A generalization of weak
commutativity between two isomorphic groups

Bruno C. R. Lima; Said N. Sidki

doi:10.1515/jgth-2016-0053

Article Publicly Available

A generalization of weak commutativity between two isomorphic groups

Bruno C. R. Lima and Said N. Sidki

Published/Copyright: December 2, 2016

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From the journal Journal of Group Theory Volume 20 Issue 4

Abstract

The operator χ of weak commutativity between isomorphic groups H and Hψ, defined by

χ⁢(H)=〈H,Hψ∣[h,hψ]=1⁢ for all ⁢h∈H〉,

is known to preserve group properties such as finiteness, solvability and polycyclicity. We introduce here the group construction

ℰ⁢(H)=〈H,Hψ∣[[h1,h2ψ],h3-1⁢h3ψ]=1⁢ for all ⁢h1,h2,h3∈H〉⁢.

The group ℰ⁢(H) maps onto χ⁢(H) and onto H⊗H, the non-abelian tensor square. The operator ℰ preserves solvability and preserves polycyclicity provided the abelianization HH′ is finite. Moreover, if H is perfect, then ℰ⁢(H) is perfect and ℰ⁢(H)≅χ⁢(H). Furthermore, ℰ⁢(H) is a finite group if and only if H is a finite perfect group.

1 Introduction

The concept of weak permutability between two groups was introduced in 1980 [9] and a finiteness criterion result was established for a group generated by two such groups. In Section 4 of the same work, the notion of weak commutativity motivated the definition of the group

χ⁢(H)=〈H,Hψ∣[h,hψ]=1⁢ for all ⁢h∈H〉⁢.

It was shown then that χ, when interpreted as an operator acting on groups, preserves the group properties: finite P-group for any finite set P of primes, solvable, finite nilpotent, perfect. Later, it was proven that if H is finitely generated nilpotent of nilpotency class c, then χ⁢(H) is nilpotent and bounds for the nilpotency degree of χ⁢(H) were found in terms of c (see [3]). Recently, the properties polycyclic and polycyclic-by-finite were added to this list [5]. Moreover, a study of homological and homotopical properties of χ⁢(H) have lead to proving that χ preserves the property of solvable of type FP∞ (see [4]).

The definition of χ⁢(H) implies that the relations

[h1,h2ψ]h3ψ=[h1,h2ψ]h3

(equivalently, [[h1,h2ψ],h3-1⁢h3ψ]=1) hold for all h1,h2,h3∈H. These same relations form an essential part of the definition of the non-abelian tensor square

H⊗H=〈h1⊗h2∣⁢h1⁢h2⊗h3=((h1)h2⊗(h3)h2)⁢(h2⊗h3),

h1⊗h2⁢h3=(h1⊗h3)⁢((h1)h2⊗(h3)h2)

for all h1,h2,h3∈H〉,

by Brown–Loday in 1987 [2] and of the definition of

ν⁢(H)=〈H,Hψ∣⁢[h1,h2ψ]h3ψ=[h1,h2ψ]h3=[h1h3,(h2h3)ψ]

for all h1,h2,h3∈H〉

by Rocco in 1991 [7] who proved the subgroup [H,Hψ] of ν⁢(H) to be isomorphic to H⊗H.

We focus here on the similarity between these constructions by defining the group

ℰ(H)=〈H,Hψ∣[[h1,h2ψ],h3-1h3ψ]=1 for all h1,h2,h3∈H)〉.

The group ℰ⁢(H) admits an automorphism h↔hψ of order 2, which will be denoted by the same symbol ψ.

It is easy to see that if H is cyclic of order 2, then ℰ⁢(H) is the infinite dihedral group. Though ℰ fails to preserve finiteness and nilpotency, still it preserves many features of χ. Some of our main results are gathered together in

Theorem 1.

The following statements hold.

Let H be a solvable group with derived series of length k. Then ℰ⁢(H) is a solvable group of length at most k+2.
Suppose H is a polycyclic group. Then ℰ⁢(H) is polycyclic if and only if the abelianization H/H′ is a finite group.
Suppose H is a perfect group. Then ℰ⁢(H) is perfect and ℰ⁢(H)≅χ⁢(H).
The group ℰ⁢(H) is finite if and only if H is a finite perfect group.

2 Preliminaries

We use standard notation as can be found in [8].

2.1 Some epimorphisms

We start by defining four epimorphisms of ℰ⁢(H) and stating some of their properties:

ρ:ℰ⁢(H)→H,h↦h,hψ↦h for all ⁢h∈H,

with

ker⁡(ρ)=ℒ⁢(H)=[H,ψ]=〈h-1⁢hψ∣h∈H〉,ℰ⁢(H)=ℒ⁢(H)⋅H,ℒ⁢(H)∩H=1⁢;

and

π:ℰ⁢(H)→H×H,h↦(h,1),hψ↦(1,h) for all ⁢h∈H,

with

ker⁡(π)=𝒟⁢(H)=[H,Hψ]=〈[h1,h2ψ]∣h1,h2∈H〉,ℰ⁢(H)=(𝒟⁢(H)⋅H).Hψ,𝒟⁢(H)∩(H⋅Hψ)=1⁢;

and

δ:ℰ⁢(H)→χ⁢(H),h↦h,hψ↦hψ for all ⁢h∈H,

with

ker⁡(δ)=Δ⁢(H)=〈[h,hψ]∣h∈H〉ℰ⁢(H);

and

ξ:ℰ⁢(H)→ν⁢(H),h↦h,hψ↦hψ for all ⁢h∈H⁢.

