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The relative rank of the endomorphism monoid of a finite GG-set

Alonso Castillo-Ramirez111Email: alonso.castillor@academicos.udg.mx,   Ramón H. Ruiz-Medina
Centro Universitario de Ciencias Exactas e Ingenierías,
Universidad de Guadalajara, Guadalajara, México.
Abstract

For a group GG acting on a set XX, let EndG(X)\mathrm{End}_{G}(X) be the monoid of all GG-equivariant transformations, or GG-endomorphisms, of XX, and let AutG(X)\mathrm{Aut}_{G}(X) be its group of units. After discussing few basic results in a general setting, we focus on the case when GG and XX are both finite in order to determine the smallest cardinality of a set WEndG(X)W\subseteq\mathrm{End}_{G}(X) such that WAutG(X)W\cup\mathrm{Aut}_{G}(X) generates EndG(X)\mathrm{End}_{G}(X); this is known in semigroup theory as the relative rank of EndG(X)\mathrm{End}_{G}(X) modulo AutG(X)\mathrm{Aut}_{G}(X).

Keywords: GG-set, equivariant transformation, GG-endomorphism monoid, GG-auto-morphism group, relative rank.

MSC 2020: 20B25, 20E22, 20M20.

1 Introduction

For any group GG, a GG-set is simply a set XX on which GG acts; this is, there exists a function :G×XX\cdot:G\times X\to X such that ex=xe\cdot x=x, for all xXx\in X, and g(hx)=ghxg\cdot(h\cdot x)=gh\cdot x, for all xXx\in X, g,hGg,h\in G. In the context of semigroup theory, GG-sets are also known as GG-acts. A GG-equivariant transformation of XX, or a GG-endomorphism of XX, is a function τ:XX\tau:X\to X such that τ(gx)=gτ(x)\tau(g\cdot x)=g\cdot\tau(x), for all gGg\in G, xXx\in X. In general, GG-equivariant maps are the standard morphisms considered in the category of GG-sets ([4, p. 86]), and they are widely used in various areas of mathematics such as equivariant topology, topological dynamics, representation theory, and even in statistical inference, via the so-called equivariant estimators.

In this setting, two basic objects, which have been previously studied from a semigroup-theoretic perspective (e.g., see [5, 6, 12, 20]), are the monoid EndG(X)\mathrm{End}_{G}(X), consisting of all GG-equivariant transformations of XX equipped with composition, and its group of units AutG(X)\mathrm{Aut}_{G}(X), consisting of all bijective GG-equivariant transformations of XX. In particular, free GG-sets (i.e., GG-sets on which all point stabilizers are trivial) and their endomorphisms have gained special interest, as the former are examples of independence algebras (see [2, 10, 14, 15, 17]).

Our work in this paper is related with the concept of rank of a monoid MM, denoted by Rank(M)\mathrm{Rank}(M), which is the smallest cardinality of a generating set of MM. For any subset NMN\subseteq M, define the relative rank of MM modulo NN by

Rank(M:N):=min{|W|:WN=M}.\mathrm{Rank}(M:N):=\min\{|W|:\langle W\cup N\rangle=M\}.

Finding the relative rank of a finite monoid MM modulo its group of units UU is a natural and important question in semigroup theory (e.g., see [1, 3, 11, 16, 18, 19, 23]), as it is linked to the rank of MM via the formula

Rank(M)=Rank(U)+Rank(M:U).\mathrm{Rank}(M)=\mathrm{Rank}(U)+\mathrm{Rank}(M:U).

Our main theorem determines the relative rank of EndG(X)\mathrm{End}_{G}(X) modulo AutG(X)\mathrm{Aut}_{G}(X) when GG and XX are both finite. Before stating it, we introduce some notation: let [H][H] be the conjugacy class of a subgroup HGH\leq G, let ConjG(X)\mathrm{Conj}_{G}(X) be the set of conjugacy classes of point stabilizers in XX, let NG(H)N_{G}(H) be the normalizer of HH in GG, and let 𝒪[H]\mathcal{O}_{[H]} be the set of all GG-orbits whose point stabilizer is conjugate to HH.

Theorem 1.

Let GG be a finite group acting on a finite set XX. Let [H1],[H2],,[Hr][H_{1}],[H_{2}],\dots,[H_{r}] be the list of different conjugacy classes of subgroups in ConjG(X)\mathrm{Conj}_{G}(X). Let Ni:=NG(Hi)N_{i}:=N_{G}(H_{i}), and for any KGK\leq G, consider the NiN_{i}-conjugacy class [K]Ni:={gKg1:gNi}[K]_{N_{i}}:=\{gKg^{-1}:g\in N_{i}\}. Then,

Rank(EndG(X):AutG(X))=i=1r|U(Hi)|κG(X),\mathrm{Rank}(\mathrm{End}_{G}(X):\mathrm{Aut}_{G}(X))=\sum_{i=1}^{r}|U(H_{i})|-\kappa_{G}(X),

where U(Hi):={[Gx]Ni:xX,HiGx}U(H_{i}):=\{[G_{x}]_{N_{i}}:x\in X,H_{i}\leq G_{x}\} and κG(X):=|{i:|𝒪[Hi]|=1}|\kappa_{G}(X):=|\{i:|\mathcal{O}_{[H_{i}]}|=1\}|.

In order to prove Theorem 1, we begin by discussing few basic results that hold for an arbitrary GG-set and are probably known in the folklore of GG-equivariant maps, but we could not find them published anywhere. For example, we observe that when the action of GG on XX is transitive, AutG(X)=EndG(X)\mathrm{Aut}_{G}(X)=\mathrm{End}_{G}(X) if and only if a point stabilizer in XX is not properly contained in any of its conjugates. Hence, the situation AutG(X)=EndG(X)\mathrm{Aut}_{G}(X)=\mathrm{End}_{G}(X) always occurs for transitive actions where point stabilizers are finite, or finite-index, or normal subgroups of GG. Besides this, for a general GG-set, we use the Imprimitive Wreath Product Embedding Theorem [22, Theorem 5.5] to decompose AutG(X)\mathrm{Aut}_{G}(X) as a direct product of wreath products:

AutG(X)[H]ConjG(X)((NG(H)/H)Sym(𝒪[H])).\mathrm{Aut}_{G}(X)\cong\prod_{[H]\in\mathrm{Conj}_{G}(X)}((N_{G}(H)/H)\wr\mathrm{Sym}(\mathcal{O}_{[H]})).

One of the motivations for our study comes from the shift action of GG on AGA^{G}, where AA is a set, and AGA^{G} is the set of all functions x:GAx:G\to A. This action, which has fundamental importance in symbolic dynamics and the theory of cellular automata (see [9]), is defined as follows: for every gGg\in G and xAGx\in A^{G},

(gx)(h):=x(g1h),hG.(g\cdot x)(h):=x(g^{-1}h),\quad\forall h\in G.

When AGA^{G} is considered as a topological space with the product topology of the discrete topology of AA, the GG-equivariant continuous transformations of AGA^{G} turn out to be precisely the cellular automata over AGA^{G} (see the Curtis-Hedlund Theorem in [9, Ch. 1]). Hence, the monoid CA(G;A)\mathrm{CA}(G;A), consisting of all cellular automata over AGA^{G}, is a submonoid of EndG(AG)\mathrm{End}_{G}(A^{G}). When GG and AA are both finite, it is clear that CA(G;A)=EndG(AG)\mathrm{CA}(G;A)=\mathrm{End}_{G}(A^{G}). This setting has been studied in [7], and the present paper generalizes and refines many of the results obtained there. In particular, Theorem 1 significantly generalizes Theorem 7 in [7], where the the relative rank of EndG(AG)\mathrm{End}_{G}(A^{G}) modulo AutG(AG)\mathrm{Aut}_{G}(A^{G}) was determined only when GG is a finite Dedekind group (i.e. all subgroups of GG are normal).

