# Representation of Group Isomorphisms: The Compact Case

Authors: Marita Ferrer; Margarita Gary; Salvador Hernández
Journal: Journal of Function Spaces
 (2015)

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## Abstract

LetG be a discrete group and let A and B be two subgroups of G-valued continuous functions defined on two 0-dimensional compact spaces X and
Y. A group isomorphism H defined between A and B is called separating when,
for each pair of maps f,g∈A satisfying that f-1eG∪g-1eG=X, it holds that
Hf-1eG∪Hg-1eG=Y. We prove that under some mild conditions every biseparating isomorphism H:A→B can be represented by means of a continuous function h:Y→X as a weighted composition operator. As a consequence we establish
the equivalence of two subgroups of continuous functions if there is a biseparating
isomorphism defined between them.


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## Body

## 1. Introduction

LetG be a discrete group and let X and Y be topological spaces. If A and B are groups of G-valued continuous maps, we say that A and B areequivalent when there is a homeomorphism h:Y→X and a continuous map ω:Y→Aut(G) such that the map H:A→B defined as Hf(y)=ω[y](f(h(y))), f∈A, y∈Y, is a group isomorphism of A onto B. Here Aut(G) is equipped with the pointwise convergence topology. We say in this case that H is represented as aweighted composition operator. There are many results that are concerned with the representation of linear operators as weighted composition maps and the equivalence of specific groups of continuous functions in the literature, which is vast in this regard. We will only mention here the classic Banach-Stone theorem that, when G is the field of real or complex numbers, establishes that if the Banach spaces of continuous functions C(X,G) and C(Y,G) are isometric, then they are equivalent and the isometry can be represented as a weighted composition map (cf. [1–10]). Another important example appears in coding theory, where the well-known MacWilliams Equivalence Theorem asserts that when G is a finite field and X and Y are finite sets, two codes (linear subspaces) A and B of GX and GY, respectively, are equivalent when they are isometric for the Hamming metric (see [11–13]). This result has been generalized to convolutional codes in [14] and it also makes sense in other areas, for example, functional analysis and linear dynamical systems (cf. [14–18]). The main motivation of this research has been to extend MacWilliams Equivalence Theorem to more general settings and explore the possible application of these methods to the study of convolutional codes or linear dynamical systems. However, throughout this paper, we will only deal with 0-dimensional compact spaces X and Y and a discrete group G. We will look at the possible application of this abstract approach elsewhere. There are many precedents in the study of the representation of group homomorphisms for group-valued continuous functions. Among them, the following ones are relevant here (cf. [19–26]). Most basic facts and notions related to topological properties may be found in [27].Throughout this paper all spaces are assumed to be Hausdorff 0-dimensional and compact. That is to say, we only deal with Hausdorff compact spaces that contain an open basis consisting of closed and open (clopen) subsets. If X is a topological space and G is a topological (discrete) group, we denote by C(X,G) the group of continuous functions from X to G. Let eG be the neutral element of G. For f∈C(X,G) thecozero of f is the set coz(f)={x∈X:f(x)≠eG} and thezero of f is the set Zf=X∖coz(f). Since G is discrete, coz(f) and Z(f) are both clopen subsets of X.LetA be a subgroup of C(X,G) and set Z(A)≝{Z(f):f∈A}. Then σ(Z(A)) denotes the minimum collection of subsets containing Z(A) that is closed under finite unions and intersections (resp., coz(A)≝{coz(f):f∈A} and σ(coz(A)) denotes the minimum collection of subsets containing coz(A) that is closed under finite unions and intersections). It is said that Aseparates points inX if for every pair of distinct points (x1,x2)∈X×X there is a map f∈A such that f(x1)≠eG and f(x2)=eG. It is said that Astrongly separates points inX if, for every pair (x1,x2)∈X×X, there are maps f1,f2∈A such that xi∈coz(fi), 1≤i≤2, and coz(f1)∩coz(f2)=∅.As groupG is finite, it could be thought that coz(A) and σ(coz(A)) should coincide. However, this is misleading as the following example shows. Obviously, by Morgan’s laws, it suffices to prove that Z(A)≠σ(Z(A)).Example 1.
LetG be the group with two elements {0,1} and take X=GN equipped with the product topology. Clearly X is a compact space homeomorphic to the Cantor set, which is 0-dimensional. Let pm∈C(X,G) be the mth projection on X and set A as the subgroup of C(X,G) generated the collection {pm:m∈N}. Take n∈N greater than 2 and Hn≝{(xj)∈X:x1=⋯=xn=0}. It is easily verified that Hn belongs to σ(Z(A)). However, take an arbitrary but fixed element f∈A. Denote by ek the element in X such that pm(ek)=0 if m≠k and pk(ek)=1 and set sup⁡(f)≝{k∈N:f(ek)=1}. If 1,…,n⊊sup⁡(f), then there is l<n such that l∉sup⁡(f). Therefore f(el)=0, which yields el∈Z(f)∖Hn. On the other hand, if {1,…,n}⊆sup⁡(f), take 1≤n1<n2≤n. We have that f(en1+en2)=0, which again yields en1+en2∈Zf∖Hn. In either case, we obtain that Z(f)≠Hn. Thus Hn∈σZA∖Z(A), which completes the proof.Denote byδx:A→G the evaluation map; that is, δx(f)=f(x) for every f∈A. It is said that A ispointwise dense when δx(A) is dense in G for all x∈X. It is said that A⊆C(X,G) iscontrollable if for every f∈A and D1,D2∈σ(Z(A)) such that D1∩D2=∅ there exist a subset U⊆σ(coz(A)) and a function g∈A such that D1⊆U⊆X∖D2, gD1=f|D1, and gZ(f)∪(X∖U)≡eG.We now formulate our main results.Theorem 2.
LetX and Y be 0-dimensional compact Hausdorff spaces and let G be a discrete group. Suppose that A and B are controllable and pointwise dense subgroups of G-valued continuous functions separating the points of X and Y, respectively. If H:A→B is a biseparating group isomorphism of A onto B, then there are continuous maps h:Y→X and ω:Y→Aut(G) satisfying the following properties.(1)
h is a homeomorphism of Y onto X.(2)
For eachy∈Y and every f∈A it holds
(1)Hf(y)=ωyfhy.(3)
H is a continuous isomorphism with respect to the pointwise convergence topology.(4)
H is a continuous isomorphism with respect to the compact open topology.Corollary 3.
LetX and Y be 0-dimensional compact Hausdorff spaces and let G be a discrete group. Suppose that A and B are controllable and pointwise dense subgroups of G-valued continuous functions separating the points of X and Y, respectively. If there is a biseparating group isomorphism H of A onto B, then A and B are equivalent.We notice that some of the requirements we have imposed on the previous results could be relaxed in general. However, this would take us to a wider setting in general. For instance, if we assume thatA does not separate points in X, then there must be some point x∈X such that f(x)=eG for all f∈A; then we can replace X by the largest subspace XA⊆X where A separates points. This subspace XA is open but not necessarily closed in general. Thus, the study of subgroups that does not separate points leads us to consider locally compact spaces. We will discuss these spaces in a subsequent paper.