The subgroups ℒ⁢(H),𝒟⁢(H),Δ⁢(H) are ψ-invariant. In addition to these, we have the normal ψ-invariant subgroups of ℰ⁢(H)

𝒲⁢(H)=𝒟⁢(H)∩ℒ⁢(H),

ℛ⁢(H)=[[H,ℒ⁢(H)],Hψ]⁢.

We mention two more normal subgroups

ℒ1⁢(H)=[ℒ⁢(H),H],ℒ2⁢(H)=[ℒ⁢(H),Hψ]⁢;

these are interchanged by ψ.

Maintaining the notation used for subgroups of χ⁢(H), we write

ℒ⁢(H)δ=L⁢(H),𝒟⁢(H)δ=D⁢(H),𝒲(H)δ =W(H),

ℛ⁢(H)δ=R⁢(H),ℒ1⁢(H)δ=L1⁢(H),ℒ2(H)δ =L2(H).

The subgroup ℛ⁢(H) is key to understanding the structure of ℰ⁢(H). We will prove later

Theorem 2.

Let ξ:E⁢(H)→ν⁢(H) be as defined above. Then ker⁡(ξ)=R⁢(H).

Remark 1.

(i) It follows directly from the definition of ℰ⁢(H) that

[𝒟⁢(H),ℒ⁢(H)]=1

and therefore 𝒲⁢(H) is central in the product 𝒟⁢(H)⁢ℒ⁢(H). We conclude from

[hψ,h]=[h⁢(h-1⁢hψ),h]=[h-1⁢hψ,h]∈𝒟⁢(H)∩ℒ⁢(H)=𝒲⁢(H)

that

Δ⁢(H)≤𝒲⁢(H)⁢.

(ii) Using [9, Proposition 4.1.4], the quotient group ℰ⁢(H)𝒲⁢(H)(≅χ⁢(H)W⁢(H)) embeds into the triple product H×H×H by

τ:𝒲⁢(H)⁢h→(h,h,1),𝒲⁢(H)⁢hψ→(1,h,h)

for all h∈H. Thus,

τ:𝒲⁢(H)⁢(h-1⁢hψ)→(h-1,1,h),

𝒲⁢(H)⁢[h1,h2ψ]→(1,[h1,h2],1),

𝒲⁢(H)⁢([h1-1⁢h1ψ,h2])→([h1-1,h2],1,1),

𝒲⁢(H)⁢([h1-1⁢h1ψ,h2ψ])→(1,1,[h1,h2])

for all h,h1,h2∈H.

2.2 Commutator relations

It is convenient to denote h-1⁢hψ by [h,ψ].

Lemma 1.

The following statements hold.

Let h1,h2,h3∈H. Then:
1. [h1,ψ] commutes with [h2,h3ψ],
2. [h1,h2ψ]=[ψ,h2,h1]⁢[h1,h2] (thus 𝒟⁢(H)≤[ℒ⁢(H),H]⁢H′),
3. in ℛ⁢(H),
  [h1,h3⁢(h3ψ)-1,h2ψ]h3ψ=[h1,h2ψ]-h3ψ⁢[h1h3,(h2h3)ψ]⁢.
Let w=w⁢(z1ϵ1,z2ϵ2,…,znϵn) be a word in z1ϵ1,z2ϵ2,…,znϵn of length n, where zi∈H, ϵk∈{1,ψ,}. Also, let w′=w⁢(z1,z2,…,zn). Then
[h1,h2ψ]w=[h1,h2ψ]w′⁢ for all ⁢h1,h2∈H⁢.
Elements of 𝒲⁢(H) have the form
[h1,h2ψ]⁢⋯⁢[hs,hs+1ψ],
where h1,h2,…,hs+1∈H and s is an odd number such that
[h1,h2]⁢⋯⁢[hs,hs+1]=1.

Proof.

(1) (i) The assertion follows from the definition of ℰ⁢(H).

(ii) Note that

[h1,h2ψ]=[h1,h2⁢[h2,ψ]]=[h1,[h2,ψ]]⁢[h1,h2][h2,ψ]

and since by item (i) [h1,h2ψ] commutes with [ψ,h2], we obtain

[h1,h2ψ]=[h1,[h2,ψ]][ψ,h2]⁢[h1,h2]=[ψ,h2,h1]⁢[h1,h2]⁢.

