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lemma corollaryequation corollary claimequation claim conjectureequation
conjecture questionequation question defnequation defn warnequation warn
exampleequation example remarkequation remark conventionequation convention

# Lifting $N_{\infty}$ operads from conjugacy data

Scott Balchin  Max Planck Institute for Mathematics<EMAIL_ADDRESS>,  Ethan MacBrough  Reed College<EMAIL_ADDRESS>and  Kyle Ormsby  Reed
College / University of Washington<EMAIL_ADDRESS>/<EMAIL_ADDRESS>
###### Abstract.

We isolate a class of groups — called _lossless groups_ — for which homotopy
classes of $G$-$N_{\infty}$ operads are in bijection with certain restricted
transfer systems on the poset of conjugacy classes $\operatorname{Sub}(G)/G$.

###### Contents

  1. 1 Introduction
  2. 2 Transfer systems on lossless groups
  3. 3 Lifting criteria
  4. 4 Examples of liftable transfer systems
  5. 5 Strategies for lossy groups

##  1\. Introduction

The concept of an $N_{\infty}$ operad, as introduced by Blumberg–Hill in
[BH15], provides an equivariant analogue of $E_{\infty}$ operads which
captures multiplicative norm maps on equivariant commutative ring spectra.
Further work by various authors proved that homotopy category of $N_{\infty}$
operads can be identified with far simpler structures called indexing systems
[BP21, GW18, Rub21a]. This was then distilled into the identification of
_transfer systems_ which are purely combinatorial representations of
$N_{\infty}$ operads on the subgroup lattice of the group in question [BBR21,
Rub21b]. This has led to computable approaches to understand the structures of
$N_{\infty}$ operads for a given group.

In [BBR21], the collection of $N_{\infty}$ operads for the cyclic groups
$C_{p^{n}}$ were classified via the use of transfer systems on the lattice
$\operatorname{Sub}(C_{p^{n}})\cong[n]$. This concept was abstracted in
[FOO+22] where the notion of transfer systems was developed for an arbitrary
finite poset, and shown in particular to be in bijection with weak
factorization systems when the poset was moreover a complete lattice. This
approach has already provided fruitful results such as classifications of
model structures on total orders in [BOOR22].

This abstraction provides a strict generalization. Indeed, in [Jak74], those
lattices $L$ which arise as $\operatorname{Sub}(G)$ for an arbitrary group $G$
are classified (this was refined to the Abelian case in [Con11]). For example,
the lattice in Figure 1 does not appear as $\operatorname{Sub}(G)$ for any
$G$.

Figure 1. A lattice which is not of the form $\operatorname{Sub}(G)$ for any
$G$.

In this paper, we move back to the roots of the classification of $N_{\infty}$
operads, with a view to improve the computational tools available. In
particular, we will make the first serious foray into the realm of non-Abelian
groups. In the definition of $N_{\infty}$ operads, conjugation must be taken
into account. One may naively hope that this is superfluous, and that one can
just work with transfer systems on the poset $\operatorname{Sub}(G)/G$, that
is, the collection of subgroups up to conjugacy. This, however, does not work
in general.

Indeed, this is already noted in an example of Rubin [Rub21b], as we now
recall. If one considers the symmetric group $\mathfrak{S}_{4}$, then there
are three conjugate copies of $D_{4}$ (the dihedral group of order $8$) living
inside it. Moreover, there are three double-transpositions in
$\mathfrak{S}_{4}$ which generate three conjugate copies of $C_{2}$. It
follows that to define a transfer system for $\mathfrak{S}_{4}$ it is not
enough to just declare that we have the relation $C_{2}\,\mathcal{R}\,D_{4}$,
we must also keep track of which copies of $C_{2}$ are related to which copies
of $D_{4}$, something that is lost when working up to conjugacy.

There are cases, however, where it is possible to work up to conjugacy; again,
such an example is observed by Rubin [Rub21b]. Moving down in the world of
symmetric groups, consider $G=\mathfrak{S}_{3}$. The subgroup lattice here
takes the form displayed in Figure 2.

Figure 2. The lattice $\operatorname{Sub}(G)$ for $G=\mathfrak{S}_{3}$. A red
node is a copy of $C_{2}$, while the blue node is a copy of $C_{3}$.

The key point is that we can treat all copies of $C_{2}$ as essentially being
the same subgroup. We refer the reader to Section 2.1 for the definition of a
transfer system for what follows. Suppose that we have a transfer system with
the relation $1\,\mathcal{R}\,\langle(12)\rangle$. As transfer systems are
required to be closed under conjugation, this implies that we necessarily have
$1\,\mathcal{R}\,\langle\tau\rangle$ for every transposition $\tau$. Dually if
we have $\langle(12)\rangle\,\mathcal{R}\,\mathfrak{S}_{3}$ then we also have
$\langle\tau\rangle\,\mathcal{R}\,\mathfrak{S}_{3}$ for every transposition
$\tau$. Next, we use that fact that in a transfer system if we have
$H\,\mathcal{R}\,K$ and $L\,\mathcal{R}\,K$ then we also have $(H\cap
L)\,\mathcal{R}\,K$. In particular, if we remember that we really have three
distinct copies of $C_{2}$, it follows that if we have
$\langle(12)\rangle\,\mathcal{R}\,\mathfrak{S}_{3}$, then we also have
$\langle(23)\rangle\,\mathcal{R}\,\mathfrak{S}_{3}$, and as such we have
$1\,\mathcal{R}\,\mathfrak{S}_{3}$ by this intersection property.

It turns out that this is the only condition that one needs to impose in this
case. So we may do exactly as we want, study transfer systems on
$\operatorname{Sub}(G)/G$, which is of the form Figure 3.

Figure 3. The lattice $\operatorname{Sub}(G)/G$ for $G=\mathfrak{S}_{3}$. The
red node is a copy of $C_{2}$ up to conjugacy, while a blue node is the copy
of $C_{3}$.

Any transfer system on this lattice (of which there are 10) lifts to a
transfer system for $\operatorname{Sub}(G)$ when we additionally satisfy that
whenever we have $\leavevmode\hbox to5.4pt{\vbox
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}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\mathcal{R}\,\mathfrak{S}_{3}$
we also have $1\,\mathcal{R}\,\mathfrak{S}_{3}$. It turns out that there are 9
such transfer systems.

The goal of this paper is to isolate a class of groups for which $G$-transfer
systems can be explicitly characterized as certain restricted transfer systems
on the poset $\operatorname{Sub}(G)/G$ in this fashion. The class of groups
that we will isolate here are the _lossless groups_ (see Section 2.1). Not
only does this provide a non-trivial structural result, it also equips us with
powerful computational tools to classify $N_{\infty}$ operads for a wide range
of non-Abelian groups, and provides a better conceptual understanding of the
structures involved.

For example, let us assume that we wish to study the collection of
$G$-$N_{\infty}$ operads for $G=D_{9}$. The lattice that we need to consider,
$\operatorname{Sub}(D_{9})$, is displayed in Figure 4.

Figure 4. The lattice $\operatorname{Sub}(G)$ for $G=D_{9}$. The colored nodes
in the rows indicate different conjugacy classes of subgroups.

Hoping to find patterns or structure on this lattice is a daunting task.
However, the results presented here will allow us to instead explore structure
in the much more manageable — and human friendly — $\operatorname{Sub}(G)/G$
as displayed in Figure 5. In [BMO22] we shall undertake this exploration, and
provide a recursive algorithm for computing $N_{\infty}$ operads for the
dihedral groups $D_{p^{n}}$ for all $n\geqslant 0$ ($p\neq 2$).

Figure 5. The lattice $\operatorname{Sub}(G)/G$ for $G=D_{9}$. The coloring of
the nodes corresponds to the coloring in Figure 4.

In Section 2 we will introduce the main object of study of this paper, the
_lossless groups_ , and prove that there is a bijection between $G$-transfer
systems for a lossless group $G$ and _liftable transfer systems_ on
$\operatorname{Sub}(G)/G$. We record the main theorem here.

######  Theorem (Section 2.1).

Let $G$ be a lossless group. Then there is a bijection between homotopy
classes of $G$-$N_{\infty}$ operads and liftable transfer systems on
$\operatorname{Sub}(G)/G$.

In fact, in a sense made precise in Section 2, the above theorem characterizes
lossless groups.

We then continue in Section 2.2 and prove that many groups of interest are in
fact lossless. In Section 3 we direct our attention to a particularly nice
class of lossless groups, namely the _metacyclic Frobenius groups_. For these
groups we have a detailed understanding of both the form of
$\operatorname{Sub}(G)/G$ and the lifting conditions required to determine a
$G$-transfer system. We apply this theory in Section 4 to demonstrate how the
theory aids computations. Finally, in Section 5 we outline a potential
strategy for dealing with lossy groups. In particular we shall focus on the
case of $G=SL_{2}(\mathbb{F}_{p})$ which is an important family of groups in
the study of topological modular forms with level structures [HL16].

### Conventions

Throughout, we shall use the following conventions for group theory:

  * •

$H\leqslant G$ designates $H$ as a subgroup of $G$.

  * •

$N\trianglelefteqslant G$ designates $N$ as a normal subgroup.

  * •

If $g\in G$ then ${}^{g}H:=gHg^{-1}$ (thus ${}^{g}({}^{h}H)={}^{(gh)}H$ for
all $g,h\in G$).

  * •

For $G$ a group we write $\operatorname{Sub}(G)/G$ for the poset of conjugacy
classes of subgroups of $G$.

  * •

For $n>2$, we write $D_{n}$ for the dihedral group of order $2n$.

### Acknowledgements

The first author would like to thank the Max Planck Institute for Mathematics
for its hospitality, and was partially supported by the European Research
Council (ERC) under Horizon Europe (grant No. 101042990). The second author
thanks Coil Technologies for their generous donation to fund his tuition,
which enabled him to conduct this research. The third author’s work was
supported by the National Science Foundation under Grant No. DMS-2204365. The
authors thank the anonymous referee for helpful comments and suggestions.

