1 Introduction
Let G be a group and q a non-negative integer.
The q-tensor square G⊗qG is a particular case of the q-tensor product G⊗qH of groups G and H which act compatibly on each other; this construction was defined by Conduché and Rodrigues-Fernandez in [6], in the context of q-crossed modules (see also [10, 13, 2]).
It reduces to Brown and Loday’s non-abelian tensor product G⊗H when q=0 (cf. [4]).
For x,y∈G, we write the conjugate of y by x as yx=x-1yx; the commutator of x and y is then written as [x,y]=x-1y-1xy.
Commutators are left normed: [x,y,z]=[[x,y],z], and so on for commutators of higher weights.
For q≥1, let 𝒢^:={k^∣k∈G} be a set of symbols, one for each element of G.
According to Ellis [10], the q-tensor square G⊗qG is then defined to be the group generated by all symbols g⊗h and k^, g,h,k∈G, subject to the defining relations
(1.1)(g⊗hh1)=(g⊗h)(gh1⊗hh1),
(1.2)gg1⊗h=(gg1⊗hg1)(g1⊗h),
(1.3)(g⊗h)k^=(gkq⊗hkq),
(1.4)kk1^=k^∏i=1q-1(k⊗(k1-i))kq-1-ik^1,
(1.5)[k^,k1^]=kq⊗k1q,
(1.6)[g,h^]=(g⊗h)q
for all g,g1,h,h1,k,k1∈G.
If q=0, then we set 𝒢^=∅ to get the group generated by the symbols g⊗h, g,h∈G, subject to relations (1.1) and (1.2) only; that is, G⊗0G is the non-abelian tensor square G⊗G.
By the defining relations (1.1)–(1.6), we see that the diagonal ∇q(G)=〈g⊗g∣g∈G〉 is a central subgroup of G⊗qG.
The q-exterior square G∧qG is by definition the factor group (see [11])
G∧qG=G⊗qG∇q(G).
We usually write g∧h for the image of g⊗h in G∧qG.
There is a homomorphism
ϱ:G⊗qG→G,g⊗h↦[g,h],k^↦kq
for all g,h,k∈G.
Clearly, ∇q(G)≤Kerϱ, and we have (see for instance [2, Proposition 18] or [5, Theorem 2.12])
Kerϱ/∇q(G)≅H2(G,ℤq),
the second homology group of G with coefficients in the trivial G-module ℤq.
The image Imϱ is the subgroup G′Gq≤G, where G′ is the derived subgroup of G, generated by all commutators [g,h] with g,h∈G, and Gq is the subgroup of G generated by all q-th powers gq, g∈G.
Thus, we get the exact sequence (cf. [2, Proposition 18])
1→H2(G,ℤq)→G∧qG→G′Gq→1.
A group G is called q-perfect in case G=G′Gq.
If this is the case, then the above sequence shows that G∧qG is a q-central extension of G, and in addition it is the unique universal q-central extension of G (see [2]).
Notice that if G is q-perfect, then G⊗qG≅G∧qG.
The q-exterior square is also helpful in deciding whether or not a group G is q-capable; recall that G is q-capable if there exists a group Q such that
Z(Q)=Zq(Q) and G≅Q/Z(Q),
where Z(Q) is the center of Q and Zq(Q) is the q-center, that is, the elements of the center Z(Q) of order dividing q.
The q-exterior center of G is the subgroup of G defined by
Zq∧(G)={g∈G∣g∧x=1∈G∧qGfor allx∈G}.
In [10, Proposition 16], Ellis proved that the group G is q-capable if, and only if, Zq∧(G)=1.
So getting a presentation for the q-tensor square of a group G and for its subfactors is an interesting task.
It is known that if G is a polycyclic group, then G⊗qG is polycyclic for all q≥0 (see for instance [5]).
In [8], the authors describe algorithms to compute the non-abelian tensor square G⊗G, the exterior square G∧G and the Schur multiplier M(G), among others, for a polycyclic group G given by a consistent polycyclic presentation; the implementation of this algorithm is available in [7].
They manage to find such an algorithm to computing G⊗G by finding a presentation of the group ν(G), as introduced for instance in [15] (see also [9]), which turns out to be an extension of G⊗G by G×G.
The present paper aims to extend these algorithms to all q≥0.
Instead of ν(G), we now consider the group νq(G), defined for instance in [5, Definition 2.1].
To ease reference, we briefly describe this group early in the next section.
The paper is organized as follows.
In Section 2, we describe some basic constructs and preliminaries results.
In Section 3, we give consistent polycyclic presentations for certain q-central extensions of G, more specifically, the groups Eq(G) and 𝔈q(G).
In Section 4, we give polycyclic presentations for the second homology group H2(G,ℤq) and for the q-exterior square G∧qG; use of the group Eq(G) is made to exemplify the computation of the q-exterior center of G.
In Section 5, we provide a consistent polycyclic presentation for the group τq(G).
Finally, in Section 6, we give an algorithm to compute a polycyclic presentation for νq(G) and the q-tensor square of a polycyclic group G.
Notation is fairly standard; for basic results on group theory, see for instance [14].
In this article, all group actions are on the right.
Basic notation and structural results concerning νq(G) can be found for instance in [5].
2 Preliminary results
We begin this section by defining the group νq(G) and giving a brief description of some of its properties.
To this end, let Gφ be an isomorphic copy of G, via an isomorphism φ such that φ:g↦gφ for all g∈G.
With these data, we immediately get the group ν(G), as mentioned before, defined as
ν(G):=〈G∪Gφ∣[g,hφ]k=[gk,(hk)φ]=[g,hφ]kφfor allg,h,k∈G〉.