(iii) We compute

[h1h3,(h2h3)ψ]=[h1h3,(h2ψ)h3ψ]=[h1h3⁢(h3ψ)-1,h2ψ]h3ψ

=[h1⁢[h1,h3⁢(h3ψ)-1],h2ψ]h3ψ

=([h1,h2ψ][h1,h3⁢(h3ψ)-1]⁢[[h1,h3⁢(h3ψ)-1],h2ψ])⁢.h3ψ

Since [h1,h2ψ]∈𝒟⁢(H) and [h1,h3⁢(h3ψ)-1]∈ℒ⁢(H), it follows that

[h1h3,(h2h3)ψ]=([h1,h2ψ]⁢[[h1,h3⁢(h3ψ)-1],h2ψ])h3ψ

=[h1,h2ψ]h3ψ⁢[[h1,h3⁢(h3ψ)-1],h2ψ]h3ψ

and therefore,

[[h1,h3⁢(h3ψ)-1],h2ψ]h3ψ=[h1,h2ψ]-h3ψ⁢[h1h3,(h2h3)ψ]⁢.

(2) By item (i),

[h1,h2ψ]z1⁢z2ψ=[h1,h2ψ]z1⁢z2⁢.

The more general statement follows by induction on n.

(3) Let ω=[h1,h2ψ]⁢⋯⁢[hs,hs+1ψ]∈𝒟⁢(H). Then by item (ii),

ω=[ψ,h2,h1]⁢[h1,h2]⁢⋯⁢[ψ,hs+1,hs]⁢[hs,hs+1]=c⁢[h1,h2]⁢⋯⁢[hs,hs+1],

where c∈ℒ⁢(H). Since H∩ℒ⁢(H)=1, we obtain

ω∈𝒲⁢(H)⇔[h1,h2]⁢⋯⁢[hs,hs+1]=1⁢.∎

2.3 The subgroups ℒ1⁢(H),ℒ2⁢(H)

Recall

ℒ1⁢(H)=[ℒ⁢(H),H],ℒ2⁢(H)=[ℒ⁢(H),Hψ]⁢.

By Remark 1 (ii),

τ:𝒲⁢(H)⁢[h1,h2ψ]→(1,[h1,h2],1),

𝒲⁢(H)⁢([h1-1⁢h1ψ,h2])→([h1-1,h2],1,1),

𝒲⁢(H)⁢([h1-1⁢h1ψ,h2ψ])→(1,1,[h1,h2])

for all h1,h2∈H. Therefore

τ:𝒟⁢(H)𝒲⁢(H)→1×H′×1,

𝒲⁢(H)⁢ℒ1⁢(H)𝒲⁢(H)→H′×1×1,

𝒲⁢(H)⁢ℒ2⁢(H)𝒲⁢(H)→1×1×H′

are epimorphisms.

Lemma 2.

We have

ℒ1⁢(H)=ℒ⁢(H)∩(𝒟⁢(H)⋅H′),

ℒ2⁢(H)=ℒ⁢(H)∩(𝒟⁢(H)⋅(H′)ψ),

ℒ1⁢(H)∩ℒ2⁢(H)=𝒲⁢(H)⁢.

Proof.

Since

[h1ψ⁢h1-1,h2]=[h1ψ,h2]h1-1⁢[h1-1,h2]

holds for all h1,h2∈H, we conclude

ℒ1⁢(H)=[ℒ⁢(H),H]≤𝒟⁢(H)⁢H′≤ℒ⁢(H)∩(𝒟⁢(H)⁢H′)⁢.

By item (ii) of the previous lemma,

𝒟⁢(H)≤[ℒ⁢(H),H]⁢H′=ℒ1⁢(H)⁢H′.

Therefore,

𝒲⁢(H)≤ℒ⁢(H)∩(𝒟⁢(H)⁢H′)≤ℒ⁢(H)∩(ℒ1⁢(H)⁢H′)

≤ℒ1⁢(H)⁢(ℒ⁢(H)∩H′)=ℒ1⁢(H)⁢.

Hence,

ℒ1⁢(H)=ℒ⁢(H)∩(𝒟⁢(H)⁢H′)⁢.

Apply ψ to ℒ1⁢(H), to obtain

ℒ2⁢(H)=ℒ⁢(H)∩(𝒟⁢(H)⁢(H′)ψ)⁢.

Clearly,

𝒲⁢(H)=ℒ⁢(H)∩𝒟⁢(H)⊆ℒ1⁢(H)∩ℒ2⁢(H).

From [9, Lemma 4.1.10 (iii)], we know that the intersection L1⁢(H)W⁢(H)∩L2⁢(H)W⁢(H) is trivial in χ⁢(H)W⁢(H)(≅ℰ⁢(H)𝒲⁢(H)); therefore,

ℒ1⁢(H)∩ℒ2⁢(H)=𝒲⁢(H)⁢.∎

3 The groups ℛ⁢(H) and ν⁢(H)

Proposition 1.

The subgroup R⁢(H) satisfies the following properties:

ℛ⁢(H)=[ℒ1⁢(H),Hψ]=[ℒ2⁢(H),H],
[𝒲⁢(H),ℰ⁢(H)]⊆ℛ⁢(H)⊆𝒲⁢(H).

Proof.

(i) By definition,

ℛ⁢(H)=[H,ℒ⁢(H),Hψ]=[ℒ1⁢(H),Hψ].

Since [𝒟⁢(H),ℒ⁢(H)]=[H,Hψ,ℒ⁢(H)]=1, we obtain, by the Hall–Witt identity, that

[Hψ,ℒ,H]=[H,ℒ,Hψ]

and consequently that ℛ⁢(H)=[ℒ2⁢(H),H].