##  2\. Transfer systems on lossless groups

In this section we will introduce the class of _lossless groups_ , which allow
us to study $G$-transfer systems using only categorical transfer systems on
$\operatorname{Sub}(G)/G$. After proving the basic structural results about
these groups, we identify several simple criteria for when a group is
lossless, and also provide several examples of how groups can fail to be
lossless.

###  2.1. General results

We first recall the definitions of $G$-transfer systems and categorical
transfer systems. We refer the reader to [BBR21, FOO+22] for further details.

######  Definition .

Let $G$ be a finite group. A ($G$-)_transfer system_ is a relation
$\mathcal{R}$ on $\operatorname{Sub}(G)$ refining inclusion satisfying the
following:

  * •

(reflexivity) $H\,\mathcal{R}\,H$ for all $H\leqslant G$;

  * •

(transitivity) $K\,\mathcal{R}\,H$ and $L\,\mathcal{R}\,K$ implies
$L\,\mathcal{R}\,H$;

  * •

(closed under conjugation) $K\,\mathcal{R}\,H$ implies that
${}^{g}K\,\mathcal{R}\,{}^{g}H$ for all $g\in G$;

  * •

(closed under restriction) $K\,\mathcal{R}\,H$ and $L\leqslant H$ implies
$(K{\cap}L)\,\mathcal{R}\,L$.

######  Proposition  ([BBR21]).

Let $G$ be a finite group. Then there is a bijection between the set of
transfer systems on $\operatorname{Sub}(G)$ and the set of $G$-$N_{\infty}$
operads.

In [FOO+22], a notion of an abstract categorical transfer system was
introduced for an arbitrary poset, but with a particular focus on when the
poset in question is a lattice. In fact, despite presenting the definition for
general posets, the authors only make serious use of their definition in the
setting of lattices. Since $\operatorname{Sub}(G)/G$ may not be a lattice when
$G$ is non-commutative, we are forced to think seriously about more general
posets. The definition we present below is _not_ equivalent to the definition
given in [FOO+22], since we require restriction closure for arbitrary maximal
lower bounds, rather than just when a unique meet exists. However, it is
straightforward to verify the two definitions coincide when the poset is a
lattice. For an element $x$ in some poset $\mathcal{P}$, let $x^{\downarrow}$
denote the down-set of $x$ in $\mathcal{P}$, i.e. the set of all $y\leqslant
x$.

######  Definition .

Let $\mathcal{P}=(\mathcal{P},\leqslant)$ be a poset. A (categorical)
_transfer system_ on $\mathcal{P}$ consists of a partial order $\mathcal{R}$
on $\mathcal{P}$ that refines $\leqslant$ and such that whenever
$x\,\mathcal{R}\,y$ and $z\leqslant y$, then for all maximal $w\in
x^{\downarrow}\cap z^{\downarrow}$ we have $w\,\mathcal{R}\,z$.

######  Remark .

Using this definition, Theorem 4.13 in [FOO+22] (categorical transfer systems
on a lattice are in natural bijection with weak factorization systems) can be
generalized to arbitrary posets. This gives some evidence that our definition
is the “morally correct” one for non-lattice posets. More pragmatically, the
stronger definition is necessary to make Section 3 work.

######  Warning .

Until this point in the literature, only transfer systems on Abelian groups
have been seriously considered. In this case, $G$-transfer systems are in
bijection with categorical transfer systems on
$\operatorname{Sub}(G)\cong\operatorname{Sub}(G)/G$, and as such there is no
distinction to be made. We are primarily concerned with non-Abelian groups in
this paper, and as such, one needs to be careful what they mean.

We now introduce the notion of a lossless group. This definition was isolated
to capture exactly the groups needed for our applications, and we have been
unable to find this class of groups studied previously in the literature.

######  Definition .

A _lossless group_ is a group $G$ such that for all pairs of subgroups
$K\leqslant H$ such that ${}^{g}K\leqslant H$ for some $g\in G$, there exists
some $h\in N_{G}(H)$ such that ${}^{h}K={}^{g}K$. A group which is not
lossless will be called _lossy_.

######  Remark .

An equivalent succinct way of phrasing this definition is that for all
$H\leqslant G$, the fusion of subgroups of $H$ is controlled by $N_{G}(H)$.
Note that we do not require fusion to be controlled on elements as is common
in group theoretic literature. This corresponds in the previous definition to
the fact that we only require $h^{-1}g\in N_{G}(K)$, rather than requiring
$h^{-1}g\in C_{G}(K)$.

In the following definition, $[H]$ denotes the conjugacy class of a subgroup
$H\leqslant G$. We also adopt the notational convention of writing $K\to
H\in\mathcal{R}$ as shorthand for $K\,\mathcal{R}\,H$; we find this notational
flexibility useful, especially as the notation for a given transfer system
becomes more complex.

######  Definition .

Let $G$ be a group and
$\pi\colon\operatorname{Sub}(G)\to\operatorname{Sub}(G)/G$ be the quotient map
of posets.

  * •

For a categorical transfer system $\mathcal{R}$ on $\operatorname{Sub}(G)/G$,
we define $\pi^{-1}(\mathcal{R})$ to be the relation on
$\operatorname{Sub}(G)$ such that $K\to H\in\pi^{-1}(\mathcal{R})$ if and only
if $[K]\to[H]\in\mathcal{R}$. We then define $\pi^{*}(\mathcal{R})$ to be the
$G$-transfer system generated by $\pi^{-1}(\mathcal{R})$.

  * •

For a $G$-transfer system $\mathcal{R}$, we define $\pi_{*}(\mathcal{R})$ to
be the relation on $\operatorname{Sub}(G)/G$ where
$[K]\to[H]\in\pi_{*}(\mathcal{R})$ if and only if there exist some pair of
subgroups $K^{\prime}\leqslant H^{\prime}$ with $[K]=[K^{\prime}]$ and
$[H]=[H^{\prime}]$, such that $K^{\prime}\to H^{\prime}\in\mathcal{R}$.

We start with an observation that we can push any $G$-transfer system to a
categorical transfer system on $\operatorname{Sub}(G)/G$ provided that $G$ is
lossless.

######  Lemma .

Let $G$ be a lossless group. Then for all $G$-transfer systems $\mathcal{R}$,
$\pi_{*}(\mathcal{R})$ is a categorical transfer system on
$\operatorname{Sub}(G)/G$.

###### Proof.

Let $\mathcal{R}^{\prime}=\pi_{*}(\mathcal{R})$. Suppose
$[K]\,\mathcal{R}^{\prime}\,[H]\,\mathcal{R}^{\prime}\,[L]$ and let there be
lifts (i.e., representatives in the conjugacy class)
$K^{\prime}\,\mathcal{R}\,H^{\prime}$ and
$H^{\prime\prime}\,\mathcal{R}\,L^{\prime}$. Then
$[H^{\prime}]=[H]=[H^{\prime\prime}]$ implies we can find $g\in G$ for which
${}^{g}H^{\prime}=H^{\prime\prime}$. Then
${}^{g}K^{\prime}\,\mathcal{R}\,L^{\prime}$ is a lift of $[K]\to[L]$. Thus
$\mathcal{R}^{\prime}$ is transitive. Note that this part does not use the
assumption that $G$ is lossless.

Now suppose $[K]\,\mathcal{R}^{\prime}\,[H]$ and $[L]\leqslant[H]$, and
suppose $[M]$ is maximal among $[M]\leqslant[K]$ and $[M]\leqslant[L]$. We can
assume without loss of generality that $K\,\mathcal{R}\,H$ is a lift and
$L\leqslant H$ and $M\leqslant K$. Let $g\in G$ such that ${}^{g}M\leqslant
L$. Since $G$ is lossless we can assume $g\in N_{G}(H)$. Thus
${}^{g}K\,\mathcal{R}\,H$ and ${}^{g}M\leqslant{}^{g}K\cap L$. But
$[{}^{g}K\cap L]\leqslant[K],[L]$, so by maximality ${}^{g}M$ must be
conjugate to ${}^{g}K\cap L$, and hence for order reasons we have
${}^{g}M={}^{g}K\cap L$. But since $\,\mathcal{R}\,$ is a transfer system we
have ${}^{g}M\,\mathcal{R}\,L$ as ${}^{g}M=({}^{g}K\cap L)$, and hence
$[M]\,\mathcal{R}^{\prime}\,[L]$. ∎

######  Proposition .

If $G$ is lossless then $\pi^{*}\dashv\pi_{*}$ is a Galois connection between
$G$-transfer systems and categorical transfer systems on
$\operatorname{Sub}(G)/G$. Furthermore, the unit of this adjunction is the
identity, i.e., for all $G$-transfer systems $\mathcal{R}$ we have
$\mathcal{R}=\pi^{-1}(\pi_{*}(\mathcal{R}))=\pi^{*}(\pi_{*}(\mathcal{R}))$.

###### Proof.

Let $\mathcal{R}$ be a $G$-transfer system and $\mathcal{R}^{\prime}$ a
categorical transfer system on $\operatorname{Sub}(G)/G$. By definition,
$\mathcal{R}\geqslant\pi^{*}(\mathcal{R}^{\prime})$ if and only if
$\mathcal{R}\geqslant\pi^{-1}(\mathcal{R}^{\prime})$ if and only if
$[K]\,\mathcal{R}^{\prime}\,[H]$ implies $K\,\mathcal{R}\,H$. On the other
hand, we have $\pi_{*}(\mathcal{R})\geqslant\mathcal{R}^{\prime}$ if and only
if $[K]\,\mathcal{R}^{\prime}\,[H]$ implies there exists some
$K^{\prime}\leqslant H^{\prime}$ with $[K^{\prime}]=[K]$ and
$[H^{\prime}]=[H]$ such that $K^{\prime}\,\mathcal{R}\,H^{\prime}$.