The structure of this paper is as follows. In Section 2, we provide all the basic results on GG-equivariant transformations that we need in order to prove Theorem 1; we believe some of these basic results may be of interest in their own right. In Section 3 we prove Theorem 1 in three steps. First, we show that the set WW, consisting of all transformations [xy][x\mapsto y], with GxGyGx\neq Gy and GxGyG_{x}\leq G_{y}, defined in (1), generates EndG(X)\mathrm{End}_{G}(X) modulo AutG(X)\mathrm{Aut}_{G}(X) (Lemma 10). Second, we show that a particular subset VV of WW also generates EndG(X)\mathrm{End}_{G}(X) modulo AutG(X)\mathrm{Aut}_{G}(X) (Lemma 11). Finally, we complete the proof by showing that any set that generates EndG(X)\mathrm{End}_{G}(X) modulo AutG(X)\mathrm{Aut}_{G}(X) must contain at least |V||V| transformations (Theorem 17). The cardinality of VV is given by the formula of Theorem 1.

2 Basic results

For the rest of this paper, let GG be a group acting on a set XX. We assume that the reader is familiar with the basic results on group actions (e.g., see [22, Sec. 2]). Besides introducing notation, in this section we prove some required basic results on GG-sets that we may not find published anywhere else.

Denote by Sym(X)\mathrm{Sym}(X) the symmetric group on XX. When XX is finite and |X|=n|X|=n, we may write Symn\mathrm{Sym}_{n} instead of Sym(X)\mathrm{Sym}(X). Denote by GxGx and GxG_{x} the GG-orbit and stabilizer of xXx\in X, respectively:

Gx:={gx:gG} and Gx:={gG:gx=x}.Gx:=\{g\cdot x:g\in G\}\quad\text{ and }\quad G_{x}:=\{g\in G:g\cdot x=x\}.

Two basic results are that the set of GG-orbits form a partition of XX, and that stabilizars are subgroups of GG.

We shall repeatedly use the conjugation action of GG on the set of its subgroups; this is defined, for any gGg\in G and any subgroup HGH\leq G, by gH:=gHg1g\cdot H:=gHg^{-1}. The GG-orbits of this action, denoted by [H][H], are called conjugacy classes, while the stabilizers, denoted by NG(H)N_{G}(H), are called normalizers; in other words, for any HGH\leq G,

[H]:={gHg1:gG} and NG(H):={gG:gHg1=H}.[H]:=\{gHg^{-1}:g\in G\}\quad\text{ and }\quad N_{G}(H):=\{g\in G:gHg^{-1}=H\}.

For a subset YXY\subseteq X, define the setwise and elementwise stabilizers of YY as the subgroups GY={gG:gY=Y}G_{Y}=\{g\in G:g\cdot Y=Y\} and G(Y)={gG:gy=y,yY}G_{(Y)}=\{g\in G:g\cdot y=y,\forall y\in Y\}, respectively.

The main object of this paper is the monoid of GG-equivariant transformations of XX, which is denoted by

EndG(X):={τ:XX:τ(gx)=gτ(x),gG,xX}.\mathrm{End}_{G}(X):=\{\tau:X\to X:\tau(g\cdot x)=g\cdot\tau(x),\forall g\in G,x\in X\}.

Let AutG(X)\mathrm{Aut}_{G}(X) be the group of units of EndG(X)\mathrm{End}_{G}(X), i.e.

AutG(X):={τEndG(X):σEndG(X),στ=τσ=id}.\mathrm{Aut}_{G}(X):=\{\tau\in\mathrm{End}_{G}(X):\exists\sigma\in\mathrm{End}_{G}(X),\ \sigma\tau=\tau\sigma=\mathrm{id}\}.

When GG is a subgroup of Sym(X)\mathrm{Sym}(X), i.e. GG is a permutation group, note that AutG(X)\mathrm{Aut}_{G}(X) is equal to the centralizer of GG in Sym(X)\mathrm{Sym}(X).

It is clear that GG-equivariant transformations map GG-orbits to GG-orbits, as τ(Gx)=Gτ(x)\tau(Gx)=G\tau(x), for all xXx\in X, τEndG(X)\tau\in\mathrm{End}_{G}(X). Moreover, it is easy to see that if τEndG(X)\tau\in\mathrm{End}_{G}(X) is bijective, then τ1\tau^{-1} is also GG-equivariant, so AutG(X)=EndG(X)Sym(X)\mathrm{Aut}_{G}(X)=\mathrm{End}_{G}(X)\cap\mathrm{Sym}(X).

Both EndG(X)\mathrm{End}_{G}(X) and AutG(X)\mathrm{Aut}_{G}(X) act naturally on XX by evaluation: for any τEndG(X)\tau\in\mathrm{End}_{G}(X) and xXx\in X, this action is defined by τx:=τ(x)\tau\cdot x:=\tau(x).

A subset YXY\subset X is GG-invariant if gyYg\cdot y\in Y, for all yYy\in Y. It is easy to prove that YXY\subseteq X is GG-invariant if and only if YY is a union of GG-orbits. We may restrict the action of GG to YY, and consider the monoid EndG(Y)\mathrm{End}_{G}(Y) and the group AutG(Y)\mathrm{Aut}_{G}(Y).

Lemma 2.

For any GG-invariant subset YXY\subseteq X, the following hold:

  1. 1.

    EndG(Y)\mathrm{End}_{G}(Y) is isomorphic to a submonoid of EndG(X)\mathrm{End}_{G}(X), and AutG(Y)\mathrm{Aut}_{G}(Y) is isomorphic to a subgroup of AutG(X)\mathrm{Aut}_{G}(X).

  2. 2.

    If YY is also AutG(X)\mathrm{Aut}_{G}(X)-invariant (i.e., τ(y)Y\tau(y)\in Y, for all yYy\in Y, τAutG(X)\tau\in\mathrm{Aut}_{G}(X)), then AutG(Y)\mathrm{Aut}_{G}(Y) is isomorphic to a normal subgroup of AutG(X)\mathrm{Aut}_{G}(X).

  3. 3.

    AutG(Y)AutG(X)Y/AutG(X)(Y)\mathrm{Aut}_{G}(Y)\cong\mathrm{Aut}_{G}(X)_{Y}/\mathrm{Aut}_{G}(X)_{(Y)}.

Proof.

The monoid EndG(Y)\mathrm{End}_{G}(Y) is embedded in EndG(X)\mathrm{End}_{G}(X) via the injective homomorphism Φ:EndG(Y)EndG(X)\Phi:\mathrm{End}_{G}(Y)\to\mathrm{End}_{G}(X) given by

Φ(τ)(x):={τ(x) if xYx otherwise.\Phi(\tau)(x):=\begin{cases}\tau(x)&\text{ if }x\in Y\\ x&\text{ otherwise.}\end{cases}

It is easy to check that indeed Φ(τ)(x)EndG(X)\Phi(\tau)(x)\in\mathrm{End}_{G}(X) using the fact that gxYg\cdot x\in Y if and only if xYx\in Y. By restricting Φ\Phi to AutG(Y)\mathrm{Aut}_{G}(Y), we show that AutG(Y)\mathrm{Aut}_{G}(Y) is embedded in AutG(X)\mathrm{Aut}_{G}(X).

Now fix σAutG(X)\sigma\in\mathrm{Aut}_{G}(X) and τAutG(Y)\tau\in\mathrm{Aut}_{G}(Y), and assume that YY is also AutG(X)\mathrm{Aut}_{G}(X)-invariant. Since σ(y)Y\sigma(y)\in Y, for all yYy\in Y, then σ|YAutG(Y)\sigma|_{Y}\in\mathrm{Aut}_{G}(Y). Therefore σΦ(τ)σ1=Φ(σ|Yτσ1|Y)Φ(AutG(Y))\sigma\Phi(\tau)\sigma^{-1}=\Phi(\sigma|_{Y}\tau\sigma^{-1}|_{Y})\in\Phi(\mathrm{Aut}_{G}(Y)), which shows that Φ(AutG(Y))\Phi(\mathrm{Aut}_{G}(Y)) is normal in AutG(X)\mathrm{Aut}_{G}(X).