## 2. Basic Notions and Facts

The following lemma is easily verified using a standard compactness argument. Recall that we are assuming that all spaces are compact and0-dimensional.Lemma 4.
LetD be a family of clopen subsets of X that is a subbase for the closed subsets of X. Then for every disjoint nonempty closed subsets A and B of X there are two disjoint subsets DA and DB in σ(D) such that A⊆DA and B⊆DB.Example 5.
A specific example where Lemma4 applies is given when A separates points in X, where D=Z(A).Next proposition shows that the notions of separating and strongly separating points are equivalent for controllable subgroups.Proposition 6.
IfA is a controllable subgroup of C(X,G) that separates the points of X, then A strongly separates the points of X.Proof.
SetD=Z(A) and take two distinct elements x1≠x2 in X. Applying Lemma 4, since A separates the points of X, there are D1,D2∈σ(D) such that x1∈D1, x2∈D2, and D1∩D2=∅. Take fi∈A such that fi(xi)≠eG, i∈{1,2}. Since A is controllable, we have U1∈σ(coz(A)) and g1∈A such that D1⊆U1⊆X∖D2, g1|D1=f1|D1, and Z(f1)∪(X∖U1)⊆Z(g1). Therefore x1∈coz(g1)⊆U1.
Applying the fact thatD is a subbase of closed subsets and using a compactness argument, we deduce that there is D1′∈σ(D) such that U1⊆D1′ and D1′∩D2=∅. By the controllability of A again, we have U2∈σ(coz(A)) and g2∈A such that D2⊆U2⊆X∖D1′, g2|D2=f2|D2, and Z(f2)∪(X∖U2)⊆Z(g2). Therefore x2∈coz(g2)⊆U2, which yields coz(g1)∩coz(g2)=∅. This completes the proof.Definition 7.
LetA be a subgroup of C(X,G) and let φ:A→G be a group homomorphism. A subset A of X is said to be asupport for φ if, given f∈A with A⊆Z(f), it holds that φ(f)=eG.Some basic properties of support subsets are shown in the next proposition. Observe that sinceA⊆A¯X⊆Z(f), we may assume without loss of generality that all support subsets are closed and therefore compact subsets of X.Proposition 8.
Letφ:A→G be a nonnull group homomorphism. The following assertions hold.(1)
X is a support for φ.(2)
IfA is a support for φ then A≠∅.(3)
IfA is a support for φ and A⊆B then B is a support for φ.(4)
LetA be a support for φ and f,g∈A such that fA=gA. Then φ(f)=φ(g).If, in addition, A is controllable and separates points in X, then we have the following.(5)
LetA and B be supports for φ; then A∩B≠∅.Proof.
Assertions (1)–(4) are obvious.
(5) LetA and B be closed supports for φ. Suppose A∩B=∅. Since A separates points in X, by Lemma 4 and Proposition 6, there are two disjoint subsets DA and DB in σ(Z(A)) containing A and B, respectively. Take f∈A such that φ(f)≠eG. Applying the controllability of A, we obtain U∈σ(coz(A)) and g∈A such that A⊆DA⊆U⊆X∖DB⊆X∖B, gDA=fDA, and Z(f)∪(X∖U)⊆Z(g). This yields a contradiction as the evaluation of φ(g) shows. Indeed, since g(x)=f(x) for all x∈A, by item (4) it follows that φ(g)=φ(f)≠eG. On the other hand, we have that g(x)=eG for all x∈B, which imples φ(g)=eG. This contradiction completes the proof.Definition 9.
A mapH:A→B is said to beseparating or disjointness preserving, if, for each pair of maps f,g∈A satisfying that f-1(eG)∪g-1(eG)=X, it holds that Hf-1(eG)∪Hg-1(eG)=Y (equivalently, if coz(f)∩coz(g)=∅ implies coz(Hf)∩coz(Hg)=∅ for all f,g∈A). In case H is bijective, the map H is said to bebiseparating if both H and H-1 are separating. Remark that this definition makes sense and extends naturally to maps φ:A→G.Remark 10.