Thus we need to show $K^{\prime}\,\mathcal{R}\,H^{\prime}$ implies
$K\,\mathcal{R}\,H$. Or in other words, we need to show that if $G$ is
lossless, then for any $G$-transfer system $\mathcal{R}$ and any two pairs
$K\leqslant H$, $K^{\prime}\leqslant H^{\prime}$ such that $[K]=[K^{\prime}]$
and $[H]=[H^{\prime}]$, we have $K\,\mathcal{R}\,H$ if and only if
$K^{\prime}\,\mathcal{R}\,H^{\prime}$. By symmetry we can suppose
$K\,\mathcal{R}\,H$ and we want to show this implies
$K^{\prime}\,\mathcal{R}\,H^{\prime}$. But $[H]=[H^{\prime}]$ implies by
definition we can find some $g\in G$ such that ${}^{g}H=H^{\prime}$, and by
conjugation closure we have ${}^{g}K\,\mathcal{R}\,{}^{g}H$, so we might as
well assume $H=H^{\prime}$. Let $h\in G$ such that ${}^{h}K=K^{\prime}$. Then
we have $K,{}^{h}K\leqslant H$, so since $G$ is lossless we can assume $h\in
N_{G}(H)$. But then again using conjugation closure we have
$(K^{\prime}={}^{h}K)\,\mathcal{R}\,({}^{h}H=H=H^{\prime})$ as claimed.

From the definition of $\pi^{-1}(-)$, this also shows
$\mathcal{R}=\pi^{-1}(\pi_{*}(\mathcal{R}))$ for any $G$-transfer system
$\mathcal{R}$. Since $\pi^{*}(\mathcal{R}^{\prime})$ is defined to be the
smallest $G$-transfer system containing $\pi^{-1}(\mathcal{R}^{\prime})$ and
$\mathcal{R}$ is a $G$-transfer system by definition, this also implies
$\mathcal{R}=\pi^{*}(\pi_{*}(\mathcal{R}))$ and hence the unit is the
identity. ∎

######  Corollary .

If $G$ is lossless then every $G$-transfer system can be lifted from a
categorical transfer system on $\operatorname{Sub}(G)/G$.

We note that a strong converse to this corollary also holds:

######  Proposition .

Let $G$ be an arbitrary (finite) group. If every $G$-transfer system can be
written as $\pi^{*}(\mathcal{R})$ for some arbitrary _relation_ $\mathcal{R}$
on $\operatorname{Sub}(G)/G$, then $G$ is lossless.

###### Proof.

Let $S$ be some set of arrows in $\operatorname{Sub}(G)$, and let
$\mathcal{R}$ be the $G$-transfer system generated by $S$. By the explicit
construction of the transfer system generated by a set of arrows given in
[Rub21b, Appendix B], one can show that $K\to H\in\mathcal{R}$ necessarily
implies that there must exist some $K^{\prime}\to H^{\prime}\in S$ and some
$g\in G$ such that $K\leqslant{}^{g}K^{\prime}$ and
$H\leqslant{}^{g}H^{\prime}$.

Now suppose $K,{}^{g}K\leqslant H$, and let $\mathcal{R}$ be the $G$-transfer
system generated by $K\to H$. Suppose
$\mathcal{R}=\pi^{*}(\mathcal{R}^{\prime})$. Since
$\pi^{*}(\mathcal{R}^{\prime})$ is generated by
$\pi^{-1}(\mathcal{R}^{\prime})$, there must be some $K^{\prime}\to
H^{\prime}\in\pi^{-1}(\mathcal{R}^{\prime})$ and some $h\in G$ such that
$K\leqslant{}^{h}K^{\prime}$ and $H\leqslant{}^{h}H^{\prime}$. But also
$\mathcal{R}$ is generated by $S=\\{K\to H\\}$, so there must exist some $k\in
G$ such that $K^{\prime}\leqslant{}^{k}K$ and
$H^{\prime}\leqslant{}^{k}H^{\prime}$. For order reasons this forces
$K={}^{h}K^{\prime}$ and $H={}^{h}H^{\prime}$, so
$[K]\to[H]=[K^{\prime}]\to[H^{\prime}]\in\mathcal{R}^{\prime}$.

But we also have $[{}^{g}K]\to[H]$ and $[K]\to[H]\in\mathcal{R}^{\prime}$ and
$[H]=[K]$, so by definition of $\pi^{-1}(\mathcal{R}^{\prime})$ we must have
${}^{g}K\to H\in\mathcal{R}$. Thus we can find $h\in G$ such that
${}^{g}K\leqslant{}^{h}K$ and $H\leqslant{}^{h}H$, and again for order reasons
this forces ${}^{g}K={}^{h}K$ and $H={}^{h}H$, i.e., $K$ and ${}^{g}K$ are
conjugate by way of $h\in N_{G}(H)$. Since $K,{}^{g}K\leqslant H$ was
arbitrary, this shows $G$ is lossless. ∎

######  Remark .

By Section 2.1 and the general theory of Galois connections, when $G$ is
lossless we have a bijection between $G$-transfer systems and categorical
transfer systems $\mathcal{R}$ on $\operatorname{Sub}(G)/G$ such that
$\mathcal{R}=\pi_{*}(\pi^{*}(\mathcal{R}))$. We call such transfer systems
_liftable_. For example, considering $G=\mathfrak{S}_{3}$, as mentioned in the
introduction a categorical transfer system is liftable if and only if whenever
we have $\tau\to\mathfrak{S}_{3}$ for any transposition $\tau$ we also have
$1\to\mathfrak{S}_{3}$ (c.f., Figure 3 and the surrounding discussion).

In general the condition $\mathcal{R}=\pi_{*}(\pi^{*}(\mathcal{R}))$ is not
very evocative. In Section 3 we give a more concrete set of conditions for a
categorical transfer system to be liftable, but for general (lossless) $G$
these conditions can still be rather opaque and difficult to verify. In
Section 3 we will consider some special cases where the lifting conditions are
simple enough to be visually intuitive, like in the case of
$G=\mathfrak{S}_{3}$.

The following corollary summarizes the results of this section.

######  Corollary .

If $G$ is a lossless group, then the function $\pi^{*}$ from liftable transfer
systems on $\operatorname{Sub}(G)/G$ to transfer systems for $G$ (i.e.,
homotopy classes of $G$-$N_{\infty}$ operads) is a bijection; furthermore, for
a general finite group $G$, bijectivity of $\pi^{*}$ (when restricted to
liftable transfer systems) implies that $G$ is lossless.

###  2.2. Examples and counterexamples of lossless groups

We continue with some observations regarding lossless groups, and provide some
non-trivial examples of interesting families of lossless groups. Note that
clearly any Abelian group is lossless.

In general it appears that the class of lossless groups is rather poorly
behaved under group-theoretic operations. However, we can prove that lossless
groups at least play nicely with quotients.

######  Lemma .

Any quotient of a lossless group is lossless.

###### Proof.

This follows directly from the standard equivariant poset isomorphism between
$\operatorname{Sub}(G/N)$ and the interval
$[N,G]\subseteq\operatorname{Sub}(G)^{N}$ for all $N\trianglelefteqslant G$. ∎

######  Remark .

The product of lossless groups can be lossy. Indeed, consider the group
$G=C_{2}\times A_{4}$, for which Figure 6 displays $\operatorname{Sub}(G)/G$.

$[e]$$[C_{2}]$$[C_{2}]$$[C_{2}]$$[C_{3}]$$[(C_{2})^{2}]$$[(C_{2})^{2}]$$[(C_{2})^{2}]$$[C_{6}]$$[A_{4}]$$[(C_{2})^{3}]$$[C_{2}\times
A_{4}]$ Figure 6. The structure of $\operatorname{Sub}(G)/G$ for
$G=C_{2}\times A_{4}$.

In $G$ we have a conjugacy class $[(C_{2})^{2}]$ which contains only two
copies of $[C_{2}]$ displayed in red in Figure 6. Since each $(C_{2})^{2}$
contains three copies of $C_{2}$, this implies the existence of some
$H=(C_{2})^{2}$ which contains two conjugate copies of $C_{2}$, $K$ and
${}^{g}K$. However, the normalizer of $(C_{2})^{2}$ is $(C_{2})^{3}$, which is
Abelian and hence $K$, ${}^{g}K$ cannot be conjugate in
$(C_{2})^{3}=N_{G}(H)$. That is, $C_{2}\times A_{4}$ fails to be lossless even
though $C_{2}$ and $A_{4}$ are lossless.

Heuristically it seems very likely that being lossless does not imply that all
(even normal) subgroups are lossless. Indeed, if $K,{}^{g}K\leqslant
H\leqslant L\leqslant G$ are such that $K,{}^{g}K$ are not conjugate in
$N_{L}(H)$ (so in particular $L$ is definitely not lossless), then it may
still be the case that $K,{}^{g}K$ are conjugate in $N_{G}(H)$ (so $G$ might
be lossless). However the authors have been unable to find an explicit example
of a lossless group with a lossy subgroup.

A substantial class of examples of lossless groups comes from the following
observation. Recall that a _T-group_ is a group in which every subnormal
subgroup is normal, i.e., $K\trianglelefteqslant H$ and $H\trianglelefteqslant
G$ implies $K\trianglelefteqslant G$ [Rob96, §13.4]. A subgroup $K\leqslant G$
is _pronormal_ if for all $g\in G$, $K$ and ${}^{g}K$ are conjugate in
$\langle K,{}^{g}K\rangle$ [DH92, §I.6]. We note that if $K,{}^{g}K\leqslant
H$ then $\langle K,{}^{g}K\rangle\leqslant H\leqslant N_{G}(H)$, so if every
subgroup of $G$ is pronormal then $G$ is lossless.

Before we continue, let us recall the theory of Hall subgroups, and Hall’s
theorem, which can be seen as a generalization of Sylow’s theorem in the
solvable case.

######  Definition .

A _Hall subgroup_ of a group $G$ is a subgroup whose order is coprime to its
index. If $\pi$ is a set of primes, then a _Hall $\pi$-subgroup_ is a subgroup
whose order is a product of primes in $\pi$.

######  Theorem  (Hall’s Theorem [Hal28]).

Let $G$ be a finite solvable group and $\pi$ any set of primes. Then $G$ has a
Hall $\pi$-subgroup, and any two such Hall $\pi$-subgroups are conjugate.
Moreover, any subgroup whose order is a product of primes in $\pi$ is
contained in some Hall $\pi$-subgroup.

######  Proposition .

Any (finite) solvable T-group is lossless.