Finally, consider the restriction homomorphism ψ:AutG(X)YAutG(Y)\psi:\mathrm{Aut}_{G}(X)_{Y}\to\mathrm{Aut}_{G}(Y) given by ϕ(τ)=τ|Y\phi(\tau)=\tau|_{Y}, for all τAutG(X)Y\tau\in\mathrm{Aut}_{G}(X)_{Y}. This is a surjective homomorphism with kernel AutG(X)(Y)\mathrm{Aut}_{G}(X)_{(Y)}, so part (3) follows by the First Isomorphism Theorem. ∎

We shall introduce notation for three particular GG-equivariant transformations of XX. Let x,yXx,y\in X with xyx\neq y.

  1. 1.

    When GxGyG_{x}\leq G_{y}, define [xy]:XX[x\mapsto y]:X\to X by

    [xy](z):={gyifz=gxzotherwise.[x\mapsto y](z):=\left\{\begin{array}[]{cl}g\cdot y&if\ z=g\cdot x\\ z&\text{otherwise.}\end{array}\right. (1)
  2. 2.

    When Gx=GyG_{x}=G_{y} and there exists kGk\in G such that y=kxy=k\cdot x, define τx,k:XX\tau_{x,k}:X\to X by

    τx,k(z)={gkxifz=gxzotherwise.\tau_{x,k}(z)=\left\{\begin{array}[]{cl}gk\cdot x&if\ z=g\cdot x\\ z&\text{otherwise.}\end{array}\right. (2)
  3. 3.

    When Gx=GyG_{x}=G_{y} and GxGyGx\neq Gy, define [x,y]:XX[x,y]:X\to X by

    [x,y](z):={gyifz=gxgxifz=gyzotherwise.[x,y](z):=\left\{\begin{array}[]{cl}g\cdot y&if\ z=g\cdot x\\ g\cdot x&if\ z=g\cdot y\\ z&\text{otherwise.}\end{array}\right. (3)

Observe that the GG-equivariant transformations given in (2) and (3) are bijections, and τx,k\tau_{x,k} is defined if and only if Gx=Gkx=kGxk1G_{x}=G_{k\cdot x}=kG_{x}k^{-1}, which is true if and only if kNG(Gx)k\in N_{G}(G_{x}).

For any τEndG(X)\tau\in\mathrm{End}_{G}(X), it is easy to check that GxGτ(x)G_{x}\leq G_{\tau(x)} for every xXx\in X. Using this simple fact combined with the previous constructions of GG-equivariant transformations, we obtain the following result.

Lemma 3.

Let x,yXx,y\in X.

  1. 1.

    There exists τAutG(X)\tau\in\mathrm{Aut}_{G}(X) such that τ(x)=y\tau(x)=y if and only if Gx=GyG_{x}=G_{y}.

  2. 2.

    If GxGyGx\neq Gy, there exists τEndG(X)AutG(X)\tau\in\mathrm{End}_{G}(X)\setminus\mathrm{Aut}_{G}(X) such that τ(x)=y\tau(x)=y if and only if GxGyG_{x}\leq G_{y}.

Recall that the action of GG on XX is transitive if there exists a unique GG-orbit, so X=GxX=Gx for any xXx\in X; in other words, if for every x,yXx,y\in X there exists gGg\in G such that y=gxy=g\cdot x. If the action of GG on XX is not transitive, Lemma 3 shows that AutG(X)\mathrm{Aut}_{G}(X) is properly contained in EndG(X)\mathrm{End}_{G}(X). On the other hand, when the action is transitive, the next result shows that AutG(X)=EndG(X)\mathrm{Aut}_{G}(X)=\mathrm{End}_{G}(X) if and only if, for any xXx\in X, the stabilizer GxG_{x} is not properly contained in any of its conjugates (i.e., GxgGxg1G_{x}\subseteq gG_{x}g^{-1} implies Gx=gGxg1G_{x}=gG_{x}g^{-1}). This last condition may seem unusual. In general, if HH is a finite, finite-index, or normal subgroup of GG, then HH cannot be properly contained in any of its conjugates; however, there are examples of infinite groups GG and infinite-index subgroups HH such that HH is properly contained in gHg1gHg^{-1} for some gGg\in G (see [21]).

Proposition 4.

Suppose that the action of GG on XX is transitive and let xXx\in X. The subgroup GxG_{x} is not properly contained in any of its conjugates if and only if

EndG(X)=AutG(X).\mathrm{End}_{G}(X)=\mathrm{Aut}_{G}(X).
Proof.

Assume that GxG_{x} is not properly contained in any of its conjugates. Fix τEndG(X)\tau\in\mathrm{End}_{G}(X). It follows by transitivity and GG-equivariance that τ\tau must be surjective. Suppose that τ(x)=τ(y)\tau(x)=\tau(y), for some x,yGx,y\in G. By transitivity, there exists gGg\in G such that y=gxy=g\cdot x. Then τ(x)=τ(gx)=gτ(x)\tau(x)=\tau(g\cdot x)=g\cdot\tau(x), implies that gGτ(x)g\in G_{\tau(x)}. By the GG-equivarance of τ\tau, we have GxGτ(x)G_{x}\leq G_{\tau(x)}, and again by transitivity there exists hGh\in G such that Gτ(x)=hGxh1G_{\tau(x)}=hG_{x}h^{-1}. Hence, by hypothesis, Gx=Gτ(x)G_{x}=G_{\tau(x)}, so gGxg\in G_{x}. Hence, y=gx=xy=g\cdot x=x, which shows that τ\tau is injective.

Conversely, suppose that there exists xXx\in X and hGh\in G such that Gx<hGxh1G_{x}<hG_{x}h^{-1}. Define τ:GxGx\tau:Gx\to Gx as follows:

τ(gx):=ghx,gG.\tau(g\cdot x):=gh\cdot x,\quad\forall g\in G.

It is clear that τEndG(X)\tau\in\mathrm{End}_{G}(X), but we shall show that τ\tau is not injective. Consider k(hGxh1)Gxk\in(hG_{x}h^{-1})-G_{x}. Then kxxk\cdot x\neq x. However, khGxh1=Ghxk\in hG_{x}h^{-1}=G_{h\cdot x} implies that

k(hx)=hxτ(kx)=τ(x).k\cdot(h\cdot x)=h\cdot x\ \Leftrightarrow\ \tau(k\cdot x)=\tau(x).

Therefore, AutG(X)\mathrm{Aut}_{G}(X) is properly contained in EndG(X)\mathrm{End}_{G}(X). ∎

Example 5.

Let GG be an infinite group such that there exists a subgroup KGK\leq G and hGh\in G with K<hKh1K<hKh^{-1}. In this situation, consider the shift action of GG on AGA^{G}, where AA is a set with at least two elements. Without loss, assume that {0,1}A\{0,1\}\subseteq A. Let x:=χKAGx:=\chi_{K}\in A^{G} be the indicator function of KK defined by

χK(g)={1 if gK0 otherwise.\chi_{K}(g)=\begin{cases}1&\text{ if }g\in K\\ 0&\text{ otherwise.}\end{cases}

It is easy to check that Gx=KG_{x}=K, and so, by Proposition 4, AutG(Gx)\mathrm{Aut}_{G}(Gx) is properly contained in EndG(Gx)\mathrm{End}_{G}(Gx).

2.1 Structure of AutG(X)\mathrm{Aut}_{G}(X)

The next result is well-known (for example, see [24, Prop. 1.8]).

Lemma 6.

If the action of GG on XX is transitive, then

AutG(X)NG(Gx)/Gx, for any xX.\mathrm{Aut}_{G}(X)\cong N_{G}(G_{x})/G_{x},\text{ for any }x\in X.
Proof.

Fix xXx\in X. The map kτx,kk\mapsto\tau_{x,k} from NG(Gx)N_{G}(G_{x}) to AutG(X)\mathrm{Aut}_{G}(X), with τx,k\tau_{x,k} defined as in Lemma 3, is a surjective homomorphism with kernel GxG_{x}. ∎

Our next goal is to describe the structure of AutG(X)\mathrm{Aut}_{G}(X) for intransitive actions.