(ii) We note

ℛ⁢(H)⊆ℒ1⁢(H)∩ℒ2⁢(H)=𝒲⁢(H)⁢;

therefore,

[𝒲⁢(H),H],[𝒲⁢(H),Hψ]≤ℛ⁢(H),

[𝒲⁢(H),ℰ⁢(H)]⊆ℛ⁢(H)⁢.∎

Proof of Theorem 2.

Recall

ν⁢(H)=〈H,Hψ∣[h1,h2ψ]h3ψ=[h1,h2ψ]h3=[h1h3,(h2h3)ψ],h1,h2,h3∈H〉⁢,

and

ξ:ℰ⁢(H)→ν⁢(H),h↦h,hψ↦hψ for all ⁢h∈H⁢.

By Lemma 1 (iii), the commutator relations

[[h1,h3⁢(h3ψ)-1],h2ψ]=(([h1,h2ψ]h3ψ)-1⁢[h1h3,(h2h3)ψ])(h3ψ)-1

hold in R⁢(H). Therefore, R⁢(H)≤ker⁡(ξ) and ξ induces an epimorphism

ξ¯:ℰ⁢(H)ℛ⁢(H)→ν⁢(H).

On the other hand, since both equalities

[h1,h2ψ]h3⁢ψ=[h1,h2ψ]h3, ⁢[h1,h2ψ]h3ψ=[h1h3,(h2h3)ψ]

hold in ℰ⁢(H) modulo ℛ⁢(H), we obtain the epimorphism

ξ′:ν⁢(H)→ℰ⁢(H)ℛ⁢(H)

defined by h→ℛ⁢(H)⁢h, hψ→ℛ⁢(H)⁢hψ. The composition ξ¯⁢ξ′ is the identity map and thus, ℰ⁢(H)ℛ⁢(H)≅ν⁢(H). With this the proof of the theorem is finished. ∎

4 Restrictions and inductions of ℰ

Proposition 2.

Let K,H1⩽H≤E⁢(H). Then:

[K,H1ψ] and [Kψ,H1]⁢⊲⁢〈H1,H1ψ〉,
〈K,Kψ〉∩𝒟⁢(H)=[K,Kψ],
〈K,Kψ〉∩ℒ⁢(H)=[K,ψ].

Proof.

(i) Since H1ψ normalizes [K,H1ψ] and

[k,h1ψ]y=[k,h1ψ]yψ=[k,h1ψ]-1⁢[k,h1ψ⁢yψ]

hold for all k∈K and h1,y∈H1, we conclude that H1 normalizes [K,H1ψ]. Analogously, [H1,Kψ]⁢⊴⁢〈H1,H1ψ〉.

(ii) Denote 〈K,Kψ〉 by K^. It is clear that [K,Kψ]⊆K^∩𝒟⁢(H). On the other hand, consider the epimorphism π:ℰ⁢(H)→H×H. Since ker⁡(π)=𝒟⁢(H), we have that π induces an epimorphism

K^K^∩𝒟⁢(H)→K×K⁢.

Consider the natural epimorphism

K×K→K^[K,Kψ]⁢.

The composition of the last two epimorphisms leads to

K^K^∩𝒟⁢(H)→K^[K,Kψ]

defined by (K^∩𝒟⁢(H))⁢x↦[K,Kψ]⁢x for all x∈K^. Therefore,

K^∩𝒟⁢(H)⊆[K,Kψ].

(iii) The proof here is similar to that of the last item, by considering the epimorphism ρ:ℰ⁢(H)→H and by taking the composition of the maps

K^K^∩ℒ→K→K^[K,ψ]⁢.∎

Proposition 3.

Let ϕ:H→K be an epimorphism, N=ker⁡(ϕ). Then ϕ extends naturally to an epimorphism ϕ^:E⁢(H)→E⁢(K) such that

ϕ^⁢(𝒟⁢(H))=𝒟⁢(K), ϕ^⁢(ℒ⁢(H))=ℒ⁢(K),
ker⁡(ϕ^)=〈N,Nψ〉⁢[N,Hψ]⁢[Nψ,H],
ker⁡(ϕ^∣𝒟⁢(H))=[N,Hψ]⁢[Nψ,H].

Proof.

(i) The extension ϕ^ of ϕ is determined by hϕ^=hϕ and (hψ)ϕ^=(hϕ)ψ for all h∈H. Then

ϕ^⁢(𝒟⁢(H))=ϕ^⁢([H,Hψ])=[ϕ^⁢(H),ϕ^⁢(Hψ)]=[K,Kψ]=𝒟⁢(K)

and

ϕ^(ℒ(H))=ϕ^([H,ψ])=[ϕ^(H),ψ])=[K,ψ].

(ii) Let M=〈N,Nψ〉⁢[N,Hψ]⁢[H,Nψ]. We have that M≤ker⁡(ϕ^), because N,Nψ⊆ker⁡(ϕ^). By the previous proposition, M is normal in ℰ⁢(H). The map

μ:K∪Kψ→ℰ⁢(H)M,k→M⁢ϕ-1⁢(k),kψ→M⁢(ϕ-1⁢(k))ψ

is well-defined, since N,Nψ⊆M. The restrictions of μ to K and Kψ are homomorphisms and they extend uniquely to a homomorphism

μ∗:K∗Kψ→ℰ⁢(H)/M.