###### Proof.

A theorem of Peng tells us that a group $G$ is a solvable T-group if and only
if every $p$-subgroup of $G$ is pronormal, and in this case $G$ is in fact
supersolvable [Pen69]. We claim that in this case all subgroups of $G$ are
pronormal, and hence $G$ is lossless. This result seems to be well-known among
group theorists (see, e.g., [FT22, dGV00, KS06]), but the authors of the
present paper were unable to find a proof in the literature so we include one
here.

Suppose that $G$ is a (super)solvable T-group, and let $K\leqslant G$. We will
prove by induction on the number of prime divisors of $|K|$ that $K$ is
pronormal. By Peng’s theorem this holds in the case where $|K|$ only has a
single prime divisor. Since $G$ is supersolvable, $K\leqslant G$ is also
supersolvable, so $K$ has a Sylow $p$-subgroup $P\leqslant K$ and a normal
Hall $p^{\prime}$-subgroup $S\trianglelefteqslant K$ for some $p$. In
particular, $P\leqslant N_{G}(S)$. By induction we can assume $S$ and $P$ are
both pronormal in $G$. By [Ros67, 1.8], this implies $K=SP$ is pronormal in
$G$. ∎

Although the solvable T-groups form a fairly large class of groups, this class
notably excludes most interesting $p$-groups. Indeed, since every subgroup of
a nilpotent group is subnormal, a $p$-group is a T-group if and only if every
subgroup is normal, so the only non-Abelian $p$-groups obtained this way are
groups of the form $Q_{8}\times(\mathbb{Z}/2)^{n}$, where $Q_{8}$ is the
ordinary quaternion group [Ded97]. In light of this, the remainder of this
section will largely focus on $p$-groups and determining conditions under
which a $p$-group is lossless.

The following result is trivial, but will show that Section 2.2 provides a
minimal example of lossy $p$-groups.

######  Proposition .

Let $G$ be any $p$-group of order at most $p^{3}$. Then $G$ is lossless.

###### Proof.

Suppose $K,{}^{g}K\leqslant H$. We want to show $K,{}^{g}K$ are conjugate in
$N_{G}(H)$. We can assume $K<H$ since otherwise $K={}^{g}K$ and there’s
nothing to show. Since $G$ is nilpotent we have $H<N_{G}(H)$, and we’re
already done if $N_{G}(H)=G$, so we must have $[G:H]\geqslant p^{2}$. But if
$|G|\leqslant p^{3}$ then this forces $K=\\{e\\}={}^{g}K$ so there’s nothing
to show. ∎

######  Example .

Let $p\neq 2$ and let $N=(\mathbb{Z}/p)^{3}$. Let a generator of $T=C_{p}$ act
on $N$ by the matrix

$A:=\begin{bmatrix}1&1&1\\\ 0&1&1\\\ 0&0&1\\\ \end{bmatrix}.$

Let $G=N\rtimes T$. Note that $|G|=p^{4}$. Let
$K=\langle(0,0,1)\rangle\subseteq N$ and
$L=\langle(0,0,1),(1,1,1)\rangle\subseteq N$. Let $g\in T$ be a generator.
Then $K,{}^{g}K\leqslant L$, but $N_{G}(L)=N$ is Abelian and hence
$K\neq{}^{g}K$ cannot be conjugate in $N_{G}(L)$. Thus $G$ is not lossless.

######  Remark .

The reason we needed to assume $p\neq 2$ in Section 2.2 is because if $p=2$
then the matrix $A$ has order $4$ instead of $2$. By an exhaustive search one
can show that every group of order $2^{4}$ is lossless.

We recall for the proof of Section 2.2 that a subgroup $H$ of $G$ is said to
be _characteristic_ if every automorphism of $G$ fixes $H$, that is,
$\phi(H)=H$ for every automorphism $\phi$ of $G$.

######  Proposition .

If $G$ has a cyclic normal subgroup of prime index, i.e., if $G$ is an
extension

$1\to C_{n}\to G\to C_{p}\to 1$

for some prime $p$, then $G$ is lossless.

###### Proof.

Let $K,{}^{g}K\leqslant H\leqslant G$, and let $N=C_{n}\trianglelefteqslant
G$. Since every subgroup of $C_{n}$ is characteristic, all subgroups of $N$
are normal in $G$. Thus if $K\leqslant H\cap N$ then $K$ is normal so
$K={}^{g}K$ and there’s nothing to show. Otherwise the quotient $H\to H/{H\cap
N}\cong C_{p}$ is non-trivial restricted to $K\leqslant H$, and hence
$H=K(H\cap N)$. Since $H\cap N\leqslant N$ is normal in $G$ this implies

${}^{g}H={}^{g}(K(H\cap N))={}^{g}K{}^{g}(H\cap N)={}^{g}K(H\cap N)\leqslant
H$

and hence $g\in N_{G}(H)$. ∎

We will now wish to discuss some families of groups which are amenable to the
above result. We will define some of the groups in question as they may not be
standard knowledge. From their description via generators and relations it is
clear that they all have cyclic normal subgroups of order 2.

######  Definition .

  * •

The _dicyclic group_ of order $4n$, denoted $\mathsf{Dic}_{n}$ is defined via
generators and relations as

$\mathsf{Dic}_{n}:=\langle r,s\mid r^{2n}=s^{4}=1,srs=r^{2n-1}\rangle.$

  * •

The _semidihedral group_ of order $2^{n}$, denoted $\mathsf{SD}_{n}$ is
defined via generators and relations as

$\mathsf{SD}_{n}:=\langle r,s\mid
r^{2^{n-1}}=s^{2}=1,srs=r^{2^{n-2}-1}\rangle.$

  * •

The _modular maximal-cyclic group_ of order $2^{n}$, denoted $\mathsf{MM}_{n}$
is defined via generators and relations as

$\mathsf{MM}_{n}:=\langle r,s\mid
r^{2^{n-1}}=s^{2}=1,srs=r^{2^{n-2}+1}\rangle.$

######  Corollary .

Any dihedral group, dicyclic (e.g., generalized quaternion) group,
semidihedral group, or modular maximal-cyclic group is lossless.

######  Corollary .

If $q$ is a prime power with $q\equiv 3\mod 4$, then the Sylow $2$-subgroup of
$\mathsf{PSL}_{3}(\mathbb{F}_{q})$ is lossless. Similarly, if $q\equiv 1\mod
4$ then the Sylow $2$-subgroup of $\mathsf{PSU}_{3}(\mathbb{F}_{q})$ is
lossless.

###### Proof.

In each of these cases the Sylow $2$-subgroup is semidihedral [ABG70]. ∎

######  Example .

Let $q\mid\phi(p^{3})/p=p(p-1)$, and let a generator $g\in T=C_{pq}$ act on
$N=\mathbb{Z}/p^{3}$ via multiplication by some element in $\mathbb{Z}/p^{3}$
of order $qp$, and let $G=N\rtimes T$. Let $N_{1}=pN$ and $N_{2}=p^{2}N$. Let
$K=T^{q}$, $H=KN_{2}$, and $L=TN_{2}$. Note that $N_{1},N_{2}$ are
characteristic in $N$ and hence normal in $G$, so $H$ and $L$ are indeed
subgroups of $G$.

A computation shows that $H$ is normal in $G$, but if $p\neq 2$ then
$N_{G}(K)=N_{G}(L)=TN_{1}<G$. Thus $G$ is not lossless, showing the assumption
that $[G:N]$ is prime in Section 2.2 is essential.

######  Proposition .

If the derived subgroup of $G$ has prime order, then $G$ is lossless.

###### Proof.

Let $K,{}^{g}K\leqslant H$. Since $G/G^{\prime}$ is Abelian, any subgroup
containing $G^{\prime}$ is normal in $G$. Thus in particular $KG^{\prime}$ is
normal, and hence ${}^{g}K\leqslant KG^{\prime}$. Since $|G^{\prime}|=p$ we
have $[KG^{\prime}:K]\leqslant p$, so $K$ is maximal in $KG^{\prime}$. Thus if
${}^{g}K\neq K$ then $\langle K,{}^{g}K\rangle=KG^{\prime}$ and hence
$G^{\prime}\leqslant KG^{\prime}\leqslant H$. Thus $H$ is normal, and as such,
there is nothing to check. ∎

######  Definition .

A $p$-group $G$ is said to be _extraspecial_ if its center $Z(G)$ is cyclic of
order $p$ and the quotient $G/Z(G)$ is a non-trivial elementary abelian
$p$-group.

######  Corollary .

Any extraspecial group is lossless.

######  Corollary .

For any prime $p$ the subgroup of upper triangular matrices in
$\mathsf{GL}_{2}(\mathbb{F}_{p})$ is lossless.

######  Proposition .

If $\gcd(m,p)=1$ and $G\cong(C_{p})^{2}\rtimes C_{m}$, then $G$ is lossless.

###### Proof.

Let $K,{}^{g}K\leqslant H$. Again any subgroup containing $G^{\prime}$ is
normal in $G$. If $K\cap G^{\prime}\neq 1$, then by the diamond identity
$[KG^{\prime}:K]=[G^{\prime}:G^{\prime}\cap KG^{\prime}]\leqslant p$ and the
same argument as before applies. Thus we can assume $K\cap G^{\prime}=1$, and
hence $K,{}^{g}K$ are contained in some $p^{\prime}$-Hall subgroups
$S,S^{\prime}$ of $H$. Thus we can find some $h\in H$ such that
${}^{h}S=S^{\prime}$ and hence ${}^{h}K\leqslant S^{\prime}$. But $S^{\prime}$
is a cyclic group so $|{}^{h}K|=|{}^{g}K|$ implies ${}^{h}K={}^{g}K$, and of
course $h\in N_{G}(H)$ as required. ∎

######  Corollary .

For any prime $p$ the subgroup of upper triangular matrices in
$\mathsf{SL}_{2}(\mathbb{F}_{p^{2}})$ is lossless.

######  Example .

If $p=2,3,$ or $5$, then $\mathsf{SL}_{2}(\mathbb{F}_{p})$ is lossless.

In fact, for $p=2,3,5$ the group $\mathsf{SL}_{2}(\mathbb{F}_{p})$ satisfies a
very strong additional property: any two isomorphic subgroups are conjugate.
We shall say that such a group is _universally lossless_. Groups like this are
quite useful for identifying “lossless pieces” of larger groups containing
them as subgroups.