First of all, we introduce some notation. Define

StabsG(X):={Gx:xX} and ConjG(X):={[Gx]:xX}.\mathrm{Stabs}_{G}(X):=\{G_{x}:x\in X\}\ \text{ and }\ \mathrm{Conj}_{G}(X):=\{[G_{x}]:x\in X\}.

For each [H]ConjG(X)[H]\in\mathrm{Conj}_{G}(X), define the set

[H]:={xX:[Gx]=[H]}.\mathcal{B}_{[H]}:=\{x\in X:\ [G_{x}]=[H]\}.

Since Ggx=gGxg1G_{g\cdot x}=gG_{x}g^{-1}, for every gGg\in G, xXx\in X, the set [H]\mathcal{B}_{[H]} is GG-invariant. Moreover, by Lemma 3, [H]\mathcal{B}_{[H]} is also AutG(X)\mathrm{Aut}_{G}(X)-invariant, so, by Lemma 2, AutG([H])\mathrm{Aut}_{G}(\mathcal{B}_{[H]}) is isomorphic to a normal subgroup of AutG(X)\mathrm{Aut}_{G}(X). The number of AutG(X)\mathrm{Aut}_{G}(X)-orbits inside [H]\mathcal{B}_{[H]} is |[H]|=[G:NG(H)]|[H]|=[G:N_{G}(H)].

By the Orbit-Stabilizer Theorem, the cardinality of each GG-orbit inside [H]\mathcal{B}_{[H]} is the index [G:H][G:H]. Let 𝒪[H]\mathcal{O}_{[H]} be the set of GG-orbits contained in [H]\mathcal{B}_{[H]}:

𝒪[H]:={GxX:[Gx]=[H],xX}, and let α[H]:=|𝒪[H]|.\mathcal{O}_{[H]}:=\{Gx\subseteq X:\ [G_{x}]=[H],x\in X\},\text{ and let }\alpha_{[H]}:=|\mathcal{O}_{[H]}|.
Example 7.

For the shift action of GG on AGA^{G}, the values of α[H]\alpha_{[H]} may be calculated using the lattice of finite-index subgroups L(G)L(G) of GG and its corresponding Möbius function μ:L(G)×L(G)\mu:L(G)\times L(G)\to\mathbb{Z}; see [8] for details.

Example 8.

Consider the shift action of 4\mathbb{Z}_{4} on {0,1}4\{0,1\}^{\mathbb{Z}_{4}}, where we represent a function x{0,1}4x\in\{0,1\}^{\mathbb{Z}_{4}} by the 44-tuple (x(0),x(1),x(2),x(3))(x(0),x(1),x(2),x(3)). In this case, StabsG(X)={H1:=1,H2:=2,H3:=0}\mathrm{Stabs}_{G}(X)=\{H_{1}:=\langle 1\rangle,H_{2}:=\langle 2\rangle,H_{3}:=\langle 0\rangle\}. Figure 1 shows all the elements of {0,1}4\{0,1\}^{\mathbb{Z}_{4}} partitioned by 4\mathbb{Z}_{4}-orbits (depicted by ellipses) and by the sets [Hi]\mathcal{B}_{[H_{i}]} (depicted by rectangles). In this case, we directly see that α[H1]=2\alpha_{[H_{1}]}=2, αH2=1\alpha_{H_{2}}=1 and α[H3]=3\alpha_{[H_{3}]}=3.

Refer to caption
Figure 1: Partitions for the shift action of 4\mathbb{Z}_{4} on {0,1}4\{0,1\}^{\mathbb{Z}_{4}}

We briefly recall the definition of wreath product of groups. For a group KK, a set Δ\Delta and a group HH acting on Δ\Delta, the (unrestricted) wreath product of KK by HH, denoted by KΔHK\wr_{\Delta}H is the semidirect product KΔϕHK^{\Delta}\rtimes_{\phi}H, where KΔK^{\Delta} is the direct product of GG indexed by Δ\Delta and homomorphism ϕ:HAut(KΔ)\phi:H\to\mathrm{Aut}(K^{\Delta}) given by ϕ(h)(f)(δ)=f(hδ)\phi(h)(f)(\delta)=f(h\cdot\delta), for all hH,fKΔ,δΔh\in H,f\in K^{\Delta},\delta\in\Delta (see [22, p. 105] for more details). In our results, we shall always consider the case when H=Sym(Δ)H=\mathrm{Sym}(\Delta) and the action of HH on Δ\Delta is the natural action by evaluation, so, in order to simplify notation, we write KSym(Δ)K\wr\mathrm{Sym}(\Delta) instead of KΔSym(Δ)K\wr_{\Delta}\mathrm{Sym}(\Delta).

For any set AA, let Tran(A)\mathrm{Tran}(A) be the full transformation monoid of AA, i.e. the set of all self-maps of AA equipped with composition. When AA is finite, it is well-known that Tran(A)\mathrm{Tran}(A) is generated by a generating set of Sym(A)\mathrm{Sym}(A) together with a map with image size |A|1|A|-1 (e.g., see [13, Ch. 3]). Analogously to the case of groups, we may define a wreath product of monoids (see [1]).

The following result uses the Imprimitive Wreath Product Embedding Theorem, which, as stated in [22, Theorem 5.5], involves a partition with finitely many blocks; however, this assumption may be dropped and the same proof of the theorem works by using the axiom of choice.

Proposition 9.

Let GG be a group acting on a set XX,

  1. 1.

    For any HStabsG(X)H\in\mathrm{Stabs}_{G}(X),

    AutG([H])(NG(H)/H)Sym(𝒪[H]).\mathrm{Aut}_{G}(\mathcal{B}_{[H]})\cong\left(N_{G}(H)/H\right)\wr\mathrm{Sym}(\mathcal{O}_{[H]}).
  2. 2.
    AutG(X)[H]ConjG(X)((NG(H)/H)Sym(𝒪[H])).\mathrm{Aut}_{G}(X)\cong\prod_{[H]\in\mathrm{Conj}_{G}(X)}((N_{G}(H)/H)\wr\mathrm{Sym}(\mathcal{O}_{[H]})).
  3. 3.

    Suppose that HStabsG(X)H\in\mathrm{Stabs}_{G}(X) is not properly contained in any of its conjugates. Then,

    EndG([H])(NG(H)/H)Tran(𝒪[H]).\mathrm{End}_{G}(\mathcal{B}_{[H]})\cong\left(N_{G}(H)/H\right)\wr\mathrm{Tran}(\mathcal{O}_{[H]}).
Proof.
  1. 1.

    The set 𝒪[H]\mathcal{O}_{[H]} of GG-orbits inside [H]\mathcal{B}_{[H]} forms a GG-invariant partition of [H]\mathcal{B}_{[H]}, with |𝒪[H]|=α[H]|\mathcal{O}_{[H]}|=\alpha_{[H]}. Using the axiom of choice, it is easy to see that the induced action of AutG([H])\mathrm{Aut}_{G}(\mathcal{B}_{[H]}) on 𝒪[H]\mathcal{O}_{[H]} is isomorphic to the whole symmetric group Sym(𝒪[H])\mathrm{Sym}(\mathcal{O}_{[H]}) (pick representatives of GG-orbits in order to define corresponding GG-equivariant transformations). By [22, Theorem 5.5]) and Lemma 2, AutG([H])\mathrm{Aut}_{G}(\mathcal{B}_{[H]}) is permutationally isomorphic to a subgroup RR of AutG(Gx)Sym(𝒪[H])\mathrm{Aut}_{G}(Gx)\wr\mathrm{Sym}(\mathcal{O}_{[H]}), with Gx𝒪[H]Gx\in\mathcal{O}_{[H]}. The kernel of the projection of RR to Sym(𝒪[H])\mathrm{Sym}(\mathcal{O}_{[H]}) is isomorphic to the direct product Aut(Gx)𝒪[H]\mathrm{Aut}(Gx)^{\mathcal{O}_{[H]}}. Therefore,

    R=AutG(Gx)Sym(𝒪[H]),R=\mathrm{Aut}_{G}(Gx)\wr\mathrm{Sym}(\mathcal{O}_{[H]}),

    and the result follows by Lemma 6.

  2. 2.

