem
8.20]. The $\operatorname{Gal}(\Omega/F)$-action on $S$ is continuous, hence
factors through a finite quotient $Q$, and by a $Q$-set $S$ we mean the
$\operatorname{Gal}(\Omega/F)$-set $S$ with the action of each
$\sigma\in\operatorname{Gal}(\Omega/F)$ given by the action of its projection
to $Q$.

For $i=1,2$, the $G$-set $[H_{i}\backslash G]$ corresponds to the étale
$\mathbf{Q}$-algebra $K_{i}$. If we view $D_{\mathfrak{p}}$ as the Galois
group of the étale $\mathbf{Q}_{p}$-algebra $L\otimes\mathbf{Q}_{p}$, the
$D_{\mathfrak{p}}$-set $[H_{i}\backslash G]$ corresponds to the étale
$\mathbf{Q}_{p}$-algebra $K_{i}\otimes\mathbf{Q}_{p}$.

The last statement follows from (iv) of Proposition 2.6, since if
$\mathfrak{p}$ is unramified then $D_{\mathfrak{p}}$ is cyclic. ∎

Finally we recall the following result on arithmetical isomorphisms which can
be found in [24, IV].

######  Proposition 2.20.

Let $G$ be a finite group with subgroups $H_{1},H_{2}\leq G$, let $R$ be an
integral domain, let $A$ be an $R[G]$-module, and let $A_{1}\coloneqq
A^{H_{1}}$ and $A_{2}\coloneqq A^{H_{2}}$ be the $R$-submodules of $A$ fixed
by $H_{1}$ and $H_{2}$, respectively. Every
$M\in\operatorname{Hom}_{R[G]}(R[H_{1}\backslash G],R[H_{2}\backslash G])$
with $\operatorname{det}M\in R^{\times}$ induces an $R[G]$-module isomorphism
$\delta_{M}\colon A_{1}\to A_{2}$.

###### Proof.

See [24, Theorem IV.1.6a]. ∎

##  3 Stronger forms of arithmetic equivalence

Recall that a finite group $K$ is said to be _cyclic modulo_ $p$ (or
$p$-_hypo-elementary_) if the quotient of $K$ by the intersection of its
$p$-Sylow subgroups (its $p$-core) is cyclic. For the sake of brevity we shall
simply call such a group _$p$ -cyclic_. The class of $p$-cyclic groups
includes all $p$-groups and all cyclic groups.

######  Proposition 3.1.

Let $G$ be a finite group and $p$ a prime. For $H_{1},H_{2}\leq G$ the
following are equivalent:

  1. (i)

There is a $G$-class preserving bijection between the sets of $p$-cyclic
subgroups of $H_{1}$ and $H_{2}$;

  2. (ii)

$\chi_{H_{1}}(K)=\chi_{H_{2}}(K)$ for every $p$-cyclic $K\leq G$;

  3. (iii)

the $G$-sets $[H_{1}\backslash G]$ and $[H_{2}\backslash G]$ are isomorphic as
$K$-sets for every $p$-cyclic $K\leq G$;

  4. (iv)

$\mathbf{Z}_{p}[H_{1}\backslash G]\simeq\mathbf{Z}_{p}[H_{2}\backslash G]$;

  5. (v)

$\mathbf{F}_{p}[H_{1}\backslash G]\simeq\mathbf{F}_{p}[H_{2}\backslash G]$;

  6. (vi)

$p\nmid d(H_{1},H_{2})$.

Moreover, in (ii) and (iii) one can replace “$K\leq G$” with “$K\leq H_{1}$ or
$K\leq H_{2}$”.

###### Proof.

The equivalence of (i), (ii), (iii) is given by Proposition 2.2. The
equivalence of (ii) and (iv) follows from [42, Proposition 3.1] (attributed to
Conlon [10]). The equivalence of (iv) and (v) is given by [17, Theorem
2.9(i)]. The equivalence of (v) and (vi) is immediate, since
$\mathbf{F}_{p}[H_{1}\backslash G]\simeq\mathbf{F}_{p}[H_{2}\backslash G]$ if
and only if there exists
$M\in\operatorname{Hom}_{\mathbf{Z}[G]}(\mathbf{Z}[H_{1}\backslash
G],\mathbf{Z}[H_{1}\backslash G])$ whose reduction modulo $p$ is invertible,
equivalently, $p\nmid\operatorname{det}M)$. That the weakened forms of (ii)
and (iii) suffice follows form Proposition 2.2 ∎

######  Definition 3.2.

Let $H_{1},H_{2}\leq G$ be finite groups. If $\mathbf{Z}_{p}[H_{1}\backslash
G]\simeq\mathbf{Z}_{p}[H_{2}\backslash G]$ for every prime $p$ then $H_{1}$
and $H_{2}$ are _locally integrally equivalent_ , and if
$\mathbf{Z}[H_{1}\backslash G]\simeq\mathbf{Z}[H_{2}\backslash G]$ then they
are _integrally equivalent_.

######  Remark 3.3.

Two $\mathbf{Z}[G]$-modules that are isomorphic as $\mathbf{Z}_{p}[G]$-modules
for every prime $p$ are said to lie in the same genus [18, 42]; subgroups
$H_{1},H_{2}\leq G$ are locally integrally equivalent if and only of the
permutation modules $\mathbf{Z}[H_{1}\backslash G]$ and
$\mathbf{Z}[H_{2}\backslash G]$ lie in the same genus.

Proposition 3.1 implies that subgroups $H_{1},H_{2}\leq G$ are locally
integrally equivalent if and only if

$d(H_{1},H_{2})=\gcd\left\\{\operatorname{det}M:M\in\operatorname{Hom}_{\mathbf{Z}[G]}(\mathbf{Z}[H_{1}\backslash
G],\mathbf{Z}[H_{2}\backslash G])\right\\}=1,$

in which case there is a finite set of matrices
$M\in\operatorname{Hom}_{\mathbf{Z}[G]}(\mathbf{Z}[H_{1}\backslash
G],\mathbf{Z}[H_{2}\backslash G]$ whose determinants have trivial GCD.
Integral equivalence holds if and only if a singleton set with this property
exists, that is, $\operatorname{det}M=\pm 1$ for some
$M\in\operatorname{Hom}_{\mathbf{Z}[G]}(\mathbf{Z}[H_{1}\backslash
G],\mathbf{Z}[H_{2}\backslash G])$. Rational equivalence only requires
$d(H_{1},H_{1})\neq 0$ and is obviously implied by local integral equivalence.

Essentially only one nontrivial example of integral equivalence is known, due
to Scott [42], in which $G\simeq{\mathbf{PSL}}_{2}(29)$ and $H_{1}$ and
$H_{2}$ are nonconjugate subgroups of $G$ isomorphic to the alternating group
$A_{5}$ that are conjugate in ${\mathbf{PGL}}_{2}(29)$; one can use this
example to construct others, but these all have a subgroup with a quotient
isomorphic to ${\mathbf{PSL}}_{2}(29)$. As noted by Scott and proved in
Theorem 3.9 below, for every prime $p\equiv\pm 29\bmod 120$ the group
${\mathbf{PSL}}_{2}(p)$ contains nonconjugate subgroups isomorphic to $A_{5}$
that are locally integrally equivalent. But with the exception of $p=29$ it is
not known whether these subgroups are also integrally equivalent.

######  Proposition 3.4.

Let $K_{1}$ and $K_{2}$ be number fields with common Galois closure $L$, and
let $H_{1}:=\operatorname{Gal}(L/K_{1})$, $H_{2}:=\operatorname{Gal}(L/K_{2})$
be locally integrally equivalent subgroups of
$G:=\operatorname{Gal}(L/\mathbf{Q})$. Then the following hold:

  1. (i)

$K_{1}$ and $K_{2}$ are arithmetically equivalent;

  2. (ii)

the class groups of $K_{1}$ and $K_{2}$ are isomorphic;

  3. (iii)

the regulators of $K_{1}$ and $K_{2}$ are equal;

  4. (iv)

for every prime $p$ the products of the ramification indices of the primes of
$K_{1}$ and $K_{2}$ above $p$ coincide.