######  Proposition .

Let $G$ be an arbitrary (finite) group, and suppose $K,{}^{g}K\leqslant
H\leqslant G$. Suppose further that $H\leqslant L\leqslant G$, where $L$ is a
universally lossless group. Then $K$ and ${}^{g}K$ are conjugate in
$N_{G}(H)$.

###### Proof.

Since $K\cong{}^{g}K$ and $K,{}^{g}K\leqslant L$, by assumption $K$ and
${}^{g}K$ are conjugate in $L$, so we can assume $g\in L$. Thus since $L$ is
lossless and $K,{}^{g}K\leqslant H\leqslant L$, we must have $K$ and ${}^{g}K$
are conjugate in $N_{L}(H)\leqslant N_{G}(H)$. ∎

##  3\. Lifting criteria

In Section 2.1 we saw that for a lossless group $G$, $G$-transfer systems are
in bijection with liftable transfer systems on $\operatorname{Sub}(G)/G$.
However as discussed in Section 2.1, explicitly identifying which transfer
systems on $\operatorname{Sub}(G)/G$ are liftable can be extremely difficult
in general. In this section we consider some special cases where the lifting
conditions are tangible. We first begin with a generality.

######  Lemma .

Let $G$ be a lossless group, and let $\mathcal{R}$ be a categorical transfer
system on $\operatorname{Sub}(G)/G$. Then $\mathcal{R}$ is liftable if and
only if for all $[K]\,\mathcal{R}\,[H]$ and any $K^{\prime}\leqslant H$ with
$[K^{\prime}]=[K]$ we have $[K\cap K^{\prime}]\,\mathcal{R}\,[H]$.

###### Proof.

Suppose $\mathcal{R}$ is liftable and $K,K^{\prime}\leqslant H$ are as in the
statement of the lemma. Then $K\to H,K^{\prime}\to H\in\pi^{*}(\mathcal{R})$
by definition, so by restriction-closure and transitivity we have $(K\cap
K^{\prime})\to K^{\prime}\to H\in\pi^{*}(\mathcal{R})$ and hence $[K\cap
K^{\prime}]\to[H]\in\pi_{*}(\pi^{*}(\mathcal{R}))=\mathcal{R}$.

Now suppose conversely that $\mathcal{R}$ is a categorical transfer system on
$\operatorname{Sub}(G)/G$ such that the condition in the statement of the
lemma holds. We claim that $\pi^{-1}(\mathcal{R})$ is a $G$-transfer system.
Note that if this claim is true then

$\pi_{*}(\pi^{*}(\mathcal{R}))=\pi_{*}(\pi^{-1}(\mathcal{R}))\leqslant\mathcal{R}\leqslant\pi_{*}(\pi^{*}(\mathcal{R}))$

where the first inequality follows directly from the definition of $\pi_{*}$
and $\pi^{-1}$ and the last inequality follows from $\pi^{*}\dashv\pi_{*}$.

Clearly $\pi^{-1}(\mathcal{R})$ is a conjugation-closed partial order refining
$\leqslant$. All we need to show is that it is restriction-closed. In other
words, we need to show that given $K\to H\in\pi^{-1}(\mathcal{R})$ and
$L\leqslant H$ we have $(K{\cap}L)\to L\in\pi^{-1}(\mathcal{R})$. We prove
this by induction on the tuple $(K,H,L)$. That is, assuming the claim is true
for all tuples $(K^{\prime},H^{\prime},L^{\prime})$ with $K^{\prime}\leqslant
K,L^{\prime}\leqslant L,H^{\prime}\leqslant H$, and at least one of these
inequalities is strict, we want to show this implies the claim for $(K,H,L)$.
The base case where $K=H=L$ is the trivial subgroup holds by reflexivity of
$\mathcal{R}$.

By the induction hypothesis with $K$ and $H$ fixed but $L^{\prime}<L$, we can
assume $(K\cap L)\to L^{\prime}\in\pi^{-1}(\mathcal{R})$ for all
$L^{\prime}<L$ with $K\cap L\leqslant L^{\prime}$. Let $M\leqslant L$ such
that $[K\cap L]\leqslant[M]$ and $[M]$ is maximal among $[M]\leqslant[K],[L]$.
Let $g\in G$ such that ${}^{g}(K\cap L)\leqslant M$. Since $G$ is lossless we
can assume $g\in N_{G}(L)$. Thus ${}^{g^{-1}}M\leqslant L$ and
$[M]=[{}^{g^{-1}}M]$, so we can assume without loss of generality that $K\cap
L\leqslant M$.

If $M<L$, then by the induction hypothesis we have $K\cap L\to
M\in\pi^{-1}(\mathcal{R})$ and hence by definition $[K\cap
L]\,\mathcal{R}\,[M]$. But since $\mathcal{R}$ is a categorical transfer
system, $[M]$ is maximal for $[M]\leqslant[K],[L]\leqslant[H]$, and
$[K]\,\mathcal{R}\,[H]$, we have $[K\cap
L]\,\mathcal{R}\,[M]\,\mathcal{R}\,[L]$, and hence $K\cap L\to
L\in\pi^{-1}(\mathcal{R})$.

On the other hand, if $M=L$ then $[L]\leqslant[K]$, so ${}^{g^{-1}}L\leqslant
K$ for some $g\in G$, and again since $G$ is lossless we can assume $g\in
N_{G}(H)$. Thus $K,{}^{g}K\leqslant H$ and $[K]\,\mathcal{R}\,[H]$, so by
hypothesis we have $[K\cap{}^{g}K]\,\mathcal{R}\,[H]$. By restriction-closure
this then implies $[K\cap{}^{g}K]\,\mathcal{R}\,[{}^{g}K]$, so
$K\cap{}^{g}K\to{}^{g}K\in\pi^{-1}(\mathcal{R})$. If ${}^{g}K<H$, then by the
induction hypothesis with $K^{\prime}=K\cap{}^{g}K$, $H^{\prime}={}^{g}K$, and
$L^{\prime}=L$ we obtain $K\cap{}^{g}K\cap L=K\cap L\to
L\in\pi^{-1}(\mathcal{R})$. But if ${}^{g}K=H$ then also $K=H$ and hence
$K\cap L=L$ so there’s nothing to show in this case. ∎

Although Section 3 is nice in its generality, and is certainly more explicit
than the basic definition, checking it still requires understanding subtle
details about the way that subgroups of $G$ embed into each other. Thankfully,
for certain groups the subgroup structure is nice enough to make this
condition particularly explicit. Some of the results described below hold in
greater generality than stated, but the purpose of this section is mainly
illustrative so we avoid excess generality. We recall that, in essence, a
_Frobenius group_ is a transitive permutation group on a finite set such that
no non-trivial element fixes more than one point, and some non-trivial element
fixes a point. They can be characterized as those groups $G$ possessing a
proper, nontrivial subgroup $T$ (called the _Frobenius complement_) such that
$T\cap{}^{g}T$ is the trivial subgroup for every $g\in G\smallsetminus T$. The
identity element along with members of $G\smallsetminus\bigcup_{g\in
G}{}^{g}T$ form the _Frobenius kernel_ of $G$.

######  Definition .

A _metacyclic Frobenius group_ (mcF group) is a Frobenius group $G$ such that
both the kernel $N\trianglelefteqslant G$ and the complement $T\leqslant G$
are cyclic groups.

######  Remark .

Any mcF group is of the form

$G\cong\mathbb{Z}/n\rtimes T$

where $T$ is a cyclic subgroup of $(\mathbb{Z}/n)^{\times}$ such that
$x-1\in(\mathbb{Z}/n)^{\times}$ for all $x\neq 1\in T$. Conversely every group
of this form is an mcF group. For a general mcF group $G$, we will assume that
we have passed through this bijection, and for clarity, write shall write $N$
for the group $(\mathbb{Z}/n)$. That is, $G\cong N\rtimes T$.

######  Remark .

Although not immediately obvious from the definition, one can show that a
group $G$ is metacyclic Frobenius if and only if $G$ is both a metacyclic
group and a Frobenius group, explaining the naming choice. This collection of
groups also implicitly appears in work of Khukhro–Makareno [KM13a, KM13b], but
the authors are not aware of any other place that they have been studied.

######  Example .

Let $n$ be odd. Then the dihedral group
$D_{n}=\mathbb{Z}/n\rtimes\mathbb{Z}/2$ of order $2n$ is a mcF group.

######  Example .

Let $p$ be any prime. Then
$\mathsf{AGL}_{1}(\mathbb{F}_{p})=\mathbb{F}_{p}\rtimes\mathbb{F}_{p}^{\times}$,
the group of affine linear transformations of the finite field
$\mathbb{F}_{p}$, is a mcF group. Here we are using Gauss’s observation that
$(\mathbb{Z}/n)^{\times}$ is cyclic when $n=p^{k}$.

######  Definition .

Let $K\leqslant G$. We call $K\cap N$ the _base_ of $K$ and write $K\cap
N=N_{K}$.

######  Lemma .

Any subgroup $K$ of an mcF group $G$ such that $N_{K}\neq\\{e\\}$ and
$K\not\leqslant N$ is itself an mcF group with kernel $N_{K}$. If
$N_{K}=\\{e\\}$ or $K\leqslant N$ then $K$ is a cyclic group.

###### Proof.

This follows immediately from [Fei57, Lemma 2.2]. ∎

######  Lemma .

A subgroup $K$ of an mcF group $G$ is normal if and only if $K\leqslant N$ or
$N_{K}=N$.

###### Proof.

If $N_{K}=N$ then $K=KN$ is the preimage of $KN/N\leqslant G/N$, and since
$G/N\cong T$ is cyclic this implies $KN/N\trianglelefteqslant G/N$ and hence
$K\trianglelefteqslant G$. If $K\leqslant N$ then $K$ is characteristic in
$N\trianglelefteqslant G$ and hence again $K\trianglelefteqslant G$.