    For each τAutG(X)\tau\in\mathrm{Aut}_{G}(X), we define a function

    F:AutG(X)[H]ConjG(X)AutG([H])F:\mathrm{Aut}_{G}(X)\to\prod_{[H]\in\mathrm{Conj}_{G}(X)}\mathrm{Aut}_{G}(\mathcal{B}_{[H]})

    by F(τ)[H]:=τ|[H]F(\tau)_{[H]}:=\tau|_{\mathcal{B}_{[H]}} for every τAutG(X)\tau\in\mathrm{Aut}_{G}(X). The function FF is injective because {[H]:[H]ConjG(X)}\{\mathcal{B}_{[H]}:[H]\in\mathrm{Conj}_{G}(X)\} is a partition of XX. To show that FF is surjective, observe that for any (τ[H]:[H]ConjG(X))(\tau_{[H]}:[H]\in\mathrm{Conj}_{G}(X)) in the product, we may define τAutG(X)\tau\in\mathrm{Aut}_{G}(X) by τ(x)=τ[H](x)\tau(x)=\tau_{[H]}(x) if and only if x[H]x\in\mathcal{B}_{[H]}. It follows that τ\tau is indeed GG-equivariant as, for every [H]ConjG(X)[H]\in\mathrm{Conj}_{G}(X), τ[H]\tau_{[H]} is GG-equivariant and [H]\mathcal{B}_{[H]} is GG-invariant. Finally, FF is a homomorphism since [H]\mathcal{B}_{[H]} is AutG(X)\mathrm{Aut}_{G}(X)-invariant so (τσ)|[H]=τ|[H]σ|[H](\tau\circ\sigma)|_{\mathcal{B}_{[H]}}=\tau|_{\mathcal{B}_{[H]}}\circ\sigma|_{\mathcal{B}_{[H]}}, for every τ,σAutG(X)\tau,\sigma\in\mathrm{Aut}_{G}(X). The result follows by Lemma 9.

  3. 3.

    With the help of Proposition 4, this follows by a very similar argument as in the proof of the first point.

3 The relative rank of EndG(X)\mathrm{End}_{G}(X) modulo AutG(X)\mathrm{Aut}_{G}(X)

For any monoid MM and UMU\subseteq M, we say that a subset WW of MM generates MM modulo UU if WU=M\langle W\cup U\rangle=M.

For this section, assume that GG is a finite group acting on a finite set XX. We shall assume that [H1],[H2],,[Hr][H_{1}],[H_{2}],\dots,[H_{r}] is the list of different conjugacy classes of subgroups in ConjG(X)\mathrm{Conj}_{G}(X). Define i:=[Hi]\mathcal{B}_{i}:=\mathcal{B}_{[H_{i}]} and αi:=α[Hi]\alpha_{i}:=\alpha_{[H_{i}]}. For nn\in\mathbb{N}, denote [n]:={1,2,,n}[n]:=\{1,2,\dots,n\}.

Lemma 10.

The set

W:={[xy]:x,yX,GxGy,GxGy}W:=\left\{[x\mapsto y]:x,y\in X,Gx\neq Gy,G_{x}\leq G_{y}\right\}

generates EndG(X)\mathrm{End}_{G}(X) modulo AutG(X)\mathrm{Aut}_{G}(X)

Proof.

Suppose that the list of conjugacy classes is ordered such that

|H1||H2||Hr|.|H_{1}|\geq|H_{2}|\geq\dots\geq|H_{r}|.

Fix τEndG(X)\tau\in\mathrm{End}_{G}(X) and for each i[r]i\in[r], define τiEndG(X)\tau_{i}\in\mathrm{End}_{G}(X) by

τi(x):={τ(x) if xix otherwise. \tau_{i}(x):=\begin{cases}\tau(x)&\text{ if }x\in\mathcal{B}_{i}\\ x&\text{ otherwise. }\end{cases}

Each τi\tau_{i} is indeed GG-equivariant as i\mathcal{B}_{i} is GG-invariant. By Lemma 3, GxGτ(x)G_{x}\leq G_{\tau(x)} for all xXx\in X, so τ(i)jij\tau(\mathcal{B}_{i})\subseteq\bigcup_{j\leq i}\mathcal{B}_{j}. This implies that we have the following factorization of τ\tau

τ=τrτr1τ2τ1,\tau=\tau_{r}\circ\tau_{r-1}\circ\dots\circ\tau_{2}\circ\tau_{1}, (4)

which holds since τi(x)\tau_{i}(x) is fixed by τj\tau_{j}, for all xix\in\mathcal{B}_{i}, j>ij>i. Now fix i[r]i\in[r] and define

i0\displaystyle\mathcal{B}_{i}^{0} :={xi:τ(x)i},\displaystyle:=\{x\in\mathcal{B}_{i}:\tau(x)\in\mathcal{B}_{i}\},
i1\displaystyle\mathcal{B}_{i}^{1} :={xi:τ(x)j with j<i}.\displaystyle:=\{x\in\mathcal{B}_{i}:\tau(x)\in\mathcal{B}_{j}\text{ with }j<i\}.

For ϵ{0,1}\epsilon\in\{0,1\}, it is easy to see that iϵ\mathcal{B}_{i}^{\epsilon} is a GG-invariant subset of XX, so we may define τiϵEndG(X)\tau_{i}^{\epsilon}\in\mathrm{End}_{G}(X) by

τiϵ(x):={τi(x) if xiϵx otherwise. \tau_{i}^{\epsilon}(x):=\begin{cases}\tau_{i}(x)&\text{ if }x\in\mathcal{B}_{i}^{\epsilon}\\ x&\text{ otherwise. }\end{cases}

It follows that τi=τi0τi1\tau_{i}=\tau_{i}^{0}\circ\tau_{i}^{1}. We finish the proof by showing that τiϵWAutG(X)\tau_{i}^{\epsilon}\in\langle W\cup\mathrm{Aut}_{G}(X)\rangle, for ϵ{0,1}\epsilon\in\{0,1\}.

  • Case ϵ=0\epsilon=0. We show that τi0WAutG(X)\tau_{i}^{0}\in\langle W\cup\mathrm{Aut}_{G}(X)\rangle. In this case, τi0\tau_{i}^{0} is contained in a submonoid of EndG(X)\mathrm{End}_{G}(X) isomorphic to EndG(i)\mathrm{End}_{G}(\mathcal{B}_{i}). If αi=1\alpha_{i}=1, then EndG(i)=EndG(Gx)=AutG(Gx)\mathrm{End}_{G}(\mathcal{B}_{i})=\mathrm{End}_{G}(Gx)=\mathrm{Aut}_{G}(Gx), for xix\in\mathcal{B}_{i}, so assume that αi>2\alpha_{i}>2. By Proposition 9, EndG(i)AutG(Gx)Tran(𝒪[Hi])\mathrm{End}_{G}(\mathcal{B}_{i})\cong\mathrm{Aut}_{G}(Gx)\wr\mathrm{Tran}(\mathcal{O}_{[H_{i}]}), for any xix\in\mathcal{B}_{i}. The monoid Tran(𝒪[Hi])\mathrm{Tran}(\mathcal{O}_{[H_{i}]}) is generated by Sym(𝒪[Hi])\mathrm{Sym}(\mathcal{O}_{[H_{i}]}) together with any mapping with image size αi1\alpha_{i}-1 (see [19, Prop. 1.2]). For any x,yix,y\in\mathcal{B}_{i} with GxGyGx\neq Gy, the map [xy][x\mapsto y] induces on 𝒪Hi\mathcal{O}_{H_{i}} a mapping with image size αi1\alpha_{i}-1. It follows that EndG(i)\mathrm{End}_{G}(\mathcal{B}_{i}) is generated by AutG(i){[xy]|i}\mathrm{Aut}_{G}(\mathcal{B}_{i})\cup\{[x\mapsto y]|_{\mathcal{B}_{i}}\}, and the result follows.