It is well known (see [15, 9]) that the subgroup [G,Gφ] of ν(G) is isomorphic to the non-abelian tensor square G⊗G so that the strategy of finding an appropriate representation of ν(G) can be useful to compute G⊗G and various of its relevant subfactors (see for instance [9, 15, 13, 1, 8]).
Now, for q≥1, let 𝒢^={k^∣k∈G} be a set of symbols, one for each element of G (for q=0, we set 𝒢^=∅, the empty set), and let F(𝒢^) be the free group on 𝒢^.
Write ν(G)∗F(𝒢^) for the free product of ν(G) and F(𝒢^).
As G and Gφ are embedded into ν(G), we shall identify the elements of G (respectively of Gφ) with their respective images in ν(G)∗F(𝒢^).
Denote by J the normal closure in ν(G)∗F(𝒢^) of the following elements for all k^,k1^∈𝒢^ and g,h∈G:
(2.1)g-1k^g(kg^)-1,
(2.2)(gφ)-1k^gφ(kg^)-1,
(2.3)(k^)-1[g,hφ]k^[gkq,(hkq)φ]-1,
(2.4)(k^)-1kk1^(k1^)-1(∏i=1q-1[k,(k1-i)φ]kq-1-i)-1,
(2.5)[k^,k1^][kq,(k1q)φ]-1,
(2.6)[g,hφ]-q.
Definition 2.1.
The group νq(G) is defined to be the factor group
νq(G):=(ν(G)∗F(𝒢^))/J.
Note that, for q=0, the sets of relations (2.1) to (2.6) are empty; in this case, we have ν0(G)=ν(G)∗F(𝒢^))/J≅ν(G).
Let R1,…,R6 be the sets of relations corresponding to (2.1)–(2.6), respectively, and let R be their union, R=⋃i=16Ri.
Therefore, νq(G) has the presentation
νq(G)=〈G,Gφ,𝒢^∣R,[g,hφ]k[gk,(hk)φ]-1,[g,hφ]kφ[gk,(hk)φ]-1for allg,h,k∈G〉.
The above presentation of νq(G) is a variant of the one given by Ellis in [10].
There is an epimorphism ρ:νq(G)↠G, g↦g, hφ↦h, k^↦kq.
On the other hand, the inclusion of G into ν(G) induces a homomorphism ı:G→νq(G).
We have gıρ=g, and thus ı is injective.
Similarly, the inclusion of Gφ into ν(G) induces a monomorphism ȷ:Gφ→νq(G).
These embeddings allow us to identify the elements g∈G and hφ∈Gφ with their respective images gı and (hφ)ȷ in νq(G).
Now let 𝔊 denote the subgroup of νq(G) generated by the images of 𝒢^.
By relations (2.3), 𝔊 normalizes the subgroup [G,Gφ] in νq(G), and hence we have that Υq(G):=[G,Gφ]𝔊 is a normal subgroup of νq(G).
Hence we get
νq(G)=Gφ⋅(G⋅Υq(G)),
where the dots mean internal semidirect products.
By [5, Proposition 2.9], there is an isomorphism μ:Υq(G)→G⊗qG such that [g,hφ]↦g⊗h, k^↦k^ for all g,h,k∈G and for all q≥0.
We then get (see [5, Corollary 2.11])
νq(G)≅G⋉(G⋉(G⊗qG)).
This decomposition of νq(G) is analogous to one due to Ellis in [10]; it generalizes a similar result found in [15] for q=0.
In view of the above isomorphism, unless otherwise stated, from now on, we will identify G⊗qG with the subgroup Υq(G)=[G,Gφ]𝔊≤νq(G) and write [g,hφ] in place of g⊗h for all g,h∈G.
Following [5], we write Δq(G) for the subgroup 〈[g,gφ]∣g∈G〉≤Υq(G), which, by Remark 2.2 below, is a central subgroup of νq(G).
The isomorphism μ restricts to an isomorphism
Δq(G)≅𝜇∇q(G)
and, consequently, the factor group Υq(G)/Δq(G) is isomorphic to the q-exterior square G∧qG.
In this case, as usual, we simply write g∧h to denote the coset [g,hφ]Δq(G) in G∧qG.
We shall eventually write 𝒯 to denote the subgroup [G,Gφ] of νq(G) in order to distinguish it from the non-abelian tensor square G⊗G≅[G,Gφ]≤ν(G) in the case q=0.
We also write τq(G) for the factor group νq(G)/Δq(G); thus we get
τq(G)≅((G∧qG)⋊G)⋊G.
Remark 2.2.
It should be noted that the actions of G and Gφ on Υq(G) are those induced by the defining relations of νq(G): for any elements g,x∈G, hφ,yφ∈Gφ and k^∈𝒢^, we have [g,hφ]x=[gx,(hx)φ] and (k^)x=(kx^).
In view of the isomorphism Υq(G)≅G⊗qG, these correspond to the action of G on G⊗qG, as given for instance in [10],
{(g⊗h)x=gx⊗hx,(k^)x=kx^.
Similarly, we get [g,hφ]yφ=[gy,(hy)φ] and (k^)yφ=(ky^).
In addition, for any τ∈Υq(G), (gτ)yφ=g[g,yφ]τyφ∈GΥq(G).
Similar actions are naturally induced on the q-exterior square G∧qG.
It is known that if G is polycyclic, then νq(G) is polycyclic for all q≥0, and thus, as mentioned before, G⊗qG is polycyclic.
In [5], the authors proved that, for a polycyclic group G given by a consistent polycyclic presentation, the defining relations of νq(G) can be reduced to relations among the polycyclic generators, with the only exception of relations (2.4) which have a more complicated handling characteristic.
Even without reducing relations (2.4), they were able to use the GAP System [18] to compute νq(G), G⊗qG and G∧qG for some small groups G and particular values of q.
In addition, in [16], an upper bound to the minimal number of generators of the q-tensor square of an n-generator nilpotent group of class 2, n>1, for all q>1 and q odd is given.