###### Proof.

As noted above, local integral equivalence implies rational equivalence, so
(i) follows from Proposition 2.6 and Theorem 2.10. Proposition 3.1 and [36,
Theorem 3] together imply that the class groups of $K_{1}$ and $K_{2}$ have
isomorphic $p$-Sylow subgroups for every prime $p$ and are therefore
isomorphic (since they are abelian), so (ii) holds. Properties (i) and (ii)
together imply (iii), by Theorem 2.12 and the analytic class number formula.
Local integral equivalence implies $d(H_{1},H_{2})=1$, which when combined
with [24, Theorem IV.2.3] implies (iv). ∎

For number fields satisfying the hypothesis of Proposition 3.4, all the
quantities that appear in the analytic class number formula (2.2) must
coincide. However, such fields need not be locally isomorphic, as shown by the
example in §4.2, and locally isomorphic number fields may have different class
numbers and regulators, as shown by the example in Remark 2.18.

We now introduce a strictly stronger notion of equivalence that implies both
local integral equivalence and local isomorphism of corresponding number
fields.

######  Definition 3.5.

Subgroups $H_{1}$ and $H_{2}$ of a finite group $G$ are _solvably equivalent_
if they satisfy the following equivalent properties (as guaranteed by
Proposition 2.2):

  1. (i)

There is a $G$-class preserving bijection between the sets of solvable
subgroups of $H_{1}$ and $H_{2}$;

  2. (ii)

$\chi_{H_{1}}(K)=\chi_{H_{2}}(K)$ for every solvable $K\leq G$;

  3. (iii)

the $G$-sets $[H_{1}\backslash G]$ and $[H_{2}\backslash G]$ are isomorphic as
$K$-sets for every solvable $K\leq G$.

Solvably equivalent subgroups are always locally integrally equivalent, since
$p$-cyclic groups are solvable, but as demonstrated by the example in §4.3,
locally integrally equivalent subgroups need not be solvably equivalent. As
shown by the example in §4.4, solvably equivalent subgroups need not be
integrally equivalent, but it is not clear whether the converse holds; the
integrally equivalent subgroups of ${\mathbf{PSL}}_{2}(29)$ in Scott’s example
are solvably equivalent, but as noted in the introduction, it is not clear
whether this is always true, nor is it clear that integral equivalence
guarantees local isomorphism of corresponding number fields (this is not true
of local integral equivalence, and if it were true for integral equivalence
then property (2) in Theorem 3 in [38] could have been included in Theorem 2
in [38]).

######  Question 3.6.

Is there a Gassmann triple $(G,H_{1},H_{2})$ in which $H_{1}$ and $H_{2}$ are
integrally equivalent but not solvably equivalent? More precisely, is there a
group $G$ containing subgroups $H_{1},H_{2}$ and a solvable subgroup $K$ such
that $\mathbf{Z}[H_{1}\backslash G]$ and $\mathbf{Z}[H_{2}\backslash G]$ are
isomorphic as $\mathbf{Z}[G]$-modules but not as $K$-sets?

######  Proposition 3.7.

Let $K_{1}$ and $K_{2}$ be number fields with the same Galois closure $L$, and
put $H_{1}:=\operatorname{Gal}(L/K_{1})$ and
$H_{2}:=\operatorname{Gal}(L/K_{2})$. If $H_{1}$ and $H_{2}$ are solvably
equivalent subgroups of $G:=\operatorname{Gal}(L/\mathbf{Q})$ then

  1. (i)

$K_{1}$ and $K_{2}$ are arithmetically equivalent;

  2. (ii)

$K_{1}$ and $K_{2}$ have isomorphic class groups and equal regulators;

  3. (iii)

$K_{1}$ and $K_{2}$ are locally isomorphic, and in particular there is a
bijection between the primes of $K_{1}$ and $K_{2}$ that preserves both
inertia degrees and ramification indices;

  4. (iv)

the adele rings $\mathbf{A}_{K_{1}}$ and $\mathbf{A}_{K_{2}}$ are isomorphic
(as topological groups and $\mathbf{A}_{\mathbf{Q}}$-algebras);

###### Proof.

Solvable equivalence implies local integral equivalence, so (i) and (ii) both
follow from Proposition 3.4. For each prime $\mathfrak{p}$ of $L$ the
decomposition subgroup
$D_{\mathfrak{p}}\subseteq\operatorname{Gal}(L/\mathbf{Q})$ is solvable, so we
have an isomorphism of $D_{\mathfrak{p}}$-sets $[H_{1}\backslash
G]\simeq[H_{2}\backslash G]$, which implies (iii), by Proposition 2.19, and
(iv) is then implied by Theorem 2.16. ∎

######  Remark 3.8.

In Proposition 3.7, the hypothesis that $H_{1}$ and $H_{2}$ are solvably
equivalent is stronger than necessary. It could be replaced, for example, by
the condition that $\chi_{H_{1}}(K)=\chi_{H_{2}}(K)$ for every $K\leq G$ with
normal subgroups $W\leq I$ such that $W$ is a $p$-group, $I/W$ is cyclic of
order prime to $p$, and $K/I$ is cyclic. Even this is stronger than necessary,
since, for example, it is satisfied by both $C_{2}^{4}$ and
${\mathbf{SL}}_{2}(3)$, neither of which occurs as the Galois group of an
extension of $\mathbf{Q}_{p}$ for any prime $p$ (the former contains too many
normal subgroups of index 2 and the latter was ruled out by Weil in [50,
§15]).

The following theorem gives an infinite family of groups each of which contain
a pair of nonconjugate solvably equivalent subgroups.

######  Theorem 3.9.

Let $p\equiv\pm 29\bmod 120$ be prime. The group
${\mathbf{SL}}_{2}(\mathbf{F}_{p})$ contains a pair of nonconjugate solvably
equivalent subgroups $H_{1},H_{2}$ whose projective images are nonconjugate
solvably equivalent subgroups of ${\mathbf{PSL}}_{2}(\mathbf{F}_{p})$
isomorphic to the alternating group $A_{5}$.

###### Proof.

It follows from [48, Lemma 3.21.3c] that for $p\equiv\pm 1\bmod 5$, up to
conjugacy in ${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$ there is a unique subgroup
$H_{1}$ of ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ with projective image
isomorphic to $A_{5}$; it is isomorphic to
${\mathbf{SL}}_{2}(\mathbf{F}_{5})$. The outer automorphism of
${\mathbf{SL}}_{2}(\mathbf{F}_{p})$ corresponds to conjugation by an element
with nonsquare determinant; let
$\sigma\coloneqq\bigl{[}\begin{smallmatrix}r&0\\\
0&1\end{smallmatrix}\bigr{]}$ be such an element, with
$r\in\mathbf{F}_{\\!p}^{\times}-\mathbf{F}_{\\!p}^{\times 2}$. Conjugation by
$\sigma$ fixes all but four of the conjugacy classes in
${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$: it interchanges the conjugacy classes
of $\bigl{[}\begin{smallmatrix}1&1\\\ 0&1\end{smallmatrix}\bigr{]}$ and
$\bigl{[}\begin{smallmatrix}1&r\\\ 0&1\end{smallmatrix}\bigr{]}$, and also
those of $\bigl{[}\begin{smallmatrix}-1&-1\\\ 0&-1\end{smallmatrix}\bigr{]}$
and $\bigl{[}\begin{smallmatrix}-1&-r\\\ 0&-1\end{smallmatrix}\bigr{]}$ (these
are the conjugacy classes of elements of order divisible by $p$).

Let $H_{2}\coloneqq\sigma H_{1}\sigma^{-1}$; the groups $H_{1}$ and $H_{2}$
are not conjugate in ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$, by [13, Theorem
4.1]. These groups do not contain any elements of order divisible by $p$,
since $p\geq 29$ and $\\#{\mathbf{SL}}_{2}(\mathbf{F}_{5})=2^{2}\cdot 3\cdot
5$. Conjugation by $\sigma$ thus defines an
${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$-conjugacy preserving bijection between
$H_{1}$ and $H_{2}$, implying that $H_{1}$ and $H_{2}$ are rationally
equivalent subgroups of ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$.