Originally, separating maps for scalar-valued continuous functions were defined as those mapsH such that fg=0 implies HfHg=0. If one interprets the null element 0 as the identity e of the group, then separating maps could be defined as those maps that satisfy f(x)g(x)=e for all x∈X which implies Hf(y)Hg(y)=e for all y∈Y or that H(f-1)=H(f)-1 for all f∈A. Obviously this definition would be vacuous here since we are assuming that H is a group homomorphism throughout this paper. Thus, this definition would take us to the more general question of representing group isomorphisms defined between groups of continuous functions without any further requirement. Unfortunately, this is not possible in general. Indeed, a remarkable result due to Milutin [28] establishes that if K is an uncountable compact metric space, then C(K,C) is linearly isomorphic to C([0,1],C). Therefore, it is essential to impose some extra algebraic or geometrical condition on the isomorphisms H in order to be able to represent them by continuous maps defined on the compact spaces K and [0,1]. In this sense, the connection with separating isomorphisms stems from [8], where it was proved that every linear isometry is a separating isomorphism.Next we will see that every nonnull separating group homomorphismφ:A→G, where A is controllable, has the smallest possible compact support set. For that purpose, set
(2)S={A⊆X:Aisacompactsupportforφ}.
There is a canonical partial order that can be defined on S: A≤B, A,B∈S, if and only if B⊆A. A standard argument shows that (S,≤) is an inductive set and, by Zorn’s lemma, S contains a ⊆-minimal element. Furthermore, this minimal element is in fact a minimum because of the next proposition.Proposition 11.
IfA separates points in X, then the minimum element S consists of a singleton.Proof.
LetS be a minimal element of S, which is nonempty by Proposition 8. Suppose now that there are two different elements x1,x2 that are contained in S. As X is Hausdorff, we can select two disjoint open subsets V1,V2 in X such that x1∈V1 and x2∈V2. Since S is minimal, the compact subset S∖Vi is not a support for φ. Hence, there are fi∈A such that S∖Vi⊆Z(fi) and φ(fi)≠eG, 1≤i≤2. Since φ is separating, it follows that A=coz(f1)∩coz(f2) is a nonempty compact subset of X.
We claim thatS∩A=∅. Otherwise, pick up an element a∈S∩A. If a∈V1 then a∈S∖V2 and a∈Z(f2), which is a contradiction; but if a∉V1 then a∈S∖V1, which implies that a∈Z(f1) and we get a contradiction again. Therefore S∩A=∅. By Lemma 4, we can take two disjoint sets DS,DA∈σ(Z(A)) such that S⊆DS and A⊆DA. Applying the fact that A is controllable to DS, DA and f1, we obtain a set U∈σ(coz(A)) and a map g∈A such that S⊆DS⊆U⊆X∖DA⊆X∖A, gS=f1|S, and g|Z(f1)∪(X∖U)≡eG. Then U∩A=∅, φ(g)=φ(f1)≠eG, and A⊆Z(g). Since φ is separating the set B=coz(g)∩coz(f2)≠∅, take b∈B. Then b∈coz(f2) and b∈coz(g)⊆coz(f1); that is, b∈coz(f1)∩coz(f2)=A. As a consequence g(b)=eG, which is a contradiction. Therefore we have proved that S=1. This completes the proof.