Our purpose in this article is to overcome in some way the difficulty of dealing with relations (2.4) and describe algorithms for deriving polycyclic presentations for the groups νq(G), G⊗qG, G∧qG and H2(G,ℤq) for all q≥0, when G is polycyclic given by a consistent polycyclic presentation.
Our approach is based on ideas of Eick and Nickel [8] for the case q=0.
The concept of a crossed pairing (biderivation) has been used in order to determine homomorphic images of the non-abelian tensor square G⊗G (see [3, Remark 3]).
We need to extend this concept in order to the context of the q-tensor square.
Definition 2.3.
Let G and L be arbitrary groups and q a non-negative integer.
A function λ:G×G×G→L is called a q-biderivation if the following properties hold:
(gg1,h,k)λ=(gg1,hg1,1)λ(g1,h,k)λ,
(g,hh1,k)λ=(g,h1,1)λ(gh1,hh1,k)λ,
((1,1,k)λ)-1(g,h,1)λ(1,1,k)λ=(gkq,hkq,1)λ,
(2.7)(1,1,kk1)λ=(1,1,k)λ∏i=1q-1{(k,(k1-i)kq-1-i,1)λ}×(1,1,k1)λ,
[(1,1,k)λ,(1,1,k1)λ]=(kq,k1q,1)λ,
(1,1,[g,h])λ=((g,h,1)λ)q
for all g,g1,h,h1,k1∈G.
By the defining relations in Definition 2.1 of νq(G), it is easy to see that a q-biderivation provides a universal property of the q-tensor square of a group G.
Proposition 2.4.
Let G and L be arbitrary groups and λ:G×G×G→L a q-biderivation.
Then there exists a unique homomorphism λ~:G⊗qG→L such that the following hold for all g,h,k∈G:
(g⊗h)λ~=(g,h,1)λ,(k^)λ~=(1,1,k)λ.
To ease reference, we include the next lemma, which relates the q-exterior square of G and the second homology group H2(G,ℤq) with an arbitrary free presentation F/R of G (see [13, 11]).
Lemma 2.5.
Let F/R be a free presentation for the group G.
Then
G∧qG≅F′Fq[R,F]Rq 𝑎𝑛𝑑 H2(G,ℤq)≅R∩F′Fq[R,F]Rq.
Thus, H2(G,ℤq)≅(G∧qG)∩Mq(G), where Mq(G)=R/[R,F]Rq is the q-multiplier.
Notice that when F is a free group, then we find that F∧qF≅F′Fq.
A similar result is also valid for projective q-crossed G-modules.
Proposition 2.6 ([13, Proposition 1.3.11]).
Given a projective q-crossed G-module δ:M→G, let F/R be a free presentation of G, with π:F→G being the natural epimorphism.
Then there exists an isomorphism
M′Mq→≅F′Fq/[R,F]Rq
such that
[m,m′]δ=[f,f′][R,F]Rq 𝑎𝑛𝑑 (mq)δ=fq[F,R]Rq,
where (m)δ=(f)π.
3 Consistent polycyclic presentations for the groups Eq(G) and 𝔈q(G), q-central extensions of G
In this section, we describe a method for computing consistent polycyclic presentations for certain q-central extensions of a polycyclic group G given by a consistent polycyclic presentation.
Our method is a generalization of the one given by Eick and Nickel in [8] for the case q=0.
Let G be a polycyclic group defined by a consistent polycyclic presentation Fn/R, where Fn is the free group in the generators g1,…,gn, and let H be a finitely presented group defined by a finite presentation Fm/S, where Fm is the free group on the generators f1,…,fm.
For our purposes, we shall assume that m≤n.
Suppose that ξ:H→G is an epimorphism such that (fi)ξ=wi, 1≤i≤m, where wi is a word in the generators g1,…,gn.
Denote by K/S the kernel Kerξ.
Thus, G≅Fm/K.
Define the groups
Eq(G):=FnRq[Fn,R] and 𝔈q(G):=FmKq[K,Fm]S
which, by construction, are q-central extensions of G.
The following result in the context of crossed modules will be helpful.
Proposition 3.1 ([13, Lemma 5.2.2]).
With the above definition, the natural epimorphism π:Eq(G)→G is a projective q-crossed G-module.
The relations of a consistent polycyclic presentation Fn/R have the form
giei=gi+1αi,i+1…gnαi,nfori∈I,
gj-1gigj=gj+1βi,j,j+1…gnβi,j,nforj<i,
gjgigj-1=gj+1γi,j,j+1…gnγi,j,nforj<iandj∉I,
for some set I⊆{1,…,n}, certain exponents ei∈ℕ for i∈I, and αi,j, βi,j,k, γi,j,k∈ℤ for all i, j and k.
To ease notation, we shall write the defining relations of G as relators, in the form r1,…,rl.
Thus, each relator rj is a word in the generators g1,…,gn; that is, rj=rj(g1,…,gn).
We now introduce l new generators t1,…,tl, one for each relator rj, and define a new group ϵ(G) to be the group generated by g1,…,gn, t1,…,tl, subject to the relators
ri(g1,…,gn)ti-1 for 1≤i≤l,
[ti,gj] for 1≤j≤n, 1≤i≤l,
Denote by Tq the q-central subgroup of ϵ(G) generated by {t1,…,tl}.
It follows directly from these relators that ϵ(G) is a q-central extension of G by Tq.
The following lemma asserts that the above relations give a polycyclic presentation of Eq(G), possibly inconsistent.
Lemma 3.2.
Let G be a polycyclic group given by a consistent polycyclic presentation Fn/R.
Then we have ϵ(G)≅Fn/Rq[R,Fn], Tq≅R/Rq[R,Fn] and ϵ(G)/Tq≅G.
Proof.