To show that $H_{1}$ and $H_{2}$ are solvably equivalent, it suffices to show
that $\sigma$ defines an ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$-conjugacy
class preserving bijection of solvable subgroups of $H_{1}$ and $H_{2}$, and
having proved rational equivalence we only need to consider the noncyclic
solvable subgroups of $H_{1}$ and $H_{2}$. Up to isomorphism, there are four
possibilities for the image of such a subgroup in in
${\mathbf{PSL}}_{2}(\mathbf{F}_{\\!p})$: $D_{2}$, $D_{3}$, $D_{5}$, and
$A_{4}$, where $D_{2}\coloneqq C_{2}\times C_{2}$ is the Klein group. It
follows from Proposition 3.13 below that there is exactly one
${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$-conjugacy class of subgroups isomorphic
to $D_{2}$, $D_{3}$, $D_{5}$, $A_{4}$ when $p\equiv\pm 3\bmod 8$, $p\equiv\pm
5\bmod 12$, $p\equiv\pm 9\bmod 20$, and $p\equiv\pm 3\bmod 8$, respectively.
These constraints are simultaneously met precisely when $p\equiv\pm 29\bmod
120$, and in this situation it is clear that $\sigma$ must define an
${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$-conjugacy class preserving bijection of
solvable subgroups of $H_{1}$ and $H_{2}$, since it preserves isomorphism
classes.

Finally, note that the conjugacy class preserving bijection between solvable
subgroups of $H_{1}$ and $H_{2}$ descends to
${\mathbf{PSL}}_{2}(\mathbf{F}_{\\!p})$, while $H_{1}$ and $H_{2}$ both
contain $-1$ and remain nonconjugate in
${\mathbf{PSL}}_{2}(\mathbf{F}_{\\!p})$. ∎

######  Remark 3.10.

Theorem 3.9 accounts for all nontrivial pairs of solvably equivalent subgroups
of ${\mathbf{SL}}_{2}(\mathbf{F}_{p})$, in fact all nontrivial pairs of
locally integrally equivalent subgroups of
${\mathbf{SL}}_{2}(\mathbf{F}_{p})$, as noted by Scott [42]. Up to a central
extension the same applies to subgroups of
${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$, since every nonsolvable subgroup of
${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$ that does not contain
${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ has projective image $A_{5}$ [43, §2].

######  Remark 3.11.

As proved by Zywina [52], the group ${\mathbf{PSL}}_{2}(\mathbf{F}_{\\!p})$
can be realized as the Galois group of a number field for every prime $p$.
This implies that there are infinitely many distinct examples of pairs of
nonisomorphic solvably equivalent number fields whose Galois groups do not
admit a common quotient.

######  Remark 3.12.

As shown in §4.4, subgroups of ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ are not
the only source of nontrivial solvably equivalent pairs of subgroups, and one
can do better than the minimal degree 203 admitted by Theorem 3.9: degree 96
is possible.

Recall that each subgroup of ${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$ of order
prime to $p$ can be classified according to the isomorphism class of its image
in ${\mathbf{PGL}}_{2}(\mathbf{F}_{\\!p})$, which must be cyclic, dihedral, or
one of $A_{4}$, $S_{4}$, $A_{5}$; see [43, §2], for example. Note that we
consider $D_{2}:=C_{2}\times C_{2}$ to be a dihedral group. The proposition
below characterizes the isomorphism classes of order prime to $p$ that arise
in ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$, up to conjugacy in
${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$; see [48, §3] for an analogous
classification for conjugacy classes of subgroups of
${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$ (including those of order divisible by
$p$), which we will use in the proof of the proposition.

We use the notation $2D_{n}$ to denote the binary dihedral group of order
$4n$, these arise as subgroups of ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$
containing $-1$ with projective image $D_{n}$, and similar define $2A_{4}$,
$2S_{4}$, $2A_{5}$. We say that a conjugacy class of subgroups of
${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ is $C_{n}$ (resp. $2D_{n}$, $2A_{4}$,
$2S_{4}$, $2A_{5}$) if it is the conjugacy class of a subgroup isomorphic to
$C_{n}$ (resp. $2D_{n}$, $2A_{4}$, $2S_{4}$, $2A_{5}$).

######  Proposition 3.13.

Let $p>3$ be prime, and let $S$ be the set of integers that divide either
$p-1$ or $p+1$. Up to conjugacy in ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ the
subgroups of ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ of order prime to $p$ are
as follows:

  * •

For each integer $n\geq 1$ with $p\equiv\pm 1\bmod n$, a single conjugacy
class $C_{n}$.

  * •

For each integer $2n>2$ with $p\equiv\pm 1\bmod 4n$, two conjugacy classes
$2D_{n}$.

  * •

For each integer $2n>2$ with $p\equiv\pm 1\bmod 2n$ and $p\not\equiv\pm 1\bmod
4n$, a single conjugacy class $2D_{n}$.

  * •

Two conjugacy classes $2A_{4}$ if $p\equiv\pm 1\bmod 8$ and one otherwise.

  * •

Two conjugacy classes $2S_{4}$ if $p\equiv\pm 1\bmod 8$ and none otherwise.

  * •

Two conjugacy classes $2A_{5}\simeq{\mathbf{SL}}(2,5)$ if $p\equiv\pm 1\bmod
5$ and none otherwise.

###### Proof.

Every cyclic subgroup of ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ order prime to
$p$ must be conjugate in ${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$ to a subgroup
of one of the two Cartan subgroups $C$: the _split Cartan_ isomorphic to
$\mathbf{F}_{\\!p}^{\times}\times\mathbf{F}_{\\!p}^{\times}$, or the _nonsplit
Cartan_ isomorphic to $\mathbf{F}_{p^{2}}^{\times}$. The intersection of $C$
with ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ is cyclic of order $p-1$ or $p+1$,
and the intersection of these groups is the cyclic group $\\{\pm 1\\}$ of
order $2=\gcd(p-1,p+1)$. It follows that up to
${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$-conjugacy there is a unique cyclic
subgroup $C_{n}$ of ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ of order $n$ for
each $n$ dividing $p-1$ or $p+1$, and [13, Theorem 4.1] implies that it is
also unique up to ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$-conjugacy.

For a Cartan subgroup $C$ of ${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$, let
$C^{+}$ denote its normalizer. It follows from [48, Lemma 3.13] that for each
subgroup $H$ of $C\cap{\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ there is at most
one subgroup $G$ of $C^{+}\cap{\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ with
dihedral image in ${\mathbf{PSL}}(2,p)$, and that subgroup must contain $-1$.
It follows from [48, Lemmas 3.16 and 3.18] that there is exactly one $G$ for
each $H\neq\\{\pm 1\\}$ that contains $-1$, up to conjugacy in
${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$. It follows that for each integer
$2n>2$ dividing $p-1$ or $p+1$ that up to
${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$-conjugacy there is a unique conjugacy
class $2D_{n}$ of ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$, and it follows from
[13, Theorem 4.1] and Remark 3.14 below that this
${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$-conjugacy class splits into two
${\mathbf{GL}}_{2}(\mathbf{F}_{\\!p})$-conjugacy classes if and only if
$p\equiv\pm 1\bmod 4n$.

The statements for $2A_{4}$, $2S_{4}$, $2A_{5}$ are immediate from [48, Lemma
3.21] and [13, Theorem 4.1]. ∎

######  Remark 3.14.

There is a minor error in the statement [13, Theorem 4.1] regarding the group
$2D_{2}$, which is denoted $BD_{4\cdot 2}$ in [13]. There are two conjugacy
classes $2D_{2}$ in ${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ when
$\sqrt{2}\in\mathbf{F}_{\\!p}$, equivalently, when $p\equiv\pm 1\bmod 8$, but
only one otherwise; this follows from the fact that the normalizer of
$BD_{4\cdot 2}$ in ${\mathbf{SL}}_{2}(\overline{\mathbf{F}}_{p})$ is $2S_{4}$
(not $2A_{4}$ as claimed in [13]), which is present in
${\mathbf{SL}}_{2}(\mathbf{F}_{\\!p})$ only when
$\sqrt{2}\in\mathbf{F}_{\\!p}$. The author is grateful to Yuval Flicker for
clarifying this point.

##  4 Computational results

In this section we present examples that realize the claims made in the
introduction, including that local integral equivalence does not imply group
isomorphism (§4.1), local isomorphism of number fields (§4.2), or integral
equivalence (§4.3), and that solvable equivalence does not imply integral
equivalence (§4.4). We also give a degree 32 example of locally integrally
equivalent number fields in §4.3 (best possible), and a degree 96 example of
solvably equivalent number fields in §4.4 (best known).