  • Case ϵ=1\epsilon=1. We show that τi1WAutG(X)\tau_{i}^{1}\in\langle W\cup\mathrm{Aut}_{G}(X)\rangle. Suppose that i1\mathcal{B}_{i}^{1} is a union of the following GG-orbits

    i1=Gx1Gx2Gxs,\mathcal{B}_{i}^{1}=Gx_{1}\cup Gx_{2}\cup\dots Gx_{s},

    for some x1,,xsXx_{1},\dots,x_{s}\in X. Write yk:=τi1(xk)y_{k}:=\tau^{1}_{i}(x_{k}), for k{1,,s}k\in\{1,\dots,s\}. As ykiy_{k}\not\in\mathcal{B}_{i}, then GxkGykGx_{k}\neq Gy_{k}. Therefore, we have the factorizacion

    τi1=[x1y1][x2y2][xsys]WAutG(X).\tau_{i}^{1}=[x_{1}\mapsto y_{1}][x_{2}\mapsto y_{2}]\dots[x_{s}\mapsto y_{s}]\in\langle W\cup\mathrm{Aut}_{G}(X)\rangle.

For any N,HGN,H\leq G, define the NN-conjugacy class of HH by [H]N:={gHg1:gN}[H]_{N}:=\{gHg^{-1}:g\in N\}.

Lemma 11.

Let GG be a finite group acting on a finite set XX. For each i[r]i\in[r], Ni:=NG(Hi)N_{i}:=N_{G}(H_{i}), and let [Ki,1]Ni,,[Ki,ri]Ni[K_{i,1}]_{N_{i}},\dots,[K_{i,r_{i}}]_{N_{i}} be the list of NiN_{i}-conjugacy classes such that Ki,jStabsG(X)K_{i,j}\in\mathrm{Stabs}_{G}(X) and Hi<Ki,jH_{i}<K_{i,j}. We fix some elements of XX as follows:

  • For each i[r]i\in[r], fix xiix_{i}\in\mathcal{B}_{i} such that Gxi=HiG_{x_{i}}=H_{i}.

  • For each i[r]i\in[r] and j[ri]j\in[r_{i}], fix yi,jXy_{i,j}\in X such that Gyi,j=Ki,jG_{y_{i,j}}=K_{i,j}.

  • For each i[r]i\in[r] such that αi2\alpha_{i}\geq 2, fix xiix_{i}^{\prime}\in\mathcal{B}_{i} such that Gxi=HiG_{x_{i}^{\prime}}=H_{i} and GxiGxiGx_{i}\neq Gx_{i}^{\prime}.

Then the set

V:={[xiyi,j]:i[r],j[ri]}{[xixi]:i[r],αi2}.V:=\left\{[x_{i}\mapsto y_{i,j}]:i\in[r],j\in[r_{i}]\right\}\cup\{[x_{i}\mapsto x_{i}^{\prime}]:i\in[r],\alpha_{i}\geq 2\}.

generates EndG(X)\mathrm{End}_{G}(X) modulo AutG(X)\mathrm{Aut}_{G}(X).

Proof.

By Lemma 10, it is enough to show that any [xy][x\mapsto y], with x,yXx,y\in X, GxGyGx\neq Gy, GxGyG_{x}\leq G_{y}, may be expressed as a product of elements of VV and AutG(X)\mathrm{Aut}_{G}(X). Without loss, assume that [Gx]=[Hi][G_{x}]=[H_{i}]. Then, there exists gGg\in G such that Ggx=gGxg1=HiG_{g\cdot x}=gG_{x}g^{-1}=H_{i}; since [xy]=[(gx)(gy)][x\mapsto y]=[(g\cdot x)\mapsto(g\cdot y)], we may assume that Gx=HiG_{x}=H_{i}. There are two cases to consider.

  • Case Gx=GyG_{x}=G_{y}. Note that Gx=Hi=Gxi=Gxi=GyG_{x}=H_{i}=G_{x_{i}}=G_{x_{i}^{\prime}}=G_{y}. We have four subcases:

    • Case GxGxiGx\neq Gx_{i} and GyGxiGy\neq Gx_{i}^{\prime}. Here we have the factorization

      [xy]=[y,xi][x,xi][xixi][x,xi][y,xi]VAutG(X).[x\mapsto y]=[y,x_{i}^{\prime}][x,x_{i}][x_{i}\mapsto x_{i}^{\prime}][x,x_{i}][y,x_{i}^{\prime}]\in\langle V\cup\mathrm{Aut}_{G}(X)\rangle.
    • Case Gx=GxiGx=Gx_{i} and GyGxiGy\neq Gx_{i}^{\prime}. There exists hGh\in G with xi=hxx_{i}=h\cdot x. As Gx=GxiG_{x}=G_{x_{i}}, we may define τx,hAutG(X)\tau_{x,h}\in\mathrm{Aut}_{G}(X) as in the proof of Lemma 3. Then

      [xy]=[xiy]τx,h,[x\mapsto y]=[x_{i}\mapsto y]\tau_{x,h},

      which holds since for every gGg\in G,

      [xiy]τx,h(gx)=[xiy](ghx)=[xiy](gxi)=gy.[x_{i}\mapsto y]\tau_{x,h}(g\cdot x)=[x_{i}\mapsto y](gh\cdot x)=[x_{i}\mapsto y](g\cdot x_{i})=g\cdot y.

      As GyGxiGy\neq Gx_{i}^{\prime}, then

      [xy]=[y,xi][xixi][y,xi]τx,hVAutG(X).[x\mapsto y]=[y,x_{i}^{\prime}][x_{i}\mapsto x_{i}^{\prime}][y,x_{i}^{\prime}]\tau_{x,h}\in\langle V\cup\mathrm{Aut}_{G}(X)\rangle.
    • Case GxGxiGx\neq Gx_{i} and Gy=GxiGy=Gx_{i}^{\prime}. There exists kGk\in G with y=kxiy=k\cdot x_{i}^{\prime}. Observe that

      [xy]=τxi,k[xxi]τxi,k1.[x\mapsto y]=\tau_{x_{i}^{\prime},k}[x\mapsto x_{i}^{\prime}]\tau_{x_{i}^{\prime},k^{-1}}.

      Therefore,

      [xy]=τxi,k[x,xi][xixi][x,xi]τxi,k1VAutG(X).[x\mapsto y]=\tau_{x_{i}^{\prime},k}[x,x_{i}][x_{i}\mapsto x_{i}^{\prime}][x,x_{i}]\tau_{x_{i}^{\prime},k^{-1}}\in\langle V\cup\mathrm{Aut}_{G}(X)\rangle.
    • Case Gx=GxiGx=Gx_{i} and Gy=GxiGy=Gx_{i}^{\prime}. With the notation of the above cases,

      [xy]=τxi,k[xixi]τx,hτxi,k1VAutG(X).[x\mapsto y]=\tau_{x_{i}^{\prime},k}[x_{i}\mapsto x_{i}^{\prime}]\tau_{x,h}\tau_{x_{i}^{\prime},k^{-1}}\in\langle V\cup\mathrm{Aut}_{G}(X)\rangle.
  • Case Gx<GyG_{x}<G_{y}. There exists Ki,j=Gyi,jK_{i,j}=G_{y_{i,j}} such that [Ki,j]Ni=[Gy]Ni[K_{i,j}]_{N_{i}}=[G_{y}]_{N_{i}}, with Ni:=NG(Hi)N_{i}:=N_{G}(H_{i}). Thus, there exists nNin\in N_{i} such that

    Ki,j=nGyn1=Gz,K_{i,j}=nG_{y}n^{-1}=G_{z},

    where z:=nyXz:=n\cdot y\in X. Now, Gx=GnxG_{x}=G_{n\cdot x} because nNin\in N_{i}, and we may define τx,nAutG(X)\tau_{x,n}\in\mathrm{Aut}_{G}(X). Hence,

    [xy]=[(n1x)y]τx,n1=[x(ny)]τx,n1=[xz]τx,n1.[x\mapsto y]=[(n^{-1}\cdot x)\mapsto y]\tau_{x,n^{-1}}=[x\mapsto(n\cdot y)]\tau_{x,n^{-1}}=[x\mapsto z]\tau_{x,n^{-1}}.