It follows by relations (a) above that ϵ(G)/Tq≅G, while, by relations (b), (c) and (d), we immediately see that Tq is a q-central subgroup of ϵ(G).
Define σ:Fn→ϵ(G) given by (gi)σ=gi, 1≤i≤n.
Relations (a) imply that σ is an epimorphism and, since ϵ(G) is a q-central extension of G, we have Rq[R,Fn]≤Ker(σ)≤R.
On the other side, there exists a well defined homomorphism from ϵ(G) to Fn/Rq[Fn,R], which is an epimorphism.
Consequently, Ker(σ)≤Rq[R,Fn]≤Ker(σ), and thus, we have ϵ(G)≅Fn/Rq[Fn,R].
Therefore, ϵ(G)≅Eq(G), where we get Tq≅R/Rq[R,Fn].
∎
By using a Smith normal form algorithm in a manner similar to that described by Eick and Nickel in [8] (see also [17, p. 424]), we can determine a consistent polycyclic presentation for Eq(G) from the (possibly inconsistent) polycyclic presentation given by Lemma 3.2.
We then get a consistent polycyclic presentation for Eq(G) in the generators g1,…,gn, t1,…,tl with the following relations:
ri(g1,…,gn)t1qi1…tlqil for 1≤i≤l,
[ti,gj] for 1≤i≤n, 1≤j≤l,
tidi for 1≤i≤l, with di∣q,
where (qij)1≤i,j≤l is an appropriate invertible matrix over ℤ.
It may happen that di=1 for some i∈{1,…,l}.
In this case, the corresponding generator ti is redundant and can be removed.
Below, we give a couple of simple examples in order to illustrate these results.
The same examples will be used in subsequent sections.
Example 3.3.
First we consider the symmetric group S3, given by the consistent polycyclic presentation
S3=〈g1,g2∣g12=1,g1-1g2g1=g22,g23=1〉.
According to the definition, we have, say for q=2,
E2(S3)=〈g1,g2,t1,t2,t3∣g12=t1,g1-1g2g1=g22t2,g23=t3,t12=1,t22=1,t32=1〉,
where t1,t2,t3 are central.
Checking for consistency, we find that t2=1.
Thus, a consistent polycyclic presentation of E2(S3) is
E2(S3)=〈g1,g2,t1,t3∣g12=t1,g1-1g2g1=g22,g23=t3,t12=1,t32=1;(t1,t3central)〉.
Example 3.4.
In this second example, we consider the infinite dihedral group, given by the consistent polycyclic presentation
D∞=〈g1,g2∣g12=1,g1-1g2g1=g2-1〉.
From this, for an arbitrary q≥2, we get
Eq(D∞)=〈g1,g2,t1,t2∣g12=t1,g1-1g2g1=g2-1t2,t1q=1,t2q=1;(t1,t2central)〉.
Checking these relations for consistency, we find that this presentation is consistent.
Now, from the polycyclic presentation of Eq(G) given earlier, we can determine a presentation for 𝔈q(G).
Lemma 3.5.
Let ς:Fm→Eq(G), given by (fi)ς=wi for 1≤i≤m, where, as before, wi=wi(g1,…,gn) is a word in the generators g1,…,gn.
Then
Proof.
(i) Notice that, by definition, Im(ς) covers G≅Eq(G)/Tq, and hence Fm/Ker(ς) is a q-central extension of G=Fm/K.
Thus, [K,Fm]Kq≤Ker(ς).
On the other hand, Fm/[K,Fm]Kq is a polycyclic q-central extension of G and, since, by construction, Eq(G) is the largest q-central extension of G with this property (by Proposition 2.6 it is a projective q-crossed G-module; see also [8, Lemma 3]), it follows that Eq(G) contains Fm/[K,Fm]Kq as a sub-factor via ς.
Thus, Ker(ς)=[Fm,K]Kq.
(ii) Now, by part (i), we get Im(ς)≅Fm[Fm,K]Kq and, by definition,
𝔈q(G)=FmS[Fm,K]Kq.
But (S)ς=S[K,Fm]Kq[K,Fm]Kq; consequently, Im(ς)(S)ς=𝔈q(G).
∎
Tuned in this way, in order to determine a presentation of 𝔈q(G), it suffices to determine generators for the subgroups Im(ς) and (S)ς of Eq(G) since standard methods for polycyclic groups can be used in order to construct a consistent polycyclic presentation for the quotient Im(ς)/(S)ς (see also [12, Chapter 8]).
Certainly, a set of generators for Im(ς) is given by w1,…,wm.
Let s1,…,sk be a set of defining relators for the finitely presented group H=Fm/S.
Then (S)ς is generated by (s1)ς,…,(sk)ς as a subgroup, once (S)ς≤Tq is central in Eq(G).
Thus, a set of generators for (S)ς can be determined by evaluating the relators s1,…,sk in Eq(G).
4 Polycyclic presentations for the q-exterior square G∧qG and for the second homology group H2(G,ℤq)
According to Lemmas 2.5 and 3.2, we have the following.
Corollary 4.1.
The following statements hold:
H2(G,ℤq)≅(Eq(G)′Eq(G)q)∩Tq.
Therefore, in order to obtain a presentation for the groups G∧qG, H2(G,ℤq), for a polycyclic group G given by a consistent polycyclic presentation, we apply standard methods to determine presentations of subgroups of polycyclic groups (see for instance [12, Chapter 8]).
By the isomorphism given in Corollary 4.1, we obtain generators for G∧qG via Eq(G).
Proposition 4.2.
The subgroup (Eq(G))′(Eq(G))q of Eq(G) is generated by the set
〈[gi,gj]ϵ,gkq∣1≤i<j≤n, 1≤k≤n〉,
where ϵ=1 if G is finite and ϵ=±1 otherwise.
Proof.