## 3. Proof of Main Results

Along this sectionA (resp., B) is a controllable subgroup of C(X,G) (resp., C(Y,G)) that separates points in X (resp., Y).LetH:A→B be a separating group homomorphism. The maps δy∘H are separating group homomorphisms of A into G for all y∈Y. Furthermore, since A is controllable and separates points in X, we can apply Proposition 11, in order to obtain that each partial ordered set Sy={A⊆X:Aisacompactsupportforδy∘H} has a minimum element, which is a singleton denoted by h(y). Therefore, by sending y∈Y to h(y)∈X for every y∈Y, we have defined thesupport map of Y into X that is associated with H.Proposition 12.
LetH:A→B be a separating group homomorphism. Then the support map h has the following properties.(1)
h is continuous.(2)
If∅≠A⊊X is open, f∈A, and A⊆Z(f), then h-1(A)⊆Z(Hf).(3)
h(coz(Hf))⊆coz(f).(4)
IfH is one-to-one, then h is onto.Moreover, when H is a bijection of A onto B, we have, in addition, the following.(5)
IfH is biseparating, then h is a homeomorphism of Y onto X.Proof.
(1) Let(yd)d∈D be a net in Y converging to y∈Y. By a standard compactness argument, we may assume without loss of generality that (h(yd))d converges to x∈X. Reasoning by contradiction, suppose h(y)≠x. Since X is Hausdorff, we can take two disjoint open neighborhoods Vh(y) and Vx of h(y) and x, respectively. Using convergence, there is d1∈D such that h(yd)∈Vx for all d≥d1.
As every support subset forδy′∘H contains h(y′), for all y′∈Y, the subset X∖Vh(y) may not be a support for δy∘H. Therefore there exists f∈A such that X∖Vh(y)⊆Z(f) and Hf(y)≠eG. Moreover, since H(f) is a continuous function, the net (Hf(yd))d converges to Hf(y) and, because G is discrete, there is d2≥d1 such that Hf(yd)≠eG for all d≥d2. If we take and index d3∈D such that d3≥d2, then the subset X∖Vx may not be a support for δyd3∘H. Thus, there exists f3∈A such that X∖Vx⊆Z(f3) and Hf3(yd3)≠eG. This means that yd3∈coz(Hf3)∩coz(Hf) and, since H is a separating map, it follows that coz(f3)∩coz(f)≠∅. But coz(f3)⊆Vx, which is disjoint from coz(f)⊆Vh(y). This is a contradiction that completes the proof.
(2) Let∅≠A⊊X be an open subset, f∈A, and A⊆Z(f). If we take y∈h-1(A), then X∖A is a nonempty compact subset that is not a support for δy∘H. Then there is g∈A such that X∖A⊆Z(g) and Hg(y)≠eG. Therefore coz(g)⊆A and coz(f)⊆X∖A. Since H is separating, we have that coz(Hg)∩coz(Hf)=∅. Therefore Hf(y)=eG.
(3) Takex∈h(coz(Hf)). Then x=h(y) for some y∈coz(Hf). Since h(y) is a support for δy∘H, it follows that x∉Z(f) or, equivalently, we have x∈coz(f).
(4) Supposeh(Y)≠X and take x∈X such that x∉h(Y). Since h is continuous and Y is compact, we have that h(Y) is a proper compact subset of X. Applying Lemma 4, there are two disjoint subsets Dx,DY∈σ(Z(A)) such that x∈Dx and h(Y)⊆DY. Moreover, as A separates points in X, there exists f∈A such that f(x)≠eG. Again, by the controllability of A, we may take a subset U⊆σ(coz(A)) and a map g∈A such that x∈Dx⊆U⊆X∖DY⊆X∖h(Y), g|Dx=f|Dx, and Z(f)∪(X∖U)⊆Z(g). As a consequence g(x)=f(x)≠eG, h(Y)⊆Z(g), and H(f)(y)=eG for all y∈Y. Then Hf≡eG. Since H is an injective group homomorphism, this yields f≡eG, which is a contradiction.