Since the relations

[k1,k2ψ]k3=[k1,k2ψ]k3ψ for all ⁢k1,k2,k3∈K

are preserved by μ∗, it induces a homomorphism ℰ⁢(K)→ℰ⁢(H)M. On the other hand, since M≤ker⁡(ϕ^), we have a homomorphism ϕ~:ℰ⁢(H)/M→ℰ⁢(K) such that ϕ~⁢(M⁢h)=ϕ^⁢(h) and ϕ~⁢(M⁢hψ)=ϕ^⁢(hψ) for all h∈H. The composition of μ∗ and ϕ~ gives us

ϕ~⁢μ∗⁢(k)=ϕ~⁢(M⁢ϕ-1⁢(k))=k,ϕ~⁢μ∗⁢(kψ)=ϕ~⁢(M⁢(ϕ-1⁢(k))ψ)=kψ

for all k∈K. So, ϕ~⁢μ∗=1ℰ⁢(K), which shows that ϕ~ is an isomorphism.

(iii) This item follows from

𝒟⁢(H)∩ker⁡ϕ^=𝒟⁢(H)∩(〈N,Nψ〉⁢[N,Hψ]⋅[H,Nψ])

=[N,Nψ]⁢[N,Hψ]⁢[Nψ,H]

=[N,Hψ]⁢[Nψ,H]⁢.∎

Proposition 4.

Let H and K be normal subgroups of a group G such that

G=H⊕K.

Then

ℰ⁢(G)=〈H,Hψ〉⁢[H,Kψ]⁢[Hψ,K]⁢〈K,Kψ〉,

ℰ⁢(H)≅〈H,Hψ〉,

ℰ⁢(K)≅〈K,Kψ〉⁢.

Proof.

By the previous proposition, the projection ϕ:G→H can be extended to an epimorphism ϕ^:ℰ⁢(G)→ℰ⁢(H) such that

ker⁡(ϕ^)=〈K,Kψ〉⁢[H⁢K,Kψ]⁢[Hψ⁢Kψ,K]⁢.

It follows that

[H⁢K,Kψ]=[H,Kψ]⁢[K,Kψ]⁢,

since by Proposition 2 the subgroups K, Kψ, H and Hψ normalize [H,Kψ]. Similarly, it follows that

[Hψ⁢Kψ,K]=[Hψ,K]⁢[K,Kψ]⁢.

Thus,

ℰ⁢(G)≅〈K,Kψ〉⁢[H,Kψ]⁢[Hψ,K]⁢ℰ⁢(H)⁢.

But as ϕ^⁢(〈H,Hψ〉)=ℰ⁢(H), and ℰ⁢(H) maps onto 〈H,Hψ〉, it follows that

ker⁡(ϕ^)∩〈H,Hψ〉=1,ℰ⁢(H)≅〈H,Hψ〉.

We conclude

ℰ⁢(G)≅〈K,Kψ〉⁢[H,Kψ]⁢[Hψ,K]⁢〈K,Kψ〉⁢.

The isomorphism ℰ⁢(K)≅〈K,Kψ〉 follows in a similar manner. ∎

5 The augmentation ideal of ℤ⁢(H/H′)

Let G~=ℤ⁢(H/H′)⋊H denote the semidirect product of the group ring ℤ⁢(H/H′) of the abelianization of H by the group H, where H acts on ℤ⁢(H/H′) by right multiplication

(∑xH′⁢h⁢H′⁢h)h1=∑xH′⁢h⁢ ⁢H′⁢h⁢h1

for all h,h1∈H. Denote the elements of G~ as pairs (w,h) with w∈ℤ⁢(H/H′), h∈H. Identify the complement H with {0}×H, write h¯ for the coset H′⁢h for any h∈H and let u=(1¯,1)∈G~. Also, let 𝒜ℤ⁢(H/H′) be the augmentation ideal of the group ring ℤ⁢(H/H′) and let G be the subgroup of G~ generated by 𝒜ℤ⁢(H/H′) and H. Note that H′ acts trivially on ℤ⁢(H/H′).

Proposition 5.

The map ε:H∪Hψ→G~ defined by h→h, hψ→hu for all h∈H extends to an epimorphism ε:E⁢(H)→G~. Moreover, if H is abelian, then ker⁡(ε)=L⁢(H)′, the commutator subgroup of L⁢(H).

Proof.

Consider

[H,u]=〈[h,u]∣h∈H〉⩽G~.

It is easy to see that

[h,u]=h-1⁢hu=u-h⁢u=(1¯-h¯,1)⁢.

Therefore, [H,u]=𝒜ℤ⁢(H/H′)×{1} and G=〈H,Hu〉. Since ℤ⁢(H/H′) is abelian and H′ acts trivially on ℤ⁢(H/H′), we find that

[[h1,h2u],[h,u]]=[h1-1⁢u-1⁢h2-1⁢u⁢h1⁢u-1⁢h2⁢u,[h,u]]

=[u-h1⁢(h1-1⁢h2-1⁢u⁢h1⁢h2)⁢u-h2⁢u,[h,u]]

=[u-h1⁢uh2⁢h1⁢[h1,h2]⁢u-h2⁢u,u-h⁢u]

=1⁢.