    This shows that [xy]VAutG(X)[x\mapsto y]\in\langle V\cup\mathrm{Aut}_{G}(X)\rangle if and only if [xz]VAutG(X)[x\mapsto z]\in\langle V\cup\mathrm{Aut}_{G}(X)\rangle, with Gz=Ki,j=Gyi,jG_{z}=K_{i,j}=G_{y_{i,j}}. Again there are four subcases, which are done analogously as in the previous case. For example, for the subcase GxGxiGx\neq Gx_{i} and GzGyi,jGz\neq Gy_{i,j}, we obtain the factorization

    [xz]=[x,xi][z,yi,j][xiyi,j][z,yi,j][x,xi]VAutG(X).[x\mapsto z]=[x,x_{i}][z,y_{i,j}][x_{i}\mapsto y_{i,j}][z,y_{i,j}][x,x_{i}]\in\langle V\cup\mathrm{Aut}_{G}(X)\rangle.

With the notation of the previous lemma, define

U(Hi):={[K]Ni:KStabsG(X),HiK} and κG(X):=|{i:αi=1}|.U(H_{i}):=\{[K]_{N_{i}}:K\in\mathrm{Stabs}_{G}(X),H_{i}\leq K\}\text{ and }\kappa_{G}(X):=|\{i:\alpha_{i}=1\}|.
Corollary 12.

With the previous notation,

Rank(EndG(X):AutG(X))|V|=i=1r|U(Hi)|κG(X).\mathrm{Rank}(\mathrm{End}_{G}(X):\mathrm{Aut}_{G}(X))\leq|V|=\sum_{i=1}^{r}|U(H_{i})|-\kappa_{G}(X).
Example 13.

When GG is a finite group acting by the shift action on AGA^{G}, with AA a finite set with at least two elements, it is not hard to show (see [7, Lemma 5]) that

κG(AG)={|{i:|G/Hi|=2}| if |A|=2,0otherwise.\kappa_{G}(A^{G})=\begin{cases}|\{i:|G/H_{i}|=2\}|&\text{ if }|A|=2,\\ 0&\text{otherwise.}\end{cases}

Before being able to determine the exact relative rank of EndG(X)\mathrm{End}_{G}(X) modulo AutG(X)\mathrm{Aut}_{G}(X), we need to introduce few definitions. For τEndG(X)\tau\in\mathrm{End}_{G}(X), define

ker(τ)={(a,b)X×X:τ(a)=τ(b)}.\ker(\tau)=\{(a,b)\in X\times X:\tau(a)=\tau(b)\}.

This is an equivalence relation on XX, and it is linked with the \mathcal{L} Green’s relation in a semigroup of transformations (see [13, Theorem 4.5.1]).

Remark 14.

We state the following elementary properties of kernels of transformations:

  1. 1.

    ker(τ)={(a,a):aX}\ker(\tau)=\{(a,a):a\in X\} if and only if τ\tau is a bijection.

  2. 2.

    τ,σEndG(X)\tau,\sigma\in\mathrm{End}_{G}(X), we have ker(τ)ker(στ)\ker(\tau)\subseteq\ker(\sigma\tau).

  3. 3.

    If τAutG(X)\tau\in\mathrm{Aut}_{G}(X), then |ker(σ)|=|ker(στ)||\ker(\sigma)|=|\ker(\sigma\tau)|, since (a,b)ker(στ)(a,b)\in\ker(\sigma\tau) if and only if (τ(a),τ(b))ker(σ)(\tau(a),\tau(b))\in\ker(\sigma).

Definition 15.

Let KStabsG(X)K\in\mathrm{Stabs}_{G}(X) be such that HiKGH_{i}\leq K\leq G. We say that τEndG(X)\tau\in\mathrm{End}_{G}(X) is an elementary collapsing of type (i,[K]Ni)(i,[K]_{N_{i}}) if there exist x,yXx,y\in X, with GxGyGx\neq Gy, such that:

  1. 1.

    Gx=HiG_{x}=H_{i} and [Gτ(x)]Ni=[Gy]Ni=[K]Ni[G_{\tau(x)}]_{N_{i}}=[G_{y}]_{N_{i}}=[K]_{N_{i}}.

  2. 2.

    ker(τ)={(gx,gy),(gy,gx):gG}{(a,a):aX}\ker(\tau)=\{(g\cdot x,g\cdot y),(g\cdot y,g\cdot x):g\in G\}\cup\{(a,a):a\in X\}.

For example, when x,yXx,y\in X satisfy that Gx=HiG_{x}=H_{i} and [Gy]Ni=[K]Ni[G_{y}]_{N_{i}}=[K]_{N_{i}}, then [xy][x\mapsto y] is an elementary collapsing of type (i,[K]Ni)(i,[K]_{N_{i}}).

Lemma 16.
  1. 1.

    Suppose τEndG(X)\tau\in\mathrm{End}_{G}(X) is an elementary collapsing of type (i,[K]Ni)(i,[K]_{N_{i}}) and type (i,[K]Ni)(i^{\prime},[K^{\prime}]_{N_{i}}). Then i=ii=i^{\prime} and [K]Ni=[K]Ni[K]_{N_{i}}=[K^{\prime}]_{N_{i}}.

  2. 2.

    The number of possible types of elementary collapsings is precisely

    i=1rU(Hi)κG(X).\sum_{i=1}^{r}U(H_{i})-\kappa_{G}(X).
Proof.

For the first part, there exist x,x,y,yXx,x^{\prime},y,y^{\prime}\in X with GxGyGx\neq Gy and GxGyGx^{\prime}\neq Gy^{\prime} satisfying Definition 15. Then

{(gx,gy),(gy,gx):gG}={(gx,gy),(gy,gx):gG}.\{(g\cdot x,g\cdot y),(g\cdot y,g\cdot x):g\in G\}=\{(g\cdot x^{\prime},g\cdot y^{\prime}),(g\cdot y^{\prime},g\cdot x^{\prime}):g\in G\}.

This shows that Gx=GxGx=Gx^{\prime} and Gy=GyGy=Gy^{\prime}. The first equality implies that [Hi]=[Gx]=[Gx]=[Hi][H_{i}]=[G_{x}]=[G_{x^{\prime}}]=[H_{i^{\prime}}], so i=ii=i^{\prime}. Moreover, there exists gGg\in G such that x=gxx=g\cdot x^{\prime}, so

Hi=Gx=Ggx=gGxg1=gHig1gNi=NG(Hi).H_{i}=G_{x}=G_{g\cdot x^{\prime}}=gG_{x^{\prime}}g^{-1}=gH_{i}g^{-1}\ \Rightarrow\ g\in N_{i}=N_{G}(H_{i}).

Now,

Gτ(x)=Gτ(gx)=Ggτ(x)=gGτ(x)g1,G_{\tau(x)}=G_{\tau(g\cdot x^{\prime})}=G_{g\cdot\tau(x^{\prime})}=gG_{\tau(x^{\prime})}g^{-1},

which implies that [Gτ(x)]Ni=[Gτ(x)]Ni[G_{\tau(x)}]_{N_{i}}=[G_{\tau(x^{\prime})}]_{N_{i}}, and so [K]Ni=[K]Ni[K]_{N_{i}}=[K^{\prime}]_{N_{i}}

The second part of the lemma follows as an elementary collapsing of type (i,[Hi]Ni)(i,[H_{i}]_{N_{i}}) exists if and only if α[Hi]1\alpha_{[H_{i}]}\neq 1. ∎

Theorem 17.

Let GG be a finite group acting on a finite set XX. Let [H1],[H2],,[Hr][H_{1}],[H_{2}],\dots,[H_{r}] be the list of different conjugacy classes of subgroups in ConjG(X)\mathrm{Conj}_{G}(X). Then,

Rank(EndG(X):AutG(X))=i=1rU(Hi)κG(X).\mathrm{Rank}(\mathrm{End}_{G}(X):\mathrm{Aut}_{G}(X))=\sum_{i=1}^{r}U(H_{i})-\kappa_{G}(X).
Proof.