(5) SinceX and Y are compact spaces, it will suffice to prove that h is one-to-one. Suppose there are two elements y1≠y2 in Y such that h(y1)=h(y2)=x0. Since B separates the points of Y, there are D1′,D2′∈σ(Z(B)) such that y1∈D1′, y2∈D2′, and D1′∩D2′=∅. Since H(A)=B, there is fi∈A such that H(fi)(yi)≠eG, for i∈{1,2}. Since B is controllable, there are U1∈σ(coz(B)), g1∈A such that y1∈D1′⊆U1⊆Y∖D2′, Hg1|D1′=Hf1|D1′, and Hg1|Z(Hf1)∪(Y∖U1)≡eG. Since D2′∩U1=∅, applying a compactness argument, there is D1′′∈σ(Z(A)) such that U1⊆D1′′ and D2′∩D1′′=∅. Now, by the controllability of B, there are U2∈σ(coz(B)) and g2∈A such that y2∈D2′⊆U2⊆Y∖D1′′, Hg2|D2′=Hf2|D2′, and Hg2|Z(Hf2)∪(Y∖U2)≡eG. Hence, since coz(Hgi)⊆Ui, U1∩U2=∅, and H is biseparating, it follows that coz(g1)∩coz(g2)=∅. On the other hand Hgi(yi)=Hfi(yi)≠eG, i∈{1,2}, and by item (3) above, we have that h(y1)=h(y2)=x0∈coz(g1)∩coz(g2), which is a contradiction.We have just seen how a separating group homomorphismH has associated a continuous map h that assigns to each point y∈Y the support subset of δy∘H. Our next goal now is to obtain a complete representation of H by means of the support map h. Having this in mind, set
(3)Gh(y)≝Im⁡δhy={fh(y):f∈A}
which is a subgroup of G for all y∈Y, and denote by Hom(Gh(y),Gy) the set of all group homomorphisms on Gh(y) into Gy. Consider now the set
(4)G≝⋃y∈YHom(Gh(y),Gy).We can think of the elements ofG as partial functions on G. That is, functions α:Dom⁡(α)⊆G→G whose domain is a (not necessarily proper) subset of G. Since the group G is discrete, we can equip G with the product (or pointwise convergence) topology as follows.Let[α;g1,…,gn]≝{β∈GG:α(gi)=β(gi),gi∈G,1≤i≤n} be a basic neighborhood of a map α∈GG. If now α is a partial map, we can restrict this basic neighborhood to G by letting [α;g1,…,gn] be the set of all partial maps β:Dom⁡(β)⊆G→G such that g1,…,gn∈Dom⁡(β) and α(gi)=β(gi),  1≤i≤n. It is easily verified that this procedure extends the pointwise convergence topology on G (cf. [29]).With this notation, we defineω:Y→G by
(5)ωyfhy≝Hfy
for each y∈Y. We will see next that ω is well defined and continuous.Proposition 13.
With the terminology established above, the following assertions are true.(1)
ω[y] is a well-defined group homomorphism of Gh(y) into Gy for all y∈Y.(2)
ω is continuous when G is equipped with the pointwise convergence topology.Proof.
(
1
) In order to prove that ω[y] is well defined, take f1,f2∈A such that f1(h(y))=f2(h(y)). By Proposition 8, we have ω[y](f1(h(y)))=Hf1(y)=Hf2(y)=ω[y](f2(h(y))). The verification that ω[y] is a group homomorphism is easy and it is left to the reader.
(
2