Thus ε extends to an epimorphism ℰ⁢(H)→𝒜ℤ⁢(H/H′)⋊H.

By [9, Theorem 2.1.1], G=𝒜ℤ⁢(H)⋅His isomorphic to the group

〈H,Hψ∣[H,ψ]′=1〉

and so, when H is abelian, we obtain ker⁡(ε)=ℒ⁢(H)′. ∎

Next, we specialize H to a finite cyclic group.

Corollary 1.

Let H=〈x〉 be a cyclic group of order n. Then

ℰ⁢(H)≅ℤn-1⋊H⁢.

Proof.

We need to show ker⁡(ε)=1; that is, ℒ⁢(H) is abelian. We note that

1=[xn,ψ]=[x,ψ]xn-1⁢[x,ψ]xn-2⁢⋯⁢[x,ψ],

[x,ψ]xn-1=[x,ψ]-xn-2⁢⋯⁢[x,ψ]-1⁢;

therefore,

ℒ⁢(H)=〈[x,ψ],[x,ψ]x2,…,[x,ψ]xn-2〉⁢.

We compute

[xi,xψ]=[xi,x.x-1xψ]=[xi,[x,ψ]]=[x,ψ]-xi[x,ψ] for all i∈ℤ.

Since [𝒟⁢(H),ℒ⁢(H)]=1, it follows that

1=[[xi,xψ],[x,ψ]]=[[x,ψ]-xi,[x,ψ]]

and thus ℒ⁢(H) is an abelian group. ∎

6 ℰ of solvable groups

Proof of Theorem 1 (i).

Since H is a solvable group with derived length k, we apply [9, Corollary 4.1.8] to conclude χ⁢(H)(k+1)=1. As ker⁡(δ)=Δ⁢(H) is abelian, we conclude further that ℰ⁢(H)(k+2)=1 and this ends the proof. ∎

6.1 Abelian groups

Given a group G we denote its center by Z⁢(G).

Proposition 6.

Let H be an abelian group. Then:

we have
𝒟⁢(H)=𝒲⁢(H)=[ℒ⁢(H),H]=[ℒ⁢(H),Hψ],ℛ⁢(H)=[𝒟⁢(H),H]=[ℒ⁢(H),H,H],
ℒ⁢(H) is nilpotent of class at most 2,
ℒ⁢(H)′⊆𝒟⁢(H)⊆Z⁢(ℒ⁢(H)),ℒ⁢(H)′⊆Z⁢(ℰ⁢(H)).

Proof.

(i) By Lemma 2,

ℒ1⁢(H)=[ℒ⁢(H),H]=ℒ⁢(H)∩𝒟⁢(H)⁢H′,

ℒ2⁢(H)=[ℒ⁢(H),Hψ]=ℒ⁢(H)∩𝒟⁢(H)⁢H′⁣ψ⁢.

Since H is abelian group, we have that

𝒲⁢(H)=ℒ⁢(H)∩𝒟⁢(H)=ℒ1⁢(H)=ℒ2⁢(H)⁢.

Since 𝒟⁢(H)⊆[ℒ⁢(H),H]⁢H′ and H abelian, we obtain

𝒟⁢(H)⊆[ℒ⁢(H),H]⊆ℒ⁢(H)∩𝒟⁢(H)=𝒲⁢(H)⁢;

therefore, 𝒟⁢(H)=[ℒ⁢(H),H]=[ℒ⁢(H),Hψ]. From this last equality, we obtain

ℛ=[H,ℒ⁢(H),Hψ]=[𝒟⁢(H),Hψ]=[𝒟⁢(H),H]=[ℒ⁢(H),H,H]⁢.

(ii) It is clear that H centralizes ℒ⁢(H) modulo [ℒ⁢(H),H](=𝒟⁢(H)) and Hψ centralizes ℒ⁢(H) modulo [ℒ⁢(H),Hψ](=𝒟⁢(H)); therefore,

ℒ⁢(H)′⊆[ℰ⁢(H),ℒ⁢(H)]⊆𝒟⁢(H)⁢.

On the other hand

[𝒟⁢(H),ℒ⁢(H)]=[ℒ⁢(H),H,ℒ⁢(H)]=[H,ℒ⁢(H),ℒ⁢(H)]=1⁢;

therefore,

[ℒ⁢(H),ℒ⁢(H),H]=[ℒ⁢(H)′,H]=1⁢.

Proceeding in the same way, we obtain

1=[𝒟⁢(H),ℒ⁢(H)]=[ℒ⁢(H),Hψ,ℒ⁢(H)]=[Hψ,ℒ⁢(H),ℒ⁢(H)]

and

[ℒ⁢(H),ℒ⁢(H),Hψ]=[ℒ⁢(H)′,Hψ]=1⁢.

We have reached the desired conclusion:

ℒ⁢(H)′⊆Z⁢(ℰ⁢(H))⁢.∎

Proposition 7.

Let H be an abelian group. Then

ℒ(H)′=〈[aψ,b][a,bψ] for all a,b∈H〉.

Proof.