As Eq(G)=〈g1,…,gn,t1,…,tl∣ri=ti,(tiq-central)〉 is polycyclic, we immediately get
(Eq(G))′=〈[gi,gj]ϵ∣1≤i<j≤n〉,
where ϵ is as above.
Now, by a simple induction on q, we see that each power gq, g∈Eq(G), is a word in the commutators [gi,gj]ϵ and in the q-th powers gkq of the generators of G.
∎
Notice that if we consider the natural epimorphism Eq(G)↠G and choose a pre-image g~∈Eq(G) for each g∈G, then, by Proposition 2.6, we can make explicit an isomorphism β:G∧qG→(Eq(G))′(Eq(G))q such that
(g∧h)β=[g~,h~] and (k^)β=(k~)q.
Remark 4.3.
We have the following observations.
As we have seen in Section 1, G acts naturally on G∧qG via
(g∧h)k=gk∧hk,(k^)g=kg^ for allg,h,k∈G.
In addition, this action is compatible with the isomorphism β, and (g∧h)k corresponds to
[g~,h~]k=[gk~,hk~],
while (k^)g corresponds to (k~q)g=(kg~)q.
The image wk of an arbitrary element w∈G∧qG is obtained by writing w as a product of q-th powers and commutators and then computing the action of k upon each factor.
By construction, the map λ:G×G×G→G∧qG, (g,h,k)↦[g~,h~](k~)q is a q-biderivation.
Applying β, it corresponds to the q-biderivation
λ:G×G×G→(Eq(G))′(Eq(G))q,(g,h,k)↦(g∧h)k^.
Note that we can determine the image of the action of G and of the q-biderivation λ (see Proposition 2.4) in the polycyclic presentation of G∧qG by using the above remark.
Example 4.4 (Continuation of Example 3.3).
We determine S3∧2S3 by identifying it with the subgroup
(E2(S3))′(E2(S3))2=〈[g1,g2],g12,g22〉=〈w∣w6〉≅C6,
where w=g22t1.
By Remark 4.3, the image of (g1∧g2)g1 in the consistent polycyclic presentation of S3∧2S3 corresponds to the element
[g1,g2]g1 of (E2(S3))′(E2(S3))2.
In turn, evaluating this element using the relations of (E2(S3))′(E2(S3))2, we obtain the element
g1-1[g1,g2]g1=g1-1g22g1=g22=w4.
Analogously, the image of (g1,g2,g1)λ in the consistent polycyclic presentation of S3∧2S3 corresponds to the element
[g1,g2]g12 of (E2(S3))′(E2(S3))2,
and thus, using the relations of (E2(S3))′(E2(S3))2, we evaluate this element to get
[g1,g2]g12=g22t1=w.
Example 4.5 (Continuation of Example 3.4).
Now we determine D∞∧2D∞ by identifying it with the subgroup
(Eq(D∞))′(Eq(D∞))2=〈[g1,g2],g12,g22〉=〈w1,w2,w3∣w12,[w1,w2],[w1,w3],w22,[w2,w3]〉≅C2×C2×C∞,
where w1=g22, w2=t1, w3=t2.
Analogous to Example 4.4, by Remark 4.3, the image of (g1∧g2)g1 in the consistent polycyclic presentation of D∞∧2D∞ corresponds to the element [g1,g2]g1 in (Eq(D∞))′(Eq(D∞))2, that is, to
g1-1[g1,g2]g1=g1-1g22t2g1=g2-1t2=w1-1w3.
Similarly, the image of (g1,g2,g1)λ corresponds to the element [g1,g2]g12 of (Eq(D∞))′(Eq(D∞))2, which results in
[g1,g2]g12=g22t2t1=w1w2w3.
5 The q-exterior center of a polycyclic group
Our next step is to show that we can easily determine the q-exterior center of a polycyclic group G given by a consistent polycyclic presentation, using a consistent polycyclic presentation for Eq(G) and standard methods for polycyclic groups (see [12, Chapter 8]).
These techniques also extend those found in [8] for the exterior center (case q=0).
Theorem 5.1.
Let G be a polycyclic group and π:Eq(G)→G the natural epimorphism.
Then Zq∧(G)=(Z(Eq(G)))π.
Proof.
For each g∈G, let g~ be a pre-image of g in Eq(G) under the epimorphism π, i.e., (g~)π=g.
Now, [g~,a~]=1 for all a∈G if, and only if, [g~,x]=1 for all x∈Eq(G).
Indeed, given x∈Eq(G), then xπ∈G, and by assumption, [g~,xπ~]=1.
On the other hand, (xπ~)π=xπ.
Thus,
(xπ~)-1x∈Ker(π)≤Z(Eq(G)).
Therefore, we have
1=[xπ~-1x,g~]=[xπ~-1,g~]x[x,g~]=[xπ~,g~]-(xπ~)-1x[x,g~]=[xπ~,g~][x,g~]=[x,g~].
Conversely, given a∈G, we have a~∈Eq(G).
By assumption, [x,g~]=1 for all x∈Eq(G) and, in particular, for x=a~.
Thus, [a~,g~]=1 for all a∈G.
Now, remember that the map β:G∧qG→(Eq(G))′(Eq(G))q given by
(g∧h)β=[g~,h~] and (k^)β=k~q
is an isomorphism, and thus we get
Zq∧(G)={g∈G∣1=g∧a∈G∧qGfor alla∈G}={g∈G∣[g~,a~]=1for alla∈G} (by usingβ)={g∈G∣[g~,x]=1for allx∈Eq(G)}={g∈G∣g~∈Z(Eq(G))}=(Z(Eq(G)))π.∎
By Theorem 5.1, the q-exterior center of a group G given by a consistent polycyclic presentation can be easily determined: First we determine a polycyclic presentation for Eq(G) and its corresponding natural epimorphism π:Eq(G)→G.