) Let (yd)d∈D be a net converging to y in Y. If g is an arbitrary element in Dom⁡(ω[y])=Gh(y), then g=f(h(y))∈Gh(y) for some f∈A. Since h continuous by Proposition 12, f∘h∈C(Y,G) and (f∘h)-1(g) is a clopen neighborhood of y∈Y. Since G is discrete, there is d1(g)∈D such that f(h(yd))=g for all d≥d1(g). Thus g∈Dom⁡(ω[yd]) for all d≥d1(g). In like manner, as ω[y](g)=Hf(y)=gy∈G and Hf∈C(Y,G), we have that (Hf)-1(gy) is a clopen neighborhood of y. As a consequence there is d2≥d1(g) such that Hf(yd)=gy for all d≥d2. Thus ω[yd](g)=Hf(yd)=gy=Hf(y)=ω[y](g) for all d≥d2. This means that the net (ω[yd])d∈D converges to ω[y] in the pointwise convergence topology over G.Observe that sinceG is discrete, the compact subsets in G are all finite. Therefore, we have also proved that ω is also continuous if we consider the compact open topology on G. We are in position now of establishing a main result in this paper.Theorem 14.
LetH:A→B be a separating group homomorphism. Then there are continuous maps
(6)h:Y⟶X,ω:Y⟶⋃y∈YHom(Gh(y),Gy)
satisfying the following properties.(1)
For eachy∈Y and every f∈A it holds
(7)Hf(y)=ωyfhy.(2)
H is continuous with respect to the pointwise convergence topology.(3)
H is continuous with respect to the compact open topology.(4)
IfH is biseparating bijection of A onto B, then h is a homeomorphism.Proof.
Item(1) is consequence of the definition of ω and (2) follows from assertion (2) in Proposition 13. Thus, only (3) needs to be verified.
Let(fd)d∈D∈A be a net converging to eG in the compact open topology. If K is a compact subset of Y, then h(K) is a compact subset in X by the continuity of h. Therefore (fd)d is eventually the constant function eG on h(K). Applying (1), it follows that (Hfd)d∈D is eventually eG on K, which completes the proof.Corollary 15.
LetH:A→B be a separating group homomorphism, where A is pointwise dense. Then there are continuous maps h:Y→X and ω:Y→End(G) satisfying the following properties.(1)
For eachy∈Y and every f∈A it holds
(8)Hf(y)=ωyfhy.(2)
H is continuous with respect to the pointwise convergence topology.(3)
H is continuous with respect to the compact open topology.(4)
IfH is a biseparating bijection of A onto B, then h is a homeomorphism.We are now in position of establishing the results formulated at Introduction.Proof of Theorem2.
After Theorem14 and Corollary 15, we only need to verify that ω[y]∈Aut(G) for all y∈Y. Applying Corollary 15 to H-1, we obtain maps
(9)ρ:X⟶EndG,k:X⟶Y
such that for every x∈X and g∈B, we have
(10)H-1g(x)=ρxgkx.
Thus, for everyf∈A and x∈X, we have
(11)fx=H-1∘(Hf)(x)=ρxHfkx=ρxωkxfhkx
which means that the support subset of δx∘(H-1∘H) is both x and h(k(x)). Since the minimum support is unique, this means that h∘k=idX, which implies that k is a right inverse of h. Analogously, for every g∈B and y∈Y, we have
(12)gy=H∘(H-1g)(y)=ωyH-1ghy=ωyρhygkhy
which means that the support subset of δy∘(H∘H-1) is both y and k(h(y)). Again, this implies that k is a left inverse of h. Since k is both a left and right inverse of h, it follows that k=h-1. Therefore
(13)fx=H-1∘(Hf)(x)=ρxHfkx=ρxωkxfx,gy=H∘(H-1g)(y)=ω[y]H-1ghy=ωyρhygy.
Applying the former equality tox=h(y), it follows that ρ[h(y)]∘ω[y]=idG for all y∈Y, and from the latter, we also have that ω[y]∘ρ[h(y)]=idG. This means that ω[y] has left and right inverse and, therefore, it is an automorphism on G, which completes the proof.Proof of Corollary3.
It follows directly from Theorem2.



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