Conversely suppose $N_{K}\neq N$ and $K\not\leqslant N$. Let $g\in
K\smallsetminus N$ and $h\in N\smallsetminus K$. Since $N$ is cyclic and
$\operatorname{ord}(h)=\operatorname{ord}(ghg^{-1})$, we can write
$ghg^{-1}=h^{m}$, and since $G$ is Frobenius and $gN\neq eN\in G/N$, $m-1$
must be a unit mod $\operatorname{ord}(h)$. But if $K$ is normal then
$h^{m-1}=h^{-1}h^{m}=(h^{-1}gh)g^{-1}\in K$ and hence $h\in K$, a
contradiction. ∎

######  Lemma .

Any mcF group $G$ is in particular a solvable T-group, and hence lossless by
Section 2.2. Furthermore, any two subgroups with the same order are conjugate
in $G$.

###### Proof.

Solvability follows from the fact that $G=N\rtimes T$ and $N,T$ are cyclic by
definition. The fact that $G$ is a T-group follows directly from Section 3 and
Section 3.

Now suppose $K,K^{\prime}\leqslant G$ and $|K|=|K^{\prime}|$. Since $N$ is a
normal Hall subgroup of $G$, we also have $|N_{K}|=|N_{K^{\prime}}|$ and hence
$N_{K}=N_{K^{\prime}}$ since $N$ is cyclic. Let $S,S^{\prime}$ be
complementary Hall subgroups of $K$ and $K^{\prime}$, respectively. Then we
can extend $S,S^{\prime}$ to Hall subgroups $T,T^{\prime}$ of $G$, and by
Hall’s theorem we can find $g\in G$ such that ${}^{g}T=T^{\prime}$. Then
${}^{g}S,S^{\prime}\leqslant T^{\prime}$ are subgroups of the cyclic group
$T^{\prime}$ and $|{}^{g}S|=|S^{\prime}|$, so ${}^{g}S=S^{\prime}$. Thus
${}^{g}K={}^{g}(N_{K}S)=N_{K}S^{\prime}=K^{\prime}$. ∎

The main property that makes the lifting conditions for mcF groups simple
comes from the following proposition.

######  Proposition .

Let $G$ be an mcF group, and let $K\leqslant G$. For all $g\in G\smallsetminus
N_{G}(K)$, we have $K\cap{}^{g}K=N_{K}$.

###### Proof.

Since $N_{K}$ is a normal subgroup of $K$, we have $N_{K}\leqslant
K\cap{}^{g}K$ for all $g\in G$. To complete the proof, we show that the
existence of $x\in(K\cap{}^{g^{-1}}K)\smallsetminus N_{K}$ implies that $g\in
N_{G}(K)$. Fix such an $x$. Then $\operatorname{ord}(x)$ cannot divide
$|N_{K}|$, so after replacing $x$ with some power we can assume
$\operatorname{ord}(x)$ is coprime to $|N_{K}|$. Let $S\leqslant K$ be a Hall
subgroup complementary to $N_{K}$ such that $x\in S$, and let $T\leqslant K$
be a Hall subgroup complementary to $N_{K}$ such that $gxg^{-1}\in T$. Let
$S^{\prime}\geqslant S$ and $T^{\prime}\geqslant T$ be Hall subgroups of $G$
complementary to $N$. Then by Hall’s theorem we can find $h\in G$ such that
${}^{hg}S^{\prime}=T^{\prime}$, and since $e\neq
gxg^{-1}\in{}^{g}S^{\prime}\cap T^{\prime}$ this implies
${}^{hg}S^{\prime}\cap{}^{g}S^{\prime}\neq\\{e\\}$. By [Fei57], this implies
we must have $T^{\prime}={}^{hg}S^{\prime}={}^{g}S^{\prime}$. Since
$T^{\prime}$ is cyclic it has a unique subgroup of order $|T|=|{}^{g}S|$, so
this then also implies $T={}^{g}S$. But then $g\in N_{G}(K)$ as

${}^{g}K={}^{g}(N_{K}S)=N_{K}T=K.$

∎

######  Corollary .

Let $G$ be an mcF group, and $\mathcal{R}$ a categorical transfer system on
$\operatorname{Sub}(G)/G$. Then $\mathcal{R}$ is liftable if and only if
whenever $[K]\,\mathcal{R}\,[H]$ with $N_{K}\neq N_{H}$, we have
$[N_{K}]\,\mathcal{R}\,[H]$ (or equivalently $[N_{K}]\,\mathcal{R}\,[K]$).

###### Proof.

By Section 3, for the if direction we need to check for all
$K,{}^{g}K\leqslant H$ that either $K={}^{g}K$ or $K\cap{}^{g}K=N_{K}$, but
this follows directly from Section 3. For the only if direction we need to
check that $N_{K}\neq N_{H}$ implies there exists some $g\in G$ with
${}^{g}K\leqslant H$ such that $N_{K}=K\cap{}^{g}K$. But $N_{K}\neq N_{H}$
implies by Section 3 that $K$ is not normal in $H$, and hence by Section 3 we
can find some $g\in H$ such that $K\cap{}^{g}K=N_{K}$, and of course
${}^{g}K\leqslant{}^{g}H=H$. ∎

######  Remark .

We can make this result more visually intuitive as follows. By Section 3, the
map $[K]\mapsto(|N_{K}|,[K:N_{K}])$ is a poset isomorphism
$\operatorname{Sub}(G)/G\cong D_{N}\times D_{T}$, where $D_{N}$ is the lattice
of divisors of $|N|$ and $D_{T}$ is the lattice of divisors of $|G/N|=|T|$.
Then Section 3 says that a categorical transfer system $\mathcal{R}$ on
$D_{N}\times D_{T}$ is liftable if and only if whenever
$(i,j)\,\mathcal{R}\,(i^{\prime},j^{\prime})$ with $j^{\prime}>j$, we must
have $(i,1)\,\mathcal{R}\,(i,j)$.

##  4\. Examples of liftable transfer systems

We will now apply the theory presented in this paper to two classes of mcF
groups.

###  4.1. Dihedral groups of prime power order

We begin by considering groups of the form
$D_{p^{k}}\cong\mathbb{Z}/p^{k}\rtimes\mathbb{Z}/2$ where $p$ is an odd prime;
these are mcF groups by Section 3. From Section 3, it follows that
$\operatorname{Sub}(G)/G\cong[k]\times[1]$ and we shall consider elements of
this lattice as pairs $(i,j)$ where $i\in[k]$ and $j\in[1]$. Here $[n]$ is the
totally ordered finite set $\\{0<1<\cdots n\\}$. It will be useful for us to
display this lattice as the horizontal ladder as in Figure 7.

$\cdots$$\cdots$$k+1$ Figure 7. The lattice $[k]\times[1]$.

Figure 8 shows the corresponding subgroups (where we have used square brackets
to denote conjugacy classes where required).

$e$$C_{p}$$\cdots$$C_{p^{k-1}}$$C_{p^{k}}$$[D_{1}]$$[D_{p}]$$\cdots$$[D_{p^{k-1}}]$$D_{p^{k}}$
Figure 8. The subgroups in the lattice
$\operatorname{Sub}(D_{p^{k}})/D_{p^{k}}$.

We can now unravel Section 3 in this specific example. We need to consider
situations where we have $[K]\,\mathcal{R}\,[H]$ with $N_{K}\neq N_{H}$. This
occurs when we move horizontally on the top row of Figure 7. In this case we
require $[N_{K}]\,\mathcal{R}\,[K]$ for it to lift to a transfer system for
$D_{p^{k}}$ itself. All in all we conclude that the conditions of Section 3
correspond to the following in terms of the group:

  * •

If $[D_{p^{i}}]\,\mathcal{R}\,[D_{p^{j}}]$ then
$C_{p^{i}}\,\mathcal{R}\,[D_{p^{i}}]$ for all $0\leqslant i<j\leqslant k$.

The following corollary rewords this condition in terms of the categorical
transfer systems on $[k]\times[1]$ using Section 3.

######  Corollary .

A $D_{p^{k}}$-transfer system is equivalent to the data of a categorical
transfer system on $[k]\times[1]$ which satisfies the following rule:

  * $\star$

If $(i,1)\,\mathcal{R}\,(i^{\prime},1)$ for $i<i^{\prime}$ then
$(i,0)\,\mathcal{R}\,(i,1)$ for all $0\leqslant i<i^{\prime}\leqslant k$.

######  Example .

Let us consider the case when $k=1$, so that the lattice in question is
$[1]\times[1]$. Then condition $(\star)$ of Section 4.1 boils down to the
single implication of Figure 9:

$\Longrightarrow$$(\star)$. Figure 9. The implications for $[1]\times[1]$.

Note that this is exactly the observation of Rubin in [Rub21b] that we
recalled in the introduction. Of the 10 transfer systems on $[1]\times[1]$,
only one of them does not satisfy condition $(\star)$, namely:

.

######  Example .

We now move to the more exotic case of $k=2$ with conjugacy lattice
$[2]\times[1]$. One can compute that there are 68 categorical transfer systems
for this lattice. This time we have three possible options for the pair
$i<i^{\prime}$ in condition $(\star)$ which are given in Figure 10.

$\Longrightarrow$$(\star_{1})$

$\Longrightarrow$$(\star_{2})$

$\Longrightarrow$$(\star_{3})$

Figure 10. The three restrictions needed for a transfer system on $D_{p^{2}}$.

Of the 68 transfer systems on $[2]\times[1]$, one can computationally verify
that 56 of them are $D_{p^{2}}$-transfer systems.

In [BMO22], the authors use these results to produce explicit recursion
formulæ for $D_{p^{k}}$-transfer systems.

###  4.2. Affine linear transformations of finite fields

In this section we will consider groups of the form
$\mathsf{AGL}_{1}(\mathbb{F}_{p})=\mathbb{F}_{p}\rtimes\mathbb{F}_{p}^{\times}$
where $p$ is prime. From Section 3, it follows that
$\operatorname{Sub}(G)/G\cong[1]\times\operatorname{Sub}(\mathbb{Z}/(p-1))$.
Of course, the prime factorization of $p-1$ follows no apparent rhyme or
reason. For the convenience of the reader we list the first few values in
Table 1.