Consider l1,l2∈ℒ⁢(H). Then

l1=[h1,ψ]⁢[h2,ψ]⁢⋯⁢[hs,ψ],

l2=[h1′,ψ]⁢[h2′,ψ]⁢⋯⁢[hr′,ψ]

for some s,r∈ℕ and h1,h2,…,hs,h1′,h2′,…,hr′∈H.

Since ℒ⁢(H)′≤𝒟⁢(H) and [ℒ⁢(H),𝒟⁢(H)]=1 we have

[l1,l2]=∏i=1s∏j=1r[[hi,ψ],[hj′,ψ]]⁢.

However,

[[hi,ψ],[hj′,ψ]]=[hi-1⁢hiψ,hj′-1⁢hj′ψ]

=[hi-1,hj′-1⁢hj′ψ]hiψ⁢[hiψ,hj′-1⁢hj′ψ]

=[hi-1,hj′ψ]hiψ⁢[hiψ,hj′-1]hj′ψ

=[hi-1,hj′ψ]hi⁢[hiψ,hj′-1]hj′

=[hi,hj′ψ]-1⁢[hiψ,hj′]-1

=[hj′ψ,hi]⁢[hj′,hiψ],

which proves the assertion. ∎

Proposition 8.

Let H be an abelian group and let a,b∈H, have orders n and m, respectively. Then the element [a,bψ]⁢[aψ,b] has order a divisor of gcd⁡(n,m).

Proof.

From

1=[an,bψ]=[a,bψ]an-1⁢[a,bψ]an-2⁢⋯⁢[a,bψ]⁢,

we obtain

1=[aψ,b](an-1)ψ⁢[aψ,b](an-2)ψ⁢⋯⁢[aψ,b]⁢,

since[𝒟⁢(H),ℒ⁢(H)]=1. Also, since 𝒟⁢(H)=𝒲⁢(H), by multiplying together these two equations, we obtain

1=[a,bψ]an-1⁢[a,bψ]an-2⁢⋯⁢[a,bψ]⁢[aψ,b](an-1)⁢[aψ,b](an-2)⁢⋯⁢[aψ,b]

=([a,bψ]⁢[aψ,b])(an-1)⁢([a,bψ]⁢[aψ,b])(an-2)⁢⋯⁢[a,bψ]⁢[aψ,b]

=([a,bψ]⁢[aψ,b])n⁢,

since ℒ′≤Z⁢(ℰ⁢(H)). Similarly, ([a,bψ]⁢[aψ,b])m=1; therefore, the order of [a,bψ]⁢[aψ,b] divides gcd⁡(n,m). ∎

The next corollary follows directly from the above proposition.

Corollary 2.

If H is a finite abelian group, then so is L⁢(H)′.

6.2 Polycyclic groups

A. McDermott defined in his thesis [6] the subgroup of the free product H∗Hψ,

K=〈[h1,h2ψ]h3⁢[h1h3,h2h3ψ]-1∣h1,h2,h3∈H〉H∗Hψ

which is clearly related to our group ℛ⁢(H). The following result is a consequence of [6, Theorem 2.2.8].

Theorem 3.

Let H be a polycyclic group with polycyclic generating sequence S={ai∣1≤ai≤n}. Then the set of generators of K can be reduced:

K=〈[ai,ajψ]ak⁢[aiak,ajakψ]-1∣ai,aj,ak∈S〉H∗Hψ⁢.

Proposition 9.

Let T be a transversal of H′ in H. Then

ℛ⁢(H)=〈[h1,h2ψ]h3⁢[h1h3,h2h3ψ]-1∣h1,h2,h3∈H〉T⁢.

Proof.

Define the subgroup

𝒥⁢(H)=〈[h1,h2ψ]h3⁢[h1h3,h2h3ψ]-1∣h1,h2,h3∈H〉ℰ⁢(H)

of ℛ⁢(H), normal in ℰ⁢(H). We recall the epimorphism ξ:ℰ⁢(H)→ν⁢(H) defined by h↦h, hψ↦hψ for all h∈H, with kernel ℛ⁢(H). Then ξ induces

ξ¯:ℰ⁢(H)𝒥⁢(H)→ν⁢(H).

On the other hand, the map σ:ν⁢(H)→ℰ⁢(H)J⁢(H) defined by

h↦𝒥⁢(H)⁢h,hψ↦𝒥⁢(H)⁢hψ

extends to an epimorphism σ¯:ν⁢(H)→ℰ⁢(H)J⁢(H), and the relations

[h1,h2ψ]h3ψ=[h1,h2ψ]h3=[h1h3,(h2h3)ψ] for all ⁢h1,h2,h3∈H

hold in ℰ⁢(H)𝒥⁢(H). Since σ¯⁢ξ¯ is the identity on ℰ⁢(H)𝒥⁢(H), we have

ℛ⁢(H)=𝒥⁢(H).

By Lemma 1 (ii),

ℛ⁢(H)≤𝒲⁢(H)≤Z⁢(𝒟⁢(H))⁢;

therefore,

ℛ⁢(H)=〈[h1,h2ψ]h3⁢[h1h3,(h2h3)ψ]-1∣h1,h2,h3∈H〉H⁢,

and

[ℛ⁢(H),H′]=[ℛ⁢(H),𝒟⁢(H)]=1⁢.