###  4.1 Locally integrally equivalent subgroups need not be isomorphic

In [38, Question 1], Prasad asks if integrally equivalent subgroups are
necessarily isomorphic. This is true in Scott’s example with two subgroups of
${\mathbf{PSL}}_{2}(\mathbf{F}_{29})$ isomorphic to the alternating group
$A_{5}$. The following example shows that locally integrally equivalent
subgroups need not be isomorphic. Let $G$ by the symmetric group $S_{21}$ and
consider the subgroups

$\displaystyle H_{1}\coloneqq\bigl{\langle}\,$ $\displaystyle(4\ 5)(6\ 15\ 7\
14)(8\ 17\ 9,16)(10\ 19\ 11\ 18)(12\ 21\ 13\ 20),$ $\displaystyle(1\ 2)(3\
5)(6\ 20\ 8\ 18)(7\ 21\ 9\ 19)(10\ 14\ 12\ 16)(11\ 15\ 13\
17)\,\bigr{\rangle},$ $\displaystyle H_{2}\coloneqq\bigl{\langle}\,$
$\displaystyle(4\ 5)(6\ 16\ 8\ 14)(7\ 17\ 9\ 15)(10\ 20\ 12\ 18)(11\ 21\ 13\
19),$ $\displaystyle(1\ 2)(3\ 5)(6\ 20\ 8\ 18)(7\ 21\ 9\ 19)(10\ 17\ 12\
15)(11\ 16\ 13\ 14)\,\bigr{\rangle},$

with GAP identifiers $\langle 48,12\rangle$ and $\langle 48,13\rangle$,
respectively. Each contains 41 subgroups that are $p$-cyclic for some prime
$p$. These fall into 15 distinct $G$-conjugacy classes and 11 distinct
isomorphism classes, which makes it easy to find a $G$-conjugacy class
preserving bijection between them (if one takes into account the isomorphism
class and the number of subgroups in each conjugacy classes, there are only 2
choices to consider). The subgroups $H_{1},H_{2}\leq G$ are thus locally
integrally equivalent, but not isomorphic. This negatively answers Question
2.11 posed by Guralnick and Weiss in [18].

This example is realized by infinitely many number fields: over $\mathbf{Q}$
the Galois group of a generic polynomial of degre 21 is $G=S_{21}$ and the
fixed fields of $H_{1}$ and $H_{2}$ are locally integrally equivalent number
fields of degree $21!/48$. It is one of many that were found by applying
Corollary 2.5 to the clas $\mathcal{P}$ of groups that are $p$-cyclic for some
prime $p$: computing $\mathcal{P}$-statistics for the isomorphism classes of
groups of order up to 255 already finds 107 pairs of isomorphism classes with
the same $\mathcal{P}$-statistics, including four isomorphism classes of
groups of order 192 with the same $\mathcal{P}$-statistics. One can often find
permutation representations of degree less than $|H_{1}|=|H_{2}|$ that also
work, as happens above.

######  Question 4.1.

Are solvably equivalent subgroups of a finite group $G$ necessarily
isomorphic?

Question 4.1 is equivalent to asking whether the isomorphism class of a
nonsolvable group determine its $\mathcal{P}$-statistics, where $\mathcal{P}$
is the class of solvable groups (by Corollary 2.5). For the 1022 isomorphism
classes of nonsolvable groups of order less than 2000, these
$\mathcal{P}$-statistics are all distinct, so any pair of nonisomorphic
solvably equivalent subgroups must have order greater than 2000.

###  4.2 Local integral equivalence does not imply local isomorphism of number
fields

Let $G$ be the group $A_{4}\times S_{5}$ with GAP identifier $\langle
1440,5846\rangle$. There is a unique pair of nonconjugate locally integrally
equivalent subgroups $H_{1},H_{2}\leq G$, both of which are isomorphic to the
dihedral group $D_{6}$ of order 12. The groups $G$, $H_{1}$, $H_{2}$ can be
explicitly represented as subgroups of $S_{9}$ via

$\displaystyle G$ $\displaystyle\coloneqq\bigl{\langle}(1\ 2\ 3)(5\ 6\ 7\ 8\
9),\,(1\ 2)(3\ 4)(5\ 6)\bigr{\rangle},$ $\displaystyle H_{1}$
$\displaystyle\coloneqq\bigl{\langle}(1\ 2)(3\ 4)(5\ 6\ 7)(8\ 9),\,(1\ 3)(2\
4)(5\ 6)\bigr{\rangle},$ $\displaystyle H_{2}$
$\displaystyle\coloneqq\bigl{\langle}(1\ 2)(3\ 4)(5\ 6\ 7)(8\ 9),\,(1\ 4)(2\
3)(5\ 6)\bigr{\rangle},$

and $H_{1}\cap H_{2}$ is cyclic of order 6. The four maximal subgroups of
$H_{1}$, isomorphic to $C_{2}^{2},S_{3},S_{3},C_{6}$, correspond to distinct
conjugacy classes of subgroups of $G$, and these are precisely the
$G$-conjugacy classes of the four maximal subgroups of $H_{2}$. There is thus
a $G$-conjugacy preserving bijection between the proper subgroups of $H_{1}$
and $H_{2}$ (all of which are $p$-cyclic for some prime $p$), and the group
$D_{6}\simeq H_{1},H_{2}$ is not $p$-cyclic for any prime $p$. It follows that
$H_{1}$ and $H_{2}$ are locally integrally equivalent subgroups of $G$.

The subgroups $H_{1}$ and $H_{2}$ are not $G$-conjugate, even though they are
$S_{9}$-conjugate, as can be verified by comparing their permutation
characters: $\chi_{H_{1}}(H_{1})=4$ differs from $\chi_{H_{2}}(H_{1})=0$, and
$\chi_{H_{1}}(H_{2})=0$ differs from $\chi_{H_{2}}(H_{2})=4$. The group
$D_{6}$ arises as a Galois group of extensions of $\mathbf{Q}_{p}$ for
$p\not\equiv 1\bmod 6$, and it follows from Proposition 2.19 that if $H_{1}$
is the decomposition group of a prime above $p$ in a Galois extension
$L/\mathbf{Q}$ with Galois group $G$, then the fixed fields $K_{1}\coloneqq
L^{H_{1}}$ and $K_{2}\coloneqq L^{H_{2}}$ are locally integrally equivalent
fields that cannot be locally isomorphic because four primes of $K_{1}$ above
$2$ must have residue field degree 1 and ramification index 1 (corresponding
to the four cosets in $[H_{1}\backslash G]$ fixed by $H_{1}$), but no primes
of $K_{2}$ above $2$ can have residue field degree 1 and ramification index 1.

To realize such an example it suffices to find a pair of linearly disjoint
$A_{4}$ and $S_{5}$ fields such that that there is a prime of the compositum
with decomposition group conjugate to $H_{1}$ or $H_{2}$. A search of $A_{4}$
and $S_{5}$ fields in the $L$-functions and modular forms database (LMFDB)
unramified away from 2,3,5,7 finds a suitable pair: we may take the Galois
closures for the fields $F_{1}\coloneqq\mathbf{Q}[x]/(x^{4}-6x^{2}-8x+60)$ and
$F_{2}\coloneqq\mathbf{Q}[x]/(x^{5}+5x^{3}+10x-2)$ with LMFDB labels
4.0.254016.2 and 5.1.500000.1, respectively. The compositum of their Galois
closures is a degree 1440 number field $L$ with Galois group $G$. The 120
primes of $L$ above $2$ all have residue degree 2, ramification index 6,
decomposition group conjugate to $H_{1}$, and inertia group conjugate to
$H_{1}\cap H_{2}$; the local algebra $L\otimes_{\mathbf{Q}}\mathbf{Q}_{2}$ is
isomorphic to $k^{120}$, where $k$ is the unique $D_{6}$-extension of
$\mathbf{Q}_{2}$ of degree 12 containing $\mathbf{Q}_{2}(\sqrt{2})$, with
LMFDB label 2.12.22.60.