Let WEndG(X)W\subseteq\mathrm{End}_{G}(X) such that AutG(X)W=EndG(X)\langle\mathrm{Aut}_{G}(X)\cup W\rangle=\mathrm{End}_{G}(X). We shall show that WW must contain at least one elementary collapsing of each possible type. Suppose that KStabsG(X)K\in\mathrm{Stabs}_{G}(X) satisfies that HiKGH_{i}\leq K\leq G. Let x,yXx,y\in X be such that Gx=HiG_{x}=H_{i} and [Gy]Ni=[K]Ni[G_{y}]_{N_{i}}=[K]_{N_{i}}. Since [xy]EndG(X)[x\mapsto y]\in\mathrm{End}_{G}(X), there exist τ1,τ2,,τsAutG(X)W\tau_{1},\tau_{2},\dots,\tau_{s}\in\mathrm{Aut}_{G}(X)\cup W such that

[xy]=τsτs1τ2τ1.[x\mapsto y]=\tau_{s}\tau_{s-1}\dots\tau_{2}\tau_{1}.

Let kk be the smallest index such that τk\tau_{k} is non-invertible. We must have τkW\tau_{k}\in W.

Claim 18.

For all m[r]{i}m\in[r]-\{i\}, we have:

  1. 1.

    τk(m)=m\tau_{k}(\mathcal{B}_{m})=\mathcal{B}_{m}, and Gz=Gτk(z)G_{z}=G_{\tau_{k}(z)} for every zmz\in\mathcal{B}_{m}.

  2. 2.

    If yiy\in\mathcal{B}_{i}, then τk(i)i\tau_{k}(\mathcal{B}_{i})\subseteq\mathcal{B}_{i}, and Gz=Gτk(z)G_{z}=G_{\tau_{k}(z)} for every ziz\in\mathcal{B}_{i}.

Proof.

Suppose there exists wmw\in\mathcal{B}_{m} such that τk(w)m\tau_{k}(w)\not\in\mathcal{B}_{m}. Let z:=τ11τk11(w)z:=\tau_{1}^{-1}\dots\tau_{k-1}^{-1}(w). By Lemma 3, zmz\in\mathcal{B}_{m}, and so [xy](z)=z[x\mapsto y](z)=z. This implies that τsτk+1\tau_{s}\dots\tau_{k+1} must map τk(w)m\tau_{k}(w)\not\in\mathcal{B}_{m} to an element of m\mathcal{B}_{m}, which is impossible by Lemma 3. This proves that τk(m)m\tau_{k}(\mathcal{B}_{m})\subseteq\mathcal{B}_{m}. If τk(m)\tau_{k}(\mathcal{B}_{m}) is properly contained in m\mathcal{B}_{m}, then there must exist w1,w2mw_{1},w_{2}\in\mathcal{B}_{m} such that τk(w1)=τk(w2)\tau_{k}(w_{1})=\tau_{k}(w_{2}). However, this implies that z1:=τ11τk11(w1)mz_{1}:=\tau_{1}^{-1}\dots\tau_{k-1}^{-1}(w_{1})\in\mathcal{B}_{m} and z2:=τ11τk11(w2)mz_{2}:=\tau_{1}^{-1}\dots\tau_{k-1}^{-1}(w_{2})\in\mathcal{B}_{m}, with z1z2z_{1}\neq z_{2}, satisfy that [xy](z1)=[xy](z2)[x\mapsto y](z_{1})=[x\mapsto y](z_{2}), which contradicts the definition of [xy][x\mapsto y]. Thus τk(m)=m\tau_{k}(\mathcal{B}_{m})=\mathcal{B}_{m}, and Gz=Gτk(z)G_{z}=G_{\tau_{k}(z)} for every zmz\in\mathcal{B}_{m}, follows by Lemma 3 and the finiteness of GG.

The second part of the claim is done analogously. ∎

We shall show that τk\tau_{k} is an elementary collapsing of type (i,[K]Ni)(i,[K]_{N_{i}}) by verifying the two properties of Definition 15.

  1. 1.

    As τk\tau_{k} is non-invertible, there exist a,bXa,b\in X, aba\neq b, such that τk(a)=τk(b)\tau_{k}(a)=\tau_{k}(b). Let a:=τ11τk11(a)a^{\prime}:=\tau_{1}^{-1}\dots\tau_{k-1}^{-1}(a) and b:=τ11τk11(b)b^{\prime}:=\tau_{1}^{-1}\dots\tau_{k-1}^{-1}(b). Then

    τsτs1τ2τ1(a)=τsτk(a)=τsτk(b)=τsτs1τ2τ1(b),\tau_{s}\tau_{s-1}\dots\tau_{2}\tau_{1}(a^{\prime})=\tau_{s}\dots\tau_{k}(a)=\tau_{s}\dots\tau_{k}(b)=\tau_{s}\tau_{s-1}\dots\tau_{2}\tau_{1}(b^{\prime}),

    implies that (a,b)ker([xy])(a^{\prime},b^{\prime})\in\ker([x\mapsto y]), so, without loss, there exists gGg\in G such that

    x\displaystyle x =ga=gτ11τk11(a)=τ11τk11(ga)\displaystyle=g\cdot a^{\prime}=g\cdot\tau_{1}^{-1}\dots\tau_{k-1}^{-1}(a)=\tau_{1}^{-1}\dots\tau_{k-1}^{-1}(g\cdot a)
    y\displaystyle y =gb=gτ11τk11(b)=τ11τk11(gb).\displaystyle=g\cdot b^{\prime}=g\cdot\tau_{1}^{-1}\dots\tau_{k-1}^{-1}(b)=\tau_{1}^{-1}\dots\tau_{k-1}^{-1}(g\cdot b).

    By Lemma 3, Hi=Gx=GgaH_{i}=G_{x}=G_{g\cdot a} and Gy=GgbG_{y}=G_{g\cdot b}. By GG-equivariance, τk(ga)=τk(gb)\tau_{k}(g\cdot a)=\tau_{k}(g\cdot b). By Claim 18, Gτk(gb)=Ggb=GyG_{\tau_{k}(g\cdot b)}=G_{g\cdot b}=G_{y}, so Gτk(ga)=GyG_{\tau_{k}(g\cdot a)}=G_{y}. Hence,

    [Gτk(ga)]Ni=[Ggb]Ni=[Gy]Ni=[K]Ni,[G_{\tau_{k}(g\cdot a)}]_{N_{i}}=[G_{g\cdot b}]_{N_{i}}=[G_{y}]_{N_{i}}=[K]_{N_{i}},

    and τk\tau_{k} satisfies part (1.) of Definition 15 with gag\cdot a and gbg\cdot b.

  2. 2.

    By GG-equivariance, τk(ha)=τk(hb)\tau_{k}(h\cdot a)=\tau_{k}(h\cdot b) for all hGh\in G. By Remark 14 (2.),

    ker(τkτk1τ2τ1)ker(τsτ1)=ker([xy]).\ker(\tau_{k}\tau_{k-1}\dots\tau_{2}\tau_{1})\subseteq\dots\ker(\tau_{s}\dots\tau_{1})=\ker([x\mapsto y]).

    Moreover, as τk1τ1AutG(X)\tau_{k-1}\dots\tau_{1}\in\mathrm{Aut}_{G}(X), we have by Remark 14 (3.), that

    |ker(τk)|=|ker(τkτk1τ2τ1)||ker([xy])|.|\ker(\tau_{k})|=|\ker(\tau_{k}\tau_{k-1}\dots\tau_{2}\tau_{1})|\leq|\ker([x\mapsto y])|.

    Therefore, ker(τk)={(ha,hb),(hb,ha):gG}{(c,c):cX}\ker(\tau_{k})=\{(h\cdot a,h\cdot b),(h\cdot b,h\cdot a):g\in G\}\cup\{(c,c):c\in X\}.

This shows that WW must contain an elementary collapsing of each possible type, and the result follows by Lemma 16 and Corollary 12. ∎

Acknowledgments

The first and second author were supported by a CONACYT Basic Science Grant (No. A1-S-8013) and a PhD CONACYT National Scholarship, respectively. Both authors sincerely thank Csaba Schneider for his kind and enriching replies to our questions on the Imprimitive Wreath Product Embedding Theorem. We also thank the anonymous referee of this paper for his insightful comments.

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