Then we compute the center Z(Eq(G)) using standard methods for polycyclically presented groups (see [12, Chapter 8]) and, finally, we apply π to obtain Zq∧(G)=(Z(Eq(G)))π.
Example 5.2 (Continuation of Example 3.3).
It follows from the consistent polycyclic presentation of E2(S3) that Z(E2(S3))=〈t1,t3〉.
Thus, Z2∧(S3)=1 and so, as one should expect, S3 is 2-capable.
In fact, the group Q given by
Q=〈a,b∣a4=1,a-1ba=b-1,b3=1〉
has center Z(Q)=〈a2〉=Z2(Q), of order 2, and S3≅Q/Z(Q).
Example 5.3 (Continuation of Example 3.4).
It follows from the polycyclic presentation of Eq(D∞) that Zq(Eq(D∞))=〈t1,t2〉.
Thus, Zq∧(D∞)=1; hence, D∞ is q-capable for all q≥0.
Indeed, the group
Q=〈a,b∣a2q=1,a-1ba=b-1〉
has center Z(Q)=〈a2〉=Zq(Q), of order q, and D∞≅Q/Z(Q).
6 A consistent polycyclic presentation for νq(G)/Δq(G)
As seen in Section 4, we can determine a consistent polycyclic presentation Fr/U for the q-exterior square G∧qG in the generator w1,…,wr and relators, say u1,…,us.
From such a presentation, we will determine a consistent polycyclic presentation for the group νq(G)/Δq(G).
Remember that
νq(G)/Δq(G)≅(G∧qG)⋊(G×G).
According to Remark 4.3, we can determine the image of the q-biderivation λ:G×G×G→G∧qG: (g,h,1)↦(g∧h) and (1,1,k)↦k^ in the consistent polycyclic presentation we obtained for G∧qG.
Analogously, we can construct the natural action of G on the presentation found for G∧qG, which is given by (g∧h)x=gx∧hx, (k^)x=(kx^).
Recall that we are given a consistent polycyclic presentation of group G; as before, G=〈g1,…,gn∣r1,…,rl〉.
Definition 6.1.
Define τq(G) to be the group generated by g1,…,gn, g1φ,…,gnφ, w1,…,wr, subject to the following defining relations:
ri(g1,…,gn)=1 for 1≤i≤l,
ri(g1φ,…,gnφ)=1 for 1≤i≤l,
ui(w1,…,wr)=1 for 1≤i≤s,
gi-1gjφgi=gjφ((gi,gj,1)λ)-1 for 1≤i,j≤n,
gigjφgi-1=gjφ((gi-1,gj,1)λ)-1 for 1≤i,j≤n, i∉I,
gj-1wigj=wigj for 1≤i≤r, 1≤j≤n;
gjwigj-1=wigj-1 for 1≤i≤r, 1≤j≤n, j∉I;
gj-φwigjφ=wigj for 1≤i≤r, 1≤j≤n;
gjφwigj-φ=wigj-1 for 1≤i≤r, 1≤j≤n, j∉I.
Notice that we can compute the right-hand side of relations (4) and (5) as words in w1,…,wr (see Remark 4.3).
Theorem 6.2.
Let W≤τq(G) be the subgroup 〈w1,…,wr〉.
W is a normal subgroup of τq(G) and τq(G)/W≅G×G.
The presentation of τq(G) in Definition 6.1 is a consistent polycyclic presentation.
The map ψ:νq(G)→τq(G) defined by
(gi)ψ=gi,(giφ)ψ=giφ 𝑎𝑛𝑑 (k^)ψ=(1,1,k)λ
for all 1≤i≤n and all k∈G extends to a well defined homomorphism (also denoted by
ψ
) such that Kerψ=Δq(G).
Proof.
The proof is mainly based on a careful analysis of the sets of defining relations (1)–(5) of τq(G), as established in Definition 6.1.
Relations (5) tell us that W is in fact a normal subgroup of τq(G).
Relations (1), (2) and (4) imply that τq(G)/W≅G×Gφ.
In addition, relations (3) show that W is a factor of G∧qG.
Thus, τq(G) satisfies the exact sequence
G∧qG→τq(G)→G×Gφ→1.
Now, relations (5) imply that G×Gφ acts by conjugation on W, in the same way as G×G acts naturally on G∧qG.
In particular, we get [w,g]=w-1wg and, analogously, [w,hφ]=w-1whφ for all words w in w1,…,wr, all words g in g1,…,gn and all words hφ in g1φ,…,gnφ.
Furthermore, the definition of a q-biderivation and relations (4) imply [g,hφ]=(g,h,1)λ for all words g in g1,…,gn and hφ in g1φ,…,gnφ.
Part (i) then follows directly from the above considerations.
Part (ii).
Relations (1)–(5) already have the form of a polycyclic presentation.
Thus, it remains to check them for consistency.
Well, all consistency relations in the generators g1,…,gn are satisfied, once relations (1) come from a consistent polycyclic presentation of G.
Analogously, for relations (2) and (3); they say that all consistency in the generators g1φ,…,gnφ and w1,…,wr are also satisfied.
Besides that, if a consistency relation involves one generator of the w1,…,wr, then it is satisfied once G×Gφ acts on W likewise G×G acts naturally on G∧qG.
Therefore, the bottom line is really to check the consistency relations in g1,…,gn, g1φ,…,gnφ, involving mixed generators gi and gjφ.
They are
gkφ(gjgi)=(gkφgj)giforj>i,
gkφ(gjφgi)=(gkφgjφ)gifork>j,
((gjφ)ej)gi=(gjφ)ej-1(gjφgi)forj∈I,
gjφ(giei)=(gjφgi)giei-1fori∈I,
gjφ=(gjφgi-1)gifori∉I.