Using the GaloisSubgroup function in Magma [5] one can compute defining
polynomials of degree 120 for the number fields $K_{1}\coloneqq L^{H_{1}}$ and
$K_{2}\coloneqq L^{H_{2}}$, and using the $p$-adic valuation extensions method
in Sage [41] one can determine the residue field degrees and ramification
indices of the primes above 2 in $K_{2}$ and $K_{2}$ by computing all
extensions of the $2$-adic valuation of $\mathbf{Q}$ to $K_{1}$ and $K_{2}$.
We have

$\displaystyle 2\mathcal{O}_{K_{1}}$
$\displaystyle=\mathfrak{p}_{1}\mathfrak{p}_{2}\mathfrak{p}_{3}\mathfrak{p}_{4}\mathfrak{p}_{5}^{6}\mathfrak{p}_{6}^{6}\mathfrak{p}_{7}^{6}\mathfrak{p}_{8}^{6}\mathfrak{p}_{9}^{6}\mathfrak{p}_{10}^{6}\mathfrak{p}_{11}^{6}\mathfrak{p}_{12}^{6}\mathfrak{p}_{13}^{2}\mathfrak{p}_{14}^{2}\mathfrak{p}_{15}^{3}\mathfrak{p}_{16}^{3}\mathfrak{p}_{17}^{6}\mathfrak{p}_{18}^{6}\mathfrak{p}_{19}^{6}\mathfrak{p}_{20}^{6},$
$\displaystyle 2\mathcal{O}_{K_{2}}$
$\displaystyle=\mathfrak{q}_{1}^{2}\mathfrak{q}_{2}^{2}\mathfrak{q}_{3}^{2}\mathfrak{q}_{4}^{2}\mathfrak{q}_{5}^{3}\mathfrak{q}_{6}^{3}\mathfrak{q}_{7}^{3}\mathfrak{q}_{8}^{3}\mathfrak{q}_{9}^{6}\mathfrak{q}_{10}^{6}\mathfrak{q}_{11}^{6}\mathfrak{q}_{12}^{6}\mathfrak{q}_{13}\mathfrak{q}_{14}\mathfrak{q}_{15}^{6}\mathfrak{q}_{16}^{6}\mathfrak{q}_{17}^{6}\mathfrak{q}_{18}^{6}\mathfrak{q}_{19}^{6}\mathfrak{q}_{20}^{6},$

where the primes $\mathfrak{p}_{i}$ of $K_{1}$ and $\mathfrak{q}_{i}$ of
$K_{2}$ have residue degree 1 for $i\leq 12$ and residue degree $2$ for
$i>12$.

######  Remark 4.2.

This example can be viewed as a refinement of the example of Mantilla-Soler
[30] noted in Remark 2.13: the sums 82 and 86 of the ramification indices
differ. But in the Mantilla-Soler example the products of the ramification
indices also differ, which is possible because the subgroups are rationally
equivalent but not locally integrally equivalent. Proposition 3.4 shows that
this is not possible when the subgroups are locally integrally equivalent. To
our knowledge, this is the first example of a pair of arithmetically
equivalent number fields and a prime $p$ for which the sums of the
ramification indices of the primes above $p$ differ but the products do not.

Finally, we note that the groups $H_{1}$ and $H_{2}$ are isomorphic to
$D_{6}$, hence solvable, but the values of the permutation characters
$\chi_{H_{1}}$ and $\chi_{H_{2}}$ differ on these groups, as noted above, so
they are not solvably equivalent, which shows that solvable equivalence is a
strictly stronger condition (as one would expect).

###  4.3 A minimal degree example of local integral equivalence

An exhaustive search of isomorphisms classes of groups of order less than 1024
in the small groups database [3] finds 74 groups $G$ that contain nonconjugate
$H_{1},H_{2}\leq G$ that are locally integrally equivalent and have trivial
normal core in $G$ (meaning that $(G,H_{1},H_{2})$ is a faithful Gassmann
triple). The order of $G$ is necessarily not a prime power, since $p$-groups
can be locally integrally equivalent only if they are conjugate, so only
1,206,112 of the 11,759,892 groups of order less than 1024 need to be checked.
Of these, two have order 384, seventeen have order 576, fifty have order 768,
and five have order 864, with the index of $H_{1},H_{2}$ in $G$ taking values
in $\\{32,48,64,72\\}$.

The two groups $G$ of order 384 have GAP identifiers $\langle
384,18046\rangle$ and $\langle 384,18050\rangle$, and are isomorphic to
transitive permutation groups of degree 32 with LMFDB labels 32T9403 and
32T9408, following the labeling convention in [7]. Both are (nonsplit)
2-extensions of $D_{4}\times S_{4}$, making it feasible to explicitly
construct examples of nonconjugate number fields $K_{1}$ and $K_{2}$ of degree
32 with common Galois closure $L$ with $G=\operatorname{Gal}(L/\mathbf{Q})$,
and $H_{1}=\operatorname{Gal}(L/K_{1})$ and
$H_{2}=\operatorname{Gal}(L/K_{2})$ locally integrally equivalent, by taking a
quadratic extension of the compositum of the Galois closure of two suitably
chosen $D_{4}$ and $S_{4}$ quartic number fields. Below we describe one such
example in detail.

The Magma computer algebra system [5] includes a database of transitive
permutation groups of degree up to 48 whose construction is described in [7,
20, 21]. An exhaustive analysis of the 40,238 transitive groups of degree less
than 32 finds none that contain a pair of locally integrally equivalent
subgroups of index equal to the degree. The following example thus achieves
the minimal possible degree $32$; for comparison, the minimal degree of
arithmetically equivalent number fields is $7$; see [6].

We begin with the $D_{4}$ field $\mathbf{Q}[x]/(x^{4}-6x^{2}-9)$ and the
$S_{4}$ field $\mathbf{Q}[x]/(x^{4}-2x^{3}-6x+3)$, with LMFDB labels
4.2.9216.1 and 4.2.3888.1, which are linearly disjoint over $\mathbf{Q}$. The
compositum of their Galois closures coincides with the splitting field of the
polynomial

$x^{16}+12x^{14}+72x^{12}+120x^{10}-234x^{8}+108x^{6}+396x^{4}-432x^{2}+81,$

which has Galois group $D_{4}\times S_{4}$. The number fields
$K_{1}\coloneqq\mathbf{Q}[x]/(f_{1}(x))$,
$K_{2}\coloneqq\mathbf{Q}[x]/(f_{2}(x))$ defined by

$\displaystyle f_{1}$ $\displaystyle\coloneqq
x^{32}+12x^{28}+72x^{24}+120x^{20}-234x^{16}+108x^{12}+396x^{8}-432x^{4}+81,$
$\displaystyle f_{2}$ $\displaystyle\coloneqq
x^{32}-12x^{28}+72x^{24}-120x^{20}-234x^{16}-108x^{12}+396x^{8}+432x^{4}+81,$

have the same Galois closure $L$ of degree 384. The group
$G\coloneqq\operatorname{Gal}(L/\mathbf{Q})$ is the transitive permutation
group 32T9403, generated by

$\displaystyle\sigma_{0}$
$\displaystyle\coloneqq(3,4,5,6,7,8)(9,10,11,12,13,14)(15,16,17,18,19,20)(21,22,23)(24,25,26)(27,28)(29,30)(31,32),$
$\displaystyle\sigma_{1}$
$\displaystyle\coloneqq(3,5)(6,8)(9,10)(11,14)(12,13)(15,17)(18,20)(21,24)(22,26)(23,25)(27,31)(28,32),$
$\displaystyle\sigma_{2}$
$\displaystyle\coloneqq(1,2)(3,17)(4,16)(5,15)(6,20)(7,19)(8,18)(9,13)(10,12)(22,23)(25,26)(29,30),$
$\displaystyle\sigma_{3}$
$\displaystyle\coloneqq(1,3,2,15)(4,9,16,12)(5,24,17,21)(6,30,18,29)(7,22,19,25)(8,14,20,11)(10,32,13,28)(23,27,26,31).$