$p$ | $\operatorname{Sub}(\mathbb{F}_{p}^{\times})$
---|---
2 | $[0]$
3 | $[1]$
5 | $[2]$
7 | $[1]\times[1]$
11 | $[1]\times[1]$
13 | $[2]\times[1]$
17 | $[4]$
Table 1. The subgroup lattice of $\mathbb{F}_{p}^{\times}$.

######  Example .

Consider $G=\mathsf{AGL}_{1}(\mathbb{F}_{3})$. Here, the subgroup lattice is
$[1]\times[1]$. Then we are in the exact same case as Section 4.1, which is
reassuring as $\mathsf{AGL}_{1}(\mathbb{F}_{3})\cong D_{3}$. In particular
there are 9 transfer systems for $G=\mathsf{AGL}_{1}(\mathbb{F}_{3})$.

######  Example .

The first non-trivial example is $G=\mathsf{AGL}_{1}(\mathbb{F}_{5})$, whose
subgroup lattice is $[1]\times[2]$. We warn the reader that this case is not
the same as Section 4.1. Indeed, even though $[1]\times[2]\cong[2]\times[1]$,
the condition $(\star)$ is not invariant under this. The lattice
$\operatorname{Sub}(G)/G$ is depicted in Figure 11.

$e$$C_{5}$$[C_{2}]$$D_{5}$$[C_{4}]$$G$ Figure 11. The subgroups in the lattice
$\operatorname{Sub}(\mathsf{AGL}_{1}(\mathbb{F}_{5}))/\mathsf{AGL}_{1}(\mathbb{F}_{5})$.

Applying Section 3 in conjunction with Section 3 we obtain our lifting
conditions that we collect in Figure 12.

$\Longrightarrow$$({\dagger}_{1})$.

$\Longrightarrow$$({\dagger}_{2})$.

Figure 12. The implications for $[1]\times[2]$.

Out of the 68 transfer systems on $[1]\times[2]$, 59 of them satisfy condition
$({\dagger}_{1})$ and $({\dagger}_{2})$.

######  Example .

Our final example is $G=\mathsf{AGL}_{1}(\mathbb{F}_{7})$. Here
$\operatorname{Sub}(G)/G\cong[1]\times([1]\times[1])$, displayed in Figure 13.

$e$$C_{7}$$[C_{2}]$$D_{7}$$[C_{3}]$$C_{7}\rtimes C_{3}$$[C_{6}]$$G$ Figure 13.
The subgroups in the lattice
$\operatorname{Sub}(\mathsf{AGL}_{1}(\mathbb{F}_{7}))/\mathsf{AGL}_{1}(\mathbb{F}_{7})$.

As with the previous examples, we apply Section 3 in conjunction with Section
3 to obtain our lifting conditions as in Figure 14.

$\Longrightarrow$$({\ddagger}_{1})$

$\Longrightarrow$$({\ddagger}_{2})$

$\Longrightarrow$$({\ddagger}_{3})$

Figure 14. The three restrictions needed for a transfer system on
$\mathsf{AGL}_{1}(\mathbb{F}_{7})$.

Note again that all other possible relations are implied by these ones due to
the usual axioms for a transfer system.

Out of the 450 transfer systems on $[1]\times[1]\times[1]$ (c.f., [BBPR21]),
one computes that 400 of these are transfer systems for
$\mathsf{AGL}_{1}(\mathbb{F}_{7})$.

##  5\. Strategies for lossy groups

In the previous sections we have explored lossless groups, and shown that they
provide a convenient computational framework for computing transfer systems,
especially when restricted to nicer subclasses such as metacyclic Frobenius
groups. Although we’ve shown in Section 2.2 that several important classes of
groups are lossless, more complicated groups that may arise in practice tend
to be lossy. Although lossy groups prevent us from working directly with
$\operatorname{Sub}(G)/G$, with some cleverness it may still be the case that
we can obtain simpler representations of transfer systems for lossy groups. In
this final section we speculatively discuss one possible strategy for dealing
with lossy groups in the case of $G=\mathsf{SL}_{2}(\mathbb{F}_{p})$ where $p$
is prime.

If $p=2,3,$ or $5$, then we have seen in Section 2.2 that $G$ is lossless, but
for $p>5$ these groups are always lossy. On the other hand, if $p\equiv\pm
3\mod 8$ then these groups are very close to being lossless. When $p\equiv\pm
1\mod 8$, the lossy binary octahedral subgroups add to the lossyness, but even
in this case $G$ only has a few deviations from losslessness. In the
speculations that follow we focus on the simple case of $p\equiv\pm 3\mod 8$,
but with some work it should be feasible to extend our constructions to work
in general. In Figure 15 we provide a schematic for $\operatorname{Sub}(G)/G$
where $G=\mathsf{SL}_{2}(\mathbb{F}_{13})$. The general goal is to represent
$G$-transfer systems as a pair of abstract transfer systems on two small
posets related to $\operatorname{Sub}(G)/G$, subject to a compatibility
condition. For $p\equiv\pm 1\mod 8$ one would likely need to use more than two
posets, but the same general technique should still apply.

Let $H_{-}\cong\mathsf{Dic}_{(p-1)/2}$ be the normalizer of the diagonal
matrices, and $H_{+}\cong\mathsf{Dic}_{(p+1)/2}$ the normalizer of some
maximal non-split torus. (Here $\mathsf{Dic}_{n}$ is the _dicyclic group_ of
order $4n$ as in Section 2.2). Let $Z=H_{1}\cap H_{2}\cong C_{2}$ be the
center of $G$.

Recall that for a group $G$, the _Frattini subgroup_ of $G$ is defined as
intersection of all maximal subgroups of $G$ [Gor80, §5.1]. The maximal
subgroups of $\mathsf{SL}_{2}(\mathbb{F}_{p})$ can be deduced from [Kin05]
since the Frattini subgroup of $\mathsf{SL}_{2}(\mathbb{F}_{p})$ is equal to
its center. When $p\equiv\pm 3\mod 8$, these maximal subgroups are

  1. (1)

normalizers of torii, which are dicyclic as described above,

  2. (2)

Borel subgroups, which are isomorphic to
$\mathbb{F}_{p}\rtimes\mathbb{F}_{p}^{\times}$ with
$x\in\mathbb{F}_{p}^{\times}$ acting on $\mathbb{F}_{p}$ as multiplication by
$x^{2}$, and

  3. (3)

binary tetrahedral or binary icosahedral subgroups.

The Borel subgroups are universally lossless, and the binary
tetrahedral/icosahedral subgroups are isomorphic to
$\mathsf{SL}_{2}(\mathbb{F}_{3})$ and $\mathsf{SL}_{2}(\mathbb{F}_{5})$ which
again are universally lossless. Furthermore, if we let $\epsilon=\pm 1$
accordingly as $p+\epsilon\equiv 4\mod 8$, then the normalizer of a torus of
order $p-\epsilon$ is a dicyclic group of order $2(p-\epsilon)\equiv 4\mod 8$,
and hence is also universally lossless. Thus the only non-universally lossless
maximal subgroups are the normalizers of torii of order $p+\epsilon$, which
are all conjugate to $H_{\epsilon}$.

By [CJ09], any two cyclic subgroups of $G$ with the same order are conjugate.
The subgroup $H_{\epsilon}$ contains three conjugacy classes of subgroups
isomorphic to $C_{4}$, and hence when we embed into $\operatorname{Sub}(G)/G$
these three copies of $C_{4}$ must be mapped to the same conjugacy class. But
if $C_{4},{}^{g}C_{4}\leqslant K\leqslant H_{\epsilon}$ is not contained in
any other maximal subgroup (e.g., $K=H_{\epsilon}$ itself), then
$N_{G}(K)\leqslant H_{\epsilon}$, so $C_{4},{}^{g}C_{4}$ cannot be conjugate
in $N_{G}(K)$. This is what causes $G$ to be lossy. On the other hand, when
$p\equiv\pm 3\mod 8$ this appears to be the only obstacle preventing
losslessness.

Let $D_{G}$ be the poset $\operatorname{Sub}(H_{\epsilon})/H_{\epsilon}$ with
an additional top vertex $[G]$. (For $G=\mathsf{SL}_{2}(\mathbb{F}_{13})$,
this is depicted in Figure 16.) Let $U_{G}\subseteq\operatorname{Sub}(G)/G$ be
the subposet on objects $[H]$ such that either $H=G$ or $H$ is contained in
some universally lossless subgroup (depicted in Figure 15). We have natural
poset maps $\psi^{D}\colon D_{G}\to\operatorname{Sub}(G)/G$ and
$\psi^{U}\colon U_{G}\to\operatorname{Sub}(G)/G$. Let
$I_{G}=(\psi^{D})^{-1}(\operatorname{im}\psi^{U})\subseteq D_{G}$ (depicted in
Figure 16 for $G=\mathsf{SL}_{2}(\mathbb{F}_{13})$). We let $\phi^{D}\colon
I_{G}\to D_{G}$ be the canonical embedding, and we let $\phi^{U}\colon
I_{G}\to U_{G}$ be the restriction of $\psi^{D}$. For any abstract transfer
system $\mathcal{R}$ on $I_{G}$, let
$\phi^{D}_{*}(\mathcal{R})=\operatorname{im}(\phi^{D}|_{\mathcal{R}})$ and
similarly for $\phi^{U}_{*}(\mathcal{R})$.

######  Definition .

A _split transfer system_ is a triple of catgorical transfer systems
$\mathcal{R}_{D}$, $\mathcal{R}_{I}$, $\mathcal{R}_{U}$ on $D_{G}$, $I_{G}$,
$U_{G}$, respectively, such that

  1. (1)

if for some $[C_{4}]\in D_{G}$ we have $[C_{4}]\,\mathcal{R}\,[G]$, then in
fact $[C_{4}]\,\mathcal{R}\,[G]$ for all conjugacy classes of $[C_{4}]\in
D_{G}$, and

  2. (2)

$\phi^{D}_{*}(\mathcal{R}_{I})=\mathcal{R}_{D}\cap\operatorname{im}\phi^{D}$
and
$\phi^{U}_{*}(\mathcal{R}_{I})=\mathcal{R}_{U}\cap\operatorname{im}\phi^{U}$.