Consider for example the first of these relations.
Supposing that
gi-1gjgi=rij(g1,…,gn)=rij
in the defining relations of G and using the fact that λ is a q-biderivation, we get
gkφ(gjgi)=(gjgi)(gkφ)gjgi=(girij)(gkφ((gjgi,gk,1)λ)-1),
(gkφgj)gi=(gjgkφ((gj,gk,1)λ)-1)gi=gjgkφgi(((gj,gk,1)λ)-1)gi=gjgi(gkφ)gi((gj,gk,1)λgi)-1=girijgkφ((gi,gk,1)λ)-1((gjgi,gkgi,1)λ)-1=girijgkφ((gjgi,gkgi,1)λ(gi,gk,1)λ)-1=girijgkφ((gjgi,gk,1)λ)-1.
The other consistency relations can be checked by similar calculations.
Thus, we obtain that τq(G) is given by a consistent polycyclic presentation.
Part (iii) follows from (ii) and from the theory of polycyclic presentations (see for instance [12, Section 8.3]), once W, as a subgroup of τq(G), has a consistent polycyclic presentation in the generators w1,…,wr and relations u1,…,us.
Therefore, W≅G∧qG.
(iv) Since λ is a q-biderivation, all relations of νq(G) hold in τq(G).
Thus, ψ:νq(G)→τq(G) is an epimorphism, once gi,gjφ∈Im(ψ) for all 1≤i,j≤n, and if a word wi∈W≅G∧qG is a product of commutators and q-th powers, then wi∈Im(ψ) for all 1≤i≤r.
Consequently, Im(ψ)=τq(G).
Besides that, ([g,hφ])ψ=[g,hφ]=(g,h,1)λ and (k^)ψ=(1,1,k)λ for all words, g in the generators g1,…,gn, hφ in g1φ,…,gnφ and all k∈G.
Therefore, the map induced by ψ on the subgroup Υq(G) coincides with the map δ:Υq(G)→G∧qG by construction.
We then get the commutative diagram
It follows that Ker(ψ)≤Υq(G) and, consequently, Ker(ψ)=Ker(δ)=Δq(G).
This completes the proof.
∎
Example 6.3 (Continuation of Example 3.3).
According to the above result, the following polycyclic presentation is a presentation of τ2(S3)=ν2(S3)/Δ2(S3) as the group generated by g1,g2,g1φ,g2φ,w subject to the relations
g12=1, g1-1g2g1=g2-1, g23,
(g1φ)2=1, (g1φ)-1g2φg1φ=(g2φ)-1, (g2φ)3,
g1-1g1φg1=g1φ,
g1-1g2φg1=g2φw2,
g2-1g1φg2=g1φw4,
g2-1g2φg2=g2φ,
g1-1wg1=w5,
g2-1wg2=w,
(g1φ)-1wg1φ=w5,
(g2φ)-1wg2φ=w.
Example 6.4 (Continuation of Example 3.4).
Again, according to Theorem 6.2, we find that τ2(D∞)=ν2(D∞)/Δ2(D∞) has the polycyclic presentation in the generators g1,g2,g1φ,g2φ,w1,w2,w3 subject to the relations
(g1φ)2=1,
(g1φ)-1g2φg1φ=(g2φ)-1,
w1-1w2w1=w2, w22,
w1-1w3w1=w3, w32,
w2-1w3w2=w3,
g1-1g1φg1=g1φ,
g1-1g2φg1=g2φw1-1w3,
g2-1g1φg2=g1φw1w3,
g2-1g2φg2=g2φ,
g2g1φg2-1=g1φw1-1w3,
g2g2φg2-1=g2φ,
g1-1w1g1=w1-1,
g1-1w2g1=w2,
g1-1w1g1=w3,
g2-1w1g2=w1,
g2-1w2g2=w2,
g2-1w3g2=w3,
g2w1g2-1=w1,
g2w2g2-1=w2,
g2w3g2-1=w3,
(g1φ)-1w1g1φ=w1-1,
(g1φ)-1w2g1φ=w2,
(g1φ)-1w1g1φ=w3,
(g2φ)-1w1g2φ=w1,
(g2φ)-1w2(g2φ)=w2,
(g2φ)-1w3(g2φ)=w3,
g2φw1(g2φ)-1=w1,
g2φw2(g2φ)-1=w2,
g2φw3(g2φ)-1=w3.
7 A polycyclic presentation for νq(G)
We can now use the consistent polycyclic presentation of τq(G) in place of Fn/R and the finite presentation of νq(G) in place of Fm/S.
Note that the epimorphism ψ:νq(G)→τq(G) has the required form.
Theorem 7.1.
νq(G)=𝔈q(τq(G)).
Proof.
We have τq(G)=Fn/R, νq(G)=Fm/S and the epimorphism
ψ:νq(G)→τq(G) with the kernelKer(ψ)=K/S.
By definition, 𝔈q(τq(G))≅Fm/Kq[K,Fm]S.
By Theorem 6.2 the group νq(G) is a q-central extension of τq(G).
Thus, [K,Fm]Kq≤S.
Since, by Lemma 3.5, 𝔈q(τq(G))=Fm/Kq[K,Fm]S, it follows that 𝔈q(τq(G))=Fm/S=νq(G), as desired.
∎
Notice that, in the above proof, we used a finite presentation for νq(G); however, the presentation coming from Definition 2.1 is finite only if G is finite.
On the other hand, the epimorphism ψ:νq(G)→τq(G) does not depend on the finiteness of the presentation of νq(G), and thus we can consider this epimorphism.
If G is an infinite polycyclic group, then, by definition, νq(G) is given by an infinite presentation, say F/S, where F is a free group on the generators of νq(G), which we denote by X, of infinite rank, and where S is the normal closure of relations (2.1)–(2.6).