Thus

ℛ⁢(H)=〈[h1,h2ψ]h3⁢[h1h3,h2h3ψ]-1∣h1,h2,h3∈H〉T⁢.∎

Proposition 10.

Let H be a polycyclic group, with polycyclic generating sequence S={a1,a2,…,an} and let T be a transversal of H′ in H. Then

ℛ⁢(H)=〈[ai,ajψ]ak⁢[aiak,ajakψ]-1∣ai,aj,ak∈S〉T⁢.

Proof.

Considering the natural epimorphism ϕ:H∗Hψ→ℰ⁢(H), we have that ϕ⁢(K)=ℛ⁢(H). By the above proposition,

ℛ⁢(H)=〈[ai,ajψ]ak⁢[aiak,ajakψ]-1∣ai,aj,ak∈S〉ℰ⁢(H)

=〈[ai,ajψ]ak⁢[aiak,ajakψ]-1∣ai,aj,ak∈S〉T⁢.∎

Proof of Theorem 1 (ii).

In one direction, let ℰ⁢(H) be polycyclic. Then, on recalling the epimorphism ε:ℰ⁢(H)→Aℤ⁢(H/H′)⁢H, we conclude that Aℤ⁢(H/H′) is finitely generated and therefore H/H′ is finite. In the other direction, on letting H/H′ be finite, we obtain that R⁢(H) is a finitely generated abelian group. Since by [1], ν⁢(H) is polycyclic and since ℰ⁢(H)ℛ⁢(H)≅ν⁢(H), we conclude that ℰ⁢(H) is polycyclic.

This ends the proof of part (ii) of Theorem 1. ∎

7 ℰ of perfect groups

When H is a perfect group, one finds a complete description of χ⁢(H) in [9, Section 4.4], where it is shown that χ⁢(H) is a certain stem extension of H×H×H by the Schur multiplier M⁢(H) of H.

Proof of Theorem 1 (iii).

(a) Since ℰ⁢(H)𝒟⁢(H)⁢L⁢(H)≅HH′, it follows that

ℰ⁢(H)=𝒟⁢(H)⁢ℒ⁢(H)

and therefore, 𝒲⁢(H)≤Z⁢(ℰ⁢(H)). From the formula

ℰ⁢(H)′=𝒟⁢(H)′⁢ℒ⁢(H)′⁢[𝒟⁢(H),ℒ⁢(H)],

we derive

ℰ⁢(H)′=𝒟⁢(H)′⁢ℒ⁢(H)′.

By [9, Lemma 4.4.6], χ⁢(H) and its subgroups D=[H,Hψ],L=[H,ψ] are all perfect groups. We conclude

Δ⁢(H)≤𝒲⁢(H)≤Z⁢(ℰ⁢(H)),

𝒟⁢(H)=Δ⁢(H)⁢𝒟⁢(H)′,

ℒ⁢(H)=Δ⁢(H)⁢ℒ⁢(H)′,

ℰ⁢(H)=𝒟⁢(H)⁢ℒ⁢(H)=Δ⁢(H)⁢𝒟⁢(H)′⁢ℒ⁢(H)′=ℰ⁢(H)′⁢.

(b) Consider ℰ⁢(H)¯=ℰ⁢(H)𝒟⁢(H)′. Then

𝒟⁢(H)¯(=[H,Hψ]¯)=𝒲⁢(H)¯,

which is contained in the center of ℰ⁢(H)¯. Therefore,

[Hψ¯,H¯,Hψ¯]=[H¯,Hψ¯,Hψ¯]=1

and by the Witt identity,

[Hψ¯,Hψ¯,H¯]=1,

[(H′)ψ¯,H¯]=[Hψ¯,H¯]=1⁢.

Thus,

𝒟⁢(H)=𝒟⁢(H)′⁢, a stem extension of ⁢H⁢.

D⁢(H)W⁢(H)≅H and W⁢(H)≅M⁢(H);

that is, D⁢(H) is a total covering of H. As H is perfect, D⁢(H) is the total covering of H. Therefore the map γ from the generating set {[h,hψ]∣h∈H} of D⁢(H) into the generating set {[h,hψ]∣h∈H} of D⁢(H) defined by

[h,hψ]↦[h,hψ],

extends to an epimorphism γ:D⁢(H)→D⁢(H). Hence, the composition

δ⁢γ:D⁢(H)→D⁢(H)

is the identity and Δ⁢(H)=1 follows.

This ends the proof of part (iii) of Theorem 1. ∎

Proof of Theorem 1 (iv).

Assume that H is a finite group. Since ℰ⁢(H) maps onto Aℤ⁢(H/H′)⋊H, on supposing ℰ⁢(H) finite, we obtain H=H′. Reciprocally, on supposing H perfect, we obtain by the previous theorem, ℰ⁢(H)≅χ⁢(H).

This proves the final part of Theorem 1. ∎

Communicated by Dessislava H. Kochloukova

Acknowledgements

The authors would like to thank the referee for suggestions which improved the readability of the original text.

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Received: 2016-4-11

Revised: 2016-9-14

Published Online: 2016-12-2

Published in Print: 2017-7-1

Articles in the same Issue

https://doi.org/10.1515/jgth-2016-0053