For every split transfer system
$(\mathcal{R}_{D},\mathcal{R}_{I},\mathcal{R}_{U})$, we can define a reflexive
relation $\mathcal{R}$ on $\operatorname{Sub}(G)$ as follows. Let $K\leqslant
H$. If $H=G$ and $K\leqslant L$ for some universally lossless maximal subgroup
$L$, then we set $K\,\mathcal{R}\,H$ if and only if
$[K]\,\mathcal{R}_{U}\,[G]$. If $K\not\leqslant L$ for any such $L$, then some
conjugate ${}^{g}K$ of $K$ is contained in $H_{\epsilon}$, and we set
$K\,\mathcal{R}\,H$ if and only if $[{}^{g}K]\,\mathcal{R}_{D}\,[G]$.

So we suppose $H<G$. If $H\leqslant L$ for some universally lossless maximal
subgroup $L\leqslant G$, then we set $K\,\mathcal{R}\,H$ if and only if
$[K]\,\mathcal{R}_{U}\,[H]$. Otherwise we can find some $g\in G$ such that
${}^{g}H\leqslant H_{\epsilon}$. Then we set $K\,\mathcal{R}\,H$ if and only
if $[{}^{g}K]\,\mathcal{R}_{D}\,[{}^{g}H]$.

Conversely, if $\mathcal{R}$ is a transfer system on $G$ then we can define
$(\mathcal{R}_{D},\mathcal{R}_{I},\mathcal{R}_{U})$ so that

  1. (1)

$\mathcal{R}_{D}=\pi_{*}(\mathcal{R}|_{\operatorname{Sub}(H_{\epsilon})})$,
where
$\pi\colon\operatorname{Sub}(H_{\epsilon})\to\operatorname{Sub}(H_{\epsilon})/H_{\epsilon}\subseteq
D_{G}$,

  2. (2)

$\mathcal{R}_{I}=\mathcal{R}_{D}|_{I_{G}}$, and

  3. (3)

$\mathcal{R}_{U}=\pi^{\prime}_{*}(\mathcal{R})|_{U_{G}}$ where
$\pi^{\prime}\colon\operatorname{Sub}(G)\to\operatorname{Sub}(G)/G$.

The discussion of this speculative section culminates in the following
conjecture. If this conjecture were true, it would provide a constructive
method for exploring $N_{\infty}$ operads for an interesting class of groups
which are not lossless. In particular, one should not despair if their
favorite group of equivariance fails to be lossless, one only needs to figure
out a way to exploit the structure of the group itself.

######  Conjecture .

Fix an arbitrary transfer system $\mathcal{R}$ on
$G=\mathsf{SL}_{2}(\mathbb{F}_{p})$ where $p>5$, $p\neq 11$, and $p\equiv\pm
3\mod 8$. Then the triple $(\mathcal{R}_{D},\mathcal{R}_{I},\mathcal{R}_{U})$
is a split transfer system, and $\mathcal{R}$ is lifted from
$(\mathcal{R}_{D},\mathcal{R}_{I},\mathcal{R}_{U})$ using the procedure above.

$[e]$$[C_{3}]$$[C_{2}]$$[C_{6}]$$[C_{4}]$$[\mathsf{Dic}_{3}]$$[\mathsf{Dic}_{3}]$$[C_{12}]$$[Q_{8}]$$[\mathsf{Dic}_{6}]$$[\mathsf{Dic}_{7}]$$[C_{14}]$$[C_{7}]$$[C_{13}]$$[C_{26}]$$[C_{13}\rtimes
C_{3}]$$[C_{2}\times C_{13}\rtimes C_{3}]$$[\mathsf{Dic}_{13}]$$[C_{13}\rtimes
C_{12}]$$[\mathsf{SL}_{2}(\mathbb{F}_{3})]$$[\mathsf{SL}_{2}(\mathbb{F}_{13})]$
Figure 15. The poset $\operatorname{Sub}(G)/G$ for
$G=\mathsf{SL}_{2}(\mathbb{F}_{13})$. The non-split torus appears as
$\mathsf{Dic}_{7}$, the split torus as $\mathsf{Dic}_{6}$, $C_{13}\rtimes
C_{12}$ is the Borel subgroup and $\mathsf{SL}_{2}(\mathbb{F}_{3})$ is the
binary tetrahedral subgroup. The objects of the sub-poset $U_{G}$ are
highlighted in red.
$[e]$$[C_{3}]$$[C_{2}]$$[C_{6}]$$[C_{4}]$$[C_{4}]$$[C_{4}]$$[\mathsf{Dic}_{3}]$$[\mathsf{Dic}_{3}]$$[C_{12}]$$[Q_{8}]$$[\mathsf{Dic}_{6}]$$[\mathsf{SL}_{2}(\mathbb{F}_{13})]$
Figure 16. The poset $D_{G}$ for $G=\mathsf{SL}_{2}(\mathbb{F}_{13})$. The
objects of the sub-poset $I_{G}$ are highlighted in red.

## References

  * [ABG70] J. L. Alperin, R. Brauer, and D. Gorenstein.  Finite groups with quasi-dihedral and wreathed Sylow $2$-subgroups.  Trans. Amer. Math. Soc., 151:1–261, 1970.
  * [BBPR21] S. Balchin, D. Bearup, C. Pech, and C. Roitzheim.  Equivariant homotopy commutativity for $C_{pqr}$.  Tbilisi Mathematical Journal, 2021.
  * [BBR21] S. Balchin, D. Barnes, and C. Roitzheim.  $N_{\infty}$-operads and associahedra.  Pacific J. Math., 315(2):285–304, 2021.
  * [BH15] A. J. Blumberg and M. A. Hill.  Operadic multiplications in equivariant spectra, norms, and transfers.  Adv. Math., 285:658–708, 2015.
  * [BMO22] S. Balchin, E. MacBrough, and K. Ormsby.  The combinatorics of $N_{\infty}$ operads for $C_{qp^{n}}$ and $D_{p^{n}}$.  2022\.
  * [BOOR22] S. Balchin, K. Ormsby, A.M. Osorno, and C. Roitzheim.  Model structures on finite total orders.  arXiv:2109.07803, 2022.
  * [BP21] P. Bonventre and L. A. Pereira.  Genuine equivariant operads.  Adv. Math., 381:Paper No. 107502, 133, 2021.
  * [CJ09] M. Costantini and E. Jabara.  On finite groups in which cyclic subgroups of the same order are conjugate.  Comm. Algebra, 37(11):3966–3990, 2009.
  * [Con11] C. Conţiu.  Conditions under which a lattice is isomorphic to the subgroup lattice of an abelian group.  Carpathian J. Math., 27(2):193–199, 2011.
  * [Ded97] R. Dedekind.  Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind.  Math. Ann., 48(4):548–561, 1897.
  * [dGV00] F. de Giovanni and G. Vincenzi.  Pronormality in infinite groups.  Math. Proc. R. Ir. Acad., 100A(2):189–203, 2000.
  * [DH92] K. Doerk and T. Hawkes.  Finite soluble groups, volume 4 of De Gruyter Expositions in Mathematics.  Walter de Gruyter & Co., Berlin, 1992.
  * [Fei57] W. Feit.  On the structure of Frobenius groups.  Canadian Journal of Mathematics, 9:587–596, 1957.
  * [FOO+22] E. E. Franchere, K. Ormsby, A. M. Osorno, W. Qin, and R. Waugh.  Self-duality of the lattice of transfer systems via weak factorization systems.  Homology Homotopy Appl., 24(2):115–134, 2022.
  * [FT22] M. Ferrara and M. Trombetti.  Groups with many pronormal subgroups.  Bull. Aust. Math. Soc., 105(1):75–86, 2022.
  * [Gor80] D. Gorenstein.  Finite groups.  Chelsea Publishing Co., New York, second edition, 1980.
  * [GW18] J. J. Gutiérrez and D. White.  Encoding equivariant commutativity via operads.  Algebr. Geom. Topol., 18(5):2919–2962, 2018.
  * [Hal28] P. Hall.  A Note on Soluble Groups.  J. London Math. Soc., 3(2):98–105, 1928.
  * [HL16] M. Hill and T. Lawson.  Topological modular forms with level structure.  Invent. Math., 203(2):359–416, 2016.
  * [Jak74] B. V. Jakovlev.  Conditions under which a lattice is isomorphic to the lattice of subgroups of a group.  Algebra i Logika, 13(6):694–712, 720, 1974.
  * [Kin05] O. H. King.  The subgroup structure of finite classical groups in terms of geometric configurations.  In Surveys in combinatorics 2005, volume 327 of London Math. Soc. Lecture Note Ser., pages 29–56. Cambridge Univ. Press, Cambridge, 2005.
  * [KM13a] E. I. Khukhro and N. Yu. Makarenko.  Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms.  J. Algebra, 386:77–104, 2013.
  * [KM13b] E. I. Khukhro and N. Yu. Makarenko.  Lie algebras that admit a metacyclic Frobenius group of automorphisms.  Sibirsk. Mat. Zh., 54(1):131–149, 2013.
  * [KS06] L. A. Kurdachenko and I. Y. Subbotin.  Transitivity of normality and pronormal subgroups.  In Combinatorial group theory, discrete groups, and number theory, volume 421 of Contemp. Math., pages 201–212. Amer. Math. Soc., Providence, RI, 2006.
  * [Pen69] T. A. Peng.  Finite groups with pro-normal subgroups.  Proc. Amer. Math. Soc., 20:232–234, 1969.
  * [Rob96] D. J. S. Robinson.  A course in the theory of groups, volume 80 of Graduate Texts in Mathematics.  Springer-Verlag, New York, second edition, 1996.
  * [Ros67] J. S. Rose.  Finite soluble groups with pronormal system normalizers.  Proc. London Math. Soc. (3), 17:447–469, 1967.
  * [Rub21a] J. Rubin.  Combinatorial $N_{\infty}$ operads.  Algebr. Geom. Topol., 21(7):3513–3568, 2021.
  * [Rub21b] J. Rubin.  Detecting Steiner and linear isometries operads.  Glasg. Math. J., 63(2):307–342, 2021.