The group $G$ contains exactly two conjugacy classes of subgroups of index 32
with trivial normal core, represented by
$H_{1}\coloneqq\langle\sigma_{1},\sigma_{2}\rangle$ and
$H_{2}\coloneqq\langle\sigma_{0},\sigma_{2}\rangle$, both isomorphic to
$D_{6}$. If we view $G$ as acting on the roots of $f_{1}(x)$, then under a
suitable ordering of roots we have $H_{1}=\operatorname{Gal}(L/K_{1})$ and
$H_{2}=\operatorname{Gal}(L/K_{2})$. The subgroups $H_{1}$ and $H_{2}$ are
locally integrally equivalent but not integrally equivalent. Indeed, for a
suitable choice of bases for $[H_{1}\backslash G]$ and $[H_{2}\backslash G]$,
every $M\in\operatorname{Hom}_{\mathbf{Z}[G]}(\mathbf{Z}[H_{1}\backslash
G],\mathbf{Z}[H_{2}\backslash G])$ has the form

$M:=\setcounter{MaxMatrixCols}{32}\begin{bmatrix}x_{8}&x_{8}&x_{5}&x_{8}&x_{8}&x_{7}&x_{8}&x_{5}&x_{8}&x_{2}&x_{1}&x_{8}&x_{6}&x_{8}&x_{8}&x_{7}&x_{7}&x_{8}&x_{7}&x_{4}&x_{3}&x_{3}&x_{1}&x_{7}&x_{8}&x_{6}&x_{2}&x_{4}&x_{5}&x_{6}&x_{7}&x_{8}\\\
x_{8}&x_{8}&x_{7}&x_{8}&x_{8}&x_{5}&x_{8}&x_{7}&x_{8}&x_{3}&x_{4}&x_{8}&x_{7}&x_{8}&x_{8}&x_{5}&x_{6}&x_{8}&x_{5}&x_{4}&x_{2}&x_{2}&x_{1}&x_{6}&x_{8}&x_{7}&x_{3}&x_{1}&x_{7}&x_{7}&x_{6}&x_{8}\\\
x_{7}&x_{6}&x_{8}&x_{3}&x_{7}&x_{8}&x_{5}&x_{1}&x_{2}&x_{8}&x_{7}&x_{5}&x_{8}&x_{3}&x_{7}&x_{8}&x_{8}&x_{2}&x_{1}&x_{5}&x_{8}&x_{8}&x_{6}&x_{8}&x_{6}&x_{4}&x_{8}&x_{7}&x_{8}&x_{8}&x_{4}&x_{7}\\\
x_{8}&x_{1}&x_{6}&x_{8}&x_{8}&x_{2}&x_{8}&x_{5}&x_{8}&x_{7}&x_{8}&x_{4}&x_{5}&x_{8}&x_{1}&x_{7}&x_{7}&x_{8}&x_{7}&x_{8}&x_{5}&x_{6}&x_{8}&x_{2}&x_{8}&x_{6}&x_{7}&x_{8}&x_{3}&x_{3}&x_{7}&x_{4}\\\
x_{8}&x_{8}&x_{6}&x_{8}&x_{8}&x_{7}&x_{8}&x_{6}&x_{8}&x_{2}&x_{4}&x_{8}&x_{5}&x_{8}&x_{8}&x_{7}&x_{7}&x_{8}&x_{7}&x_{1}&x_{3}&x_{3}&x_{4}&x_{7}&x_{8}&x_{5}&x_{2}&x_{1}&x_{6}&x_{5}&x_{7}&x_{8}\\\
x_{5}&x_{7}&x_{8}&x_{2}&x_{6}&x_{8}&x_{7}&x_{4}&x_{3}&x_{8}&x_{5}&x_{7}&x_{8}&x_{2}&x_{5}&x_{8}&x_{8}&x_{3}&x_{1}&x_{7}&x_{8}&x_{8}&x_{7}&x_{8}&x_{7}&x_{1}&x_{8}&x_{6}&x_{8}&x_{8}&x_{4}&x_{6}\\\
x_{8}&x_{4}&x_{7}&x_{8}&x_{8}&x_{3}&x_{8}&x_{7}&x_{8}&x_{5}&x_{8}&x_{1}&x_{7}&x_{8}&x_{1}&x_{5}&x_{6}&x_{8}&x_{6}&x_{8}&x_{7}&x_{7}&x_{8}&x_{3}&x_{8}&x_{7}&x_{6}&x_{8}&x_{2}&x_{2}&x_{5}&x_{4}\\\
x_{8}&x_{8}&x_{7}&x_{8}&x_{8}&x_{6}&x_{8}&x_{7}&x_{8}&x_{3}&x_{1}&x_{8}&x_{7}&x_{8}&x_{8}&x_{6}&x_{5}&x_{8}&x_{6}&x_{1}&x_{2}&x_{2}&x_{4}&x_{5}&x_{8}&x_{7}&x_{3}&x_{4}&x_{7}&x_{7}&x_{5}&x_{8}\\\
x_{5}&x_{7}&x_{8}&x_{7}&x_{6}&x_{8}&x_{7}&x_{8}&x_{6}&x_{1}&x_{3}&x_{7}&x_{8}&x_{7}&x_{6}&x_{8}&x_{8}&x_{5}&x_{8}&x_{2}&x_{1}&x_{4}&x_{2}&x_{8}&x_{7}&x_{8}&x_{4}&x_{3}&x_{8}&x_{8}&x_{8}&x_{5}\\\
x_{4}&x_{8}&x_{2}&x_{8}&x_{1}&x_{6}&x_{4}&x_{7}&x_{8}&x_{5}&x_{8}&x_{8}&x_{2}&x_{8}&x_{8}&x_{3}&x_{3}&x_{8}&x_{5}&x_{8}&x_{7}&x_{7}&x_{8}&x_{5}&x_{1}&x_{7}&x_{6}&x_{8}&x_{7}&x_{7}&x_{6}&x_{8}\\\
x_{7}&x_{3}&x_{8}&x_{5}&x_{7}&x_{1}&x_{6}&x_{8}&x_{7}&x_{8}&x_{7}&x_{3}&x_{8}&x_{6}&x_{2}&x_{8}&x_{8}&x_{7}&x_{8}&x_{5}&x_{8}&x_{8}&x_{6}&x_{4}&x_{5}&x_{8}&x_{8}&x_{7}&x_{1}&x_{4}&x_{8}&x_{2}\\\
x_{4}&x_{8}&x_{3}&x_{8}&x_{1}&x_{7}&x_{1}&x_{6}&x_{8}&x_{7}&x_{8}&x_{8}&x_{3}&x_{8}&x_{8}&x_{2}&x_{2}&x_{8}&x_{7}&x_{8}&x_{5}&x_{6}&x_{8}&x_{7}&x_{4}&x_{5}&x_{7}&x_{8}&x_{5}&x_{6}&x_{7}&x_{8}\\\
x_{7}&x_{5}&x_{8}&x_{3}&x_{7}&x_{8}&x_{6}&x_{4}&x_{2}&x_{8}&x_{7}&x_{6}&x_{8}&x_{3}&x_{7}&x_{8}&x_{8}&x_{2}&x_{4}&x_{6}&x_{8}&x_{8}&x_{5}&x_{8}&x_{5}&x_{1}&x_{8}&x_{7}&x_{8}&x_{8}&x_{1}&x_{7}\\\
x_{8}&x_{4}&x_{5}&x_{8}&x_{8}&x_{2}&x_{8}&x_{6}&x_{8}&x_{7}&x_{8}&x_{1}&x_{6}&x_{8}&x_{4}&x_{7}&x_{7}&x_{8}&x_{7}&x_{8}&x_{6}&x_{5}&x_{8}&x_{2}&x_{8}&x_{5}&x_{7}&x_{8}&x_{3}&x_{3}&x_{7}&x_{1}\\\
x_{7}&x_{5}&x_{8}&x_{5}&x_{7}&x_{8}&x_{5}&x_{8}&x_{7}&x_{1}&x_{2}&x_{6}&x_{8}&x_{6}&x_{7}&x_{8}&x_{8}&x_{7}&x_{8}&x_{3}&x_{4}&x_{1}&x_{3}&x_{8}&x_{6}&x_{8}&x_{4}&x_{2}&x_{8}&x_{8}&x_{8}&x_{7}\\\
x_{1}&x_{8}&x_{3}&x_{8}&x_{4}&x_{7}&x_{4}&x_{5}&x_{8}&x_{7}&x_{8}&x_{8}&x_{3}&x_{8}&x_{8}&x_{2}&x_{2}&x_{8}&x_{7}&x_{8}&x_{6}&x_{5}&x_{8}&x_{7}&x_{1}&x_{6}&x_{7}&x_{8}&x_{6}&x_{5}&x_{7}&x_{8}\\\
x_{5}&x_{2}&x_{8}&x_{7}&x_{6}&x_{4}&x_{7}&x_{8}&x_{5}&x_{8}&x_{6}&x_{2}&x_{8}&x_{7}&x_{3}&x_{8}&x_{8}&x_{6}&x_{8}&x_{7}&x_{8}&x_{8}&x_{7}&x_{1}&x_{7}&x_{8}&x_{8}&x_{5}&x_{1}&x_{4}&x_{8}&x_{3}\\\
x_{1}&x_{8}&x_{2}&x_{8}&x_{4}&x_{5}&x_{1}&x_{7}&x_{8}&x_{6}&x_{8}&x_{8}&x_{2}&x_{8}&x_{8}&x_{3}&x_{3}&x_{8}&x_{6}&x_{8}&x_{7}&x_{7}&x_{8}&x_{6}&x_{4}&x_{7}&x_{5}&x_{8}&x_{7}&x_{7}&x_{5}&x_{8}\\\
x_{6}&x_{7}&x_{8}&x_{2}&x_{5}&x_{8}&x_{7}&x_{1}&x_{3}&x_{8}&x_{6}&x_{7}&x_{8}&x_{2}&x_{6}&x_{8}&x_{8}&x_{3}&x_{4}&x_{7}&x_{8}&x_{8}&x_{7}&x_{8}&x_{7}&x_{4}&x_{8}&x_{5}&x_{8}&x_{8}&x_{1}&x_{5}\\\
x_{8}&x_{1}&x_{7}&x_{8}&x_{8}&x_{3}&x_{8}&x_{7}&x_{8}&x_{6}&x_{8}&x_{4}&x_{7}&x_{8}&x_{4}&x_{6}&x_{5}&x_{8}&x_{5}&x_{8}&x_{7}&x_{7}&x_{8}&x_{3}&x_{8}&x_{7}&x_{5}&x_{8}&x_{2}&x_{2}&x_{6}&x_{1}\\\
x_{8}&x_{8}&x_{5}&x_{1}&x_{8}&x_{7}&x_{8}&x_{3}&x_{1}&x_{7}&x_{8}&x_{8}&x_{6}&x_{4}&x_{8}&x_{7}&x_{7}&x_{4}&x_{2}&x_{8}&x_{5}&x_{6}&x_{8}&x_{7}&x_{8}&x_{3}&x_{7}&x_{8}&x_{6}&x_{5}&x_{2}&x_{8}\\\
x_{3}&x_{7}&x_{4}&x_{7}&x_{3}&x_{8}&x_{2}&x_{8}&x_{5}&x_{8}&x_{5}&x_{7}&x_{1}&x_{7}&x_{6}&x_{1}&x_{4}&x_{6}&x_{8}&x_{7}&x_{8}&x_{8}&x_{7}&x_{8}&x_{2}&x_{8}&x_{8}&x_{6}&x_{8}&x_{8}&x_{8}&x_{5}\\\
x_{8}&x_{8}&x_{7}&x_{4}&x_{8}&x_{5}&x_{8}&x_{2}&x_{1}&x_{5}&x_{8}&x_{8}&x_{7}&x_{1}&x_{8}&x_{6}&x_{5}&x_{4}&x_{3}&x_{8}&x_{7}&x_{7}&x_{8}&x_{6}&x_{8}&x_{2}&x_{6}&x_{8}&x_{7}&x_{7}&x_{3}&x_{8}\\\
x_{2}&x_{6}&x_{4}&x_{5}&x_{2}&x_{8}&x_{3}&x_{8}&x_{7}&x_{8}&x_{7}&x_{5}&x_{1}&x_{6}&x_{7}&x_{4}&x_{1}&x_{7}&x_{8}&x_{6}&x_{8}&x_{8}&x_{5}&x_{8}&x_{3}&x_{8}&x_{8}&x_{7}&x_{8}&x_{8}&x_{8}&x_{7}\\\
x_{8}&x_{8}&x_{6}&x_{4}&x_{8}&x_{7}&x_{8}&x_{3}&x_{4}&x_{7}&x_{8}&x_{8}&x_{5}&x_{1}&x_{8}&x_{7}&x_{7}&x_{1}&x_{2}&x_{8}&x_{6}&x_{5}&x_{8}&x_{7}&x_{8}&x_{3}&x_{7}&x_{8}&x_{5}&x_{6}&x_{2}&x_{8}\\\
x_{6}&x_{7}&x_{8}&x_{7}&x_{5}&x_{8}&x_{7}&x_{8}&x_{5}&x_{4}&x_{3}&x_{7}&x_{8}&x_{7}&x_{5}&x_{8}&x_{8}&x_{6}&x_{8}&x_{2}&x_{4}&x_{1}&x_{2}&x_{8}&x_{7}&x_{8}&x_{1}&x_{3}&x_{8}&x_{8}&x_{8}&x_{6}\\\
x_{7}&x_{3}&x_{8}&x_{6}&x_{7}&x_{4}&x_{5}&x_{8}&x_{7}&x_{8}&x_{7}&x_{3}&x_{8}&x_{5}&x_{2}&x_{8}&x_{8}&x_{7}&x_{8}&x_{6}&x_{8}&x_{8}&x_{5}&x_{1}&x_{6}&x_{8}&x_{8}&x_{7}&x_{4}&x_{1}&x_{8}&x_{2}\\\
x_{6}&x_{2}&x_{8}&x_{7}&x_{5}&x_{1}&x_{7}&x_{8}&x_{6}&x_{8}&x_{5}&x_{2}&x_{8}&x_{7}&x_{3}&x_{8}&x_{8}&x_{5}&x_{8}&x_{7}&x_{8}&x_{8}&x_{7}&x_{4}&x_{7}&x_{8}&x_{8}&x_{6}&x_{4}&x_{1}&x_{8}&x_{3}\\\
x_{2}&x_{5}&x_{1}&x_{6}&x_{2}&x_{8}&x_{3}&x_{8}&x_{7}&x_{8}&x_{7}&x_{6}&x_{4}&x_{5}&x_{7}&x_{1}&x_{4}&x_{7}&x_{8}&x_{5}&x_{8}&x_{8}&x_{6}&x_{8}&x_{3}&x_{8}&x_{8}&x_{7}&x_{8}&x_{8}&x_{8}&x_{7}\\\
x_{3}&x_{7}&x_{1}&x_{7}&x_{3}&x_{8}&x_{2}&x_{8}&x_{6}&x_{8}&x_{6}&x_{7}&x_{4}&x_{7}&x_{5}&x_{4}&x_{1}&x_{5}&x_{8}&x_{7}&x_{8}&x_{8}&x_{7}&x_{8}&x_{2}&x_{8}&x_{8}&x_{5}&x_{8}&x_{8}&x_{8}&x_{6}\\\
x_{8}&x_{8}&x_{7}&x_{1}&x_{8}&x_{6}&x_{8}&x_{2}&x_{4}&x_{6}&x_{8}&x_{8}&x_{7}&x_{4}&x_{8}&x_{5}&x_{6}&x_{1}&x_{3}&x_{8}&x_{7}&x_{7}&x_{8}&x_{5}&x_{8}&x_{2}&x_{5}&x_{8}&x_{7}&x_{7}&x_{3}&x_{8}\\\
x_{7}&x_{6}&x_{8}&x_{6}&x_{7}&x_{8}&x_{6}&x_{8}&x_{7}&x_{4}&x_{2}&x_{5}&x_{8}&x_{5}&x_{7}&x_{8}&x_{8}&x_{7}&x_{8}&x_{3}&x_{1}&x_{4}&x_{3}&x_{8}&x_{5}&x_{8}&x_{1}&x_{2}&x_{8}&x_{8}&x_{8}&x_{7}\\\
\end{bmatrix}$

for some $x_{1},\ldots,x_{8}\in\mathbf{Z}$ corresponding to the decomposition
of $G$ into eight double cosets $H_{1}gH_{2}$, consisting of
$2,2,2,2,3,3,6,12$ right cosets of $H_{1}$, respectively. A (nontrivial)
calculation finds that

$\displaystyle\operatorname{det}M=-\,$
$\displaystyle(2(x_{2}-x_{3})^{2}+3(x_{5}-x_{6})^{2})^{8}$
$\displaystyle\cdot\ $
$\displaystyle(2(x_{1}-x_{4})+(x_{5}+x_{6}-2x_{7}))^{6}$ $\displaystyle\cdot\
$ $\displaystyle(2(x_{1}+x_{2}+x_{3}+x_{4})-(x_{5}+x_{6}+2x_{7}+4x_{8}))^{3}$
$\displaystyle\cdot\ $
$\displaystyle(2(x_{1}-x_{2}-x_{3}+x_{4})-(x_{5}+x_{6}+2x_{7}-4x_{8}))^{3}$
$\displaystyle\cdot\ $
$\displaystyle(2(x_{1}-x_{4})-3(x_{5}+x_{6}-2x_{7}))^{2}$ $\displaystyle\cdot\
$ $\displaystyle(2(x_{1}+x_{2}+x_{3}+x_{4})+3(x_{5}+x_{6}+2x_{7}+4x_{8}))$
$\displaystyle\cdot\ $
$\displaystyle(2(x_{1}-x_{2}-x_{3}+x_{4})+3(x_{5}+x_{6}+2x_{7}-4x_{8})).$

The assignment

$x_{1}=x_{2}=1,\quad x_{3}=-1,\quad x_{4}=x_{5}=x_{6}=x_{7}=x_{8}=0$

gives $\operatorname{det}M=2^{32}$, while the assignment

$x_{5}=1,\quad x_{1}=x_{2}=x_{3}=x_{4}=x_{6}=x_{7}=x_{8}=0$

gives $\operatorname{det}M=3^{12}$; thus $d(H_{1},H_{2})=1$. It follows that
$H_{1}$ and $H_{2}$ are locally integrally equivalent, by Proposition 3.1. But
no assignment of $x_{1},\ldots,x_{8}\in\mathbf{Z}$ makes
$\operatorname{det}M=\pm 1$; indeed, any such assignment would require all 7
factors of $\operatorname{det}M$ listed above to have values in $\\{\pm 1\\}$,
which is not possible. Thus $H_{1}$ and $H_{2}$ are not integrally equivalent;
as noted in the introduction, this negatively answers Question 2.10 of
Guralnick and Weiss in [18].

There are infinitely many nonisomorphic variations of this example; replacing
$f_{1}(x)$ and $f_{2}(x)$ with $f_{1}(x\sqrt{T})$ and $f_{2}(x\sqrt{T})$
yields polynomials with Galois group $G$ over $\mathbf{Q}(T)$; for almost all
squarefree $a\in\mathbf{Z}$ the substitution $T=a$ yields nonisomorphic
$K_{1},K_{2}$ ramified at primes dividing $a$.

###  4.4 A degree 96 example of solvable equivalence

The results of §4.3 imply that any group $G$ that contains nonconjugate
solvably equivalent subgroups must have order at least $32\cdot 60=1920$,
since nonconjugate locally integrally equivalent subgroups must have index at
least 32, and nonsolvable groups must have order at least 60. A search of the
small groups database shows that there are no such $G$ of order 1920 or 1980,
and a search of transitive groups of degree up to 48 and order at most
$48\cdot 60=2880$ finds no such $G$, which implies a lower bound of 2940.

An exhaustive search of transitive groups of degree up to 48 and order at most
48,000 finds transitive groups of degrees 12, 16, 20, 24, 30, 32, 36, and 40
that contain nonconjugate solvably equivalent subgroups, including examples of
index 96, 192, 384, 576, 672, and 768. The first example of index 96 occurs
for the transitive group 16T1654 of order 5760, which is the smallest order we
found. This group $G$ contains five conjugacy classes of subgroups isomorphic
to $A_{5}$, of which exactly two have representatives $H_{1}$ and $H_{2}$ with
the property that every proper subgroup of $H_{1}$ is also a proper subgroup
of $H_{2}$. The groups $H_{1}$ are thus solvably equivalent subgroups of index
96. There are 5 double cosets $H_{1}gH_{2}$, comprised of $5,6,10,15,60$ right
cosets of $H_{1}$, respectively; each
$M\in\operatorname{Hom}_{\mathbf{Z}[G]}(\mathbf{Z}[H_{1}\backslash
G],\mathbf{Z}[H_{2}\backslash G])$ can thus be viewed as a matrix in
indeterminates $x_{1},x_{2},x_{3},x_{4},x_{5}\in\mathbf{Z}$, and we have

$\displaystyle\operatorname{det}M=-\,$
$\displaystyle(5x_{1}+6x_{2}+10x_{3}+15x_{4}+60x_{5})$ $\displaystyle\cdot\ $
$\displaystyle(x_{1}-6x_{2}-10x_{3}+3x_{4}+12x_{5})^{5}$ $\displaystyle\cdot\
$ $\displaystyle(3x_{1}+2x_{2}-2x_{3}-7x_{4}+4x_{5})^{15}$
$\displaystyle\cdot\ $ $\displaystyle(3x_{1}-2x_{2}+2x_{3}+x_{4}-4x_{5})^{30}$
$\displaystyle\cdot\ $ $\displaystyle(x_{1}+2x_{2}-2x_{3}+3x_{4}-4x_{5})^{45}$

By solving 32 systems of linear equations, one finds that no assignment of
$x_{1},x_{2},x_{3},x_{4},x_{5}\in\mathbf{Z}$ makes every factor in
$\operatorname{det}M$ equal to $\pm 1$. Thus $H_{1}$ and $H_{2}$ are not
integrally equivalent.

The regular inverse Galois problem for 16T1654 is known (it is a quotient of
12T277), thus there are infinitely many pairs of solvably equivalent number
fields $K_{1}$ and $K_{2}$ with Galois closure $L$ that satisfy
$\operatorname{Gal}(L/\mathbf{Q})=G$, $\operatorname{Gal}(L/K_{1})=H_{1}$,
$\operatorname{Gal}(L/K_{2})=H_{2}$. For example, we may take $L$ as the
splitting field of

$x^{16}-2x^{15}+3x^{14}-16x^{13}+18x^{12}-10x^{10}+40x^{9}-39x^{8}+54x^{7}+23x^{6}+16x^{5}-140x^{4}-188x^{3}-28x^{2}+104x-4,$

corresponding to the number field with LMFDB label
16.4.711702043399998895292416.2. The field $L$ contains solvable equivalent
subfields $K_{1}$ and $K_{2}$ of degree 96 that are necessarily arithmetically
equivalent, locally isomorphic, and have isomorphic class groups, by
Proposition 3.7. One can find 190 examples of 16T1654 number fields in the
Klüners and Malle Database of Number Fields [25].

######  Remark 4.3.

In [42, Remark 4.3] Scott raises several questions related to integral
permutation modules that lie in the same genus, which in our setting
corresponds to local integral equivalence. The _rank_ of a group $G$ acting on
a finite set $\Omega$ is the number of orbits of the diagonal action on
$\Omega\times\Omega$. Scott shows that if the rank of $G$ acting on $\Omega$
is 2 or 3 then local integral equivalence of $\mathbf{Z}[G]$-modules $\Omega$
and $\Omega^{\prime}$ implies an isomorphism of $G$-sets [42, Proposition
4.1]. His example with $G={\mathbf{PSL}}_{2}(29)$ proves that this does not
hold when the rank is $8$. The example in §4.4 shows that this also fails to
hold when the rank is $5$.

## Acknowledgments

The author would like to thank Alex Bartel for introducing him to the notion
of a Gassmann triple at an LMFDB workshop at Oregon State University in 2015.
The author is also grateful to Yuval Flicker for the clarification noted in
Remark 3.14, and to David Roe and Raymond van Bommel for their assistance in
finding a particularly efficient method to compute ramification indices that
is exploited in §4.2.

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[avs] Andrew V. Sutherland
Department of Mathematics
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, Massachusetts 02139
USA
<EMAIL_ADDRESS>
https://math.mit.edu/~drew