But, being polycyclic, νq(G) has a finite polycyclic presentation 〈X0∣S0〉, where X0⊆X and S is the normal closure of S0, S0¯=S.
Thus, we can use the results in Lemma 3.5, and it suffices to prove that the image of ς is generated by the elements g1,…,gn, g1φ,…,gnφ, gi^,…,gn^ and that (S)ς is generated by the defining relations of νq(G) evaluated only on the polycyclic generators of G.
Proposition 7.2.
Consider the subgroup L of Eq(τq(G)) given by
L=〈g1,…,gn,g1φ,…,gnφ,(1,1,g1)λ,…,(1,1,gn)λ〉.
Then, Im(ς)=L.
In addition, (S)ς is generated by the defining relations of νq(G) in the polycyclic generators of G and Gφ in Eq(τq(G)).
Proof.
By definition of L, to show that Im(ς)=L, it suffices to show that
(1,1,k)λ∈L for allk∈G.
We prove this by induction on the number of polycyclic generators of G.
If n=1, then G=〈g1〉, so k=g1α for some α∈ℤ.
Thus, (1,1,k)λ=(1,1,g1α)λ.
For α≥2, using relation (2.7) in Definition 2.3, we have
(1,1,k)λ=(1,1,g12)λ=(1,1,g1)λ⏟∈L∏i=1q-1(((g1,g1-i,1)λ)g1q-1-i)⏟=[g1,g1-iφ]∈L(1,1,g1)λ⏟∈L∈L.
By assuming it for α-1, then, analogously,
(1,1,k)λ=(1,1,g1α)λ=(1,1,g1)λ∏i=1q-1(((g1,g1-i(n-1),1)λ)g1q-1-i)(1,1,g1n-1)λ∈L.
In addition, again by the very definition of λ, as above, we obtain (1,1,1)λ=1 and (1,1,k-1)λ=(∏i=1q-1((x,xi,1)λ))-1((1,1,x)λ)-1, which are elements of L.
This completes the case n=1.
Suppose n≥1 and that our assertion is true for n-1.
If k=g1α1…gnαn, then the same argument used above gives
(1,1,k)λ=(1,1,g1α1…gnαn)λ=(1,1,g1α1)λ∏i=1q-1(((g1α1,(g2α2…gnαn)-i,1)λ)(g1α1)q-1-i)×(1,1,g2α2…gnαn)λ∈L.
Thus, Im(ς)=L.
∎
Therefore, having obtained a consistent polycyclic presentation of τq(G), we can extend it by adding new (q-central) generators ti, one for each relator ri of τq(G), and changing each relator ri by riti-1.
Then, we evaluate the consistency relations among the relators of νq(G) in this new presentation and apply Lemma 3.5 (ii).
The following result can be used in order to reduce the number of new generators added and the number of relators evaluated in this process.
Lemma 7.3.
It is redundant to add new generators corresponding to relations (1) and (2) in the definition of τq(G).
If these generators are not introduced, then it is redundant to evaluate the relators (1) and (2) in the definition of νq(G).
Proof.
The relators (1) and (2) in the definition of νq(G) coincide with the relators (1) and (2) in the definition of τq(G).
Therefore, if we add new generators corresponding to those relators in (1) and (2) of Definition 6.1 and then we evaluate the relators (1) and (2) in the definition of νq(G), then, as a result, we obtain the corresponding generators.
This means that the corresponding generators are eliminated in the process of the constructing the factor group as described in Lemma 3.5 (ii).
This proves the result.
∎
Example 7.4 (Continuation of Example 3.3).
We compute a polycyclic presentation of ν2(S3) as a central extension 𝔈2(τ2(S3)) of τ2(S3).
There are a lot of calculations to get such a presentation (by hand), so we omit the details.
We obtain a polycyclic presentation for ν2(S3) in the generators g1,g2,g1φ,g2φ,w,t and defining relations given by
g12=1, g1-1g2g1=g2-1, g23,
(g1φ)2=1,
(g1φ)-1g2φg1φ=(g2φ)-1,
(g2φ)3,
g1-1g1φg1=g1φw6,
g1-1g2φg1=g2φw8,
g2-1g1φg2=g1φw4,
g2-1g2φg2=g2φ,
g1-1wg1=w5,
g2-1wg2=w,
g1-1wg1=w5,
g2-1wg2=w.
From this, we get the 2-tensor square
S3⊗2S3≅〈w〉≤ν2(S3),
that is, S3⊗2S3≅ℤ12.
In addition, we immediately find that Δ2(S3)≅ℤ2.
Example 7.5.
Here we compute a polycyclic presentation for ν3(D∞) as the group generated by g1,g2, g1φ,g2φ, w1,w2 subject to the relations
(g1φ)2=1,
(g1φ)-1g2φg1φ=(g2φ)-1,
g1-1g1φg1=g1φ,
g1-1g2φg1=g2φw2-2,
g2-1g1φg2=g1φw22,
g2-1g2φg2=g2φ,
g2g1φg2-1=g1φw2-2,
g2g2φg2-1=g2φ,
g1-1w1g1=w1,
g1-1w2g1=w2-1,
g2-1w1g2=w1w22,
g2-1w2g2=w2,
g2w1g2-1=w1w2-2,
g2w2g2-1=w2,
(g1φ)-1w1g1φ=w1,
(g1φ)-1w2g1φ=w2-1,
(g2φ)-1w1g2φ=w1w22,
(g2φ)-1w2(g2φ)=w2,
g2φw1(g2φ)-1=w1w2-2,
g2φw2(g2φ)-1=w2.
According to this presentation, we find that D∞⊗3D∞≅D∞.
Notice that the computation of a presentation of νq(G) becomes relatively simple if the group G is q-perfect, according to Theorem 7.6 below.
We shall continue using the same notation as before.
More specifically, let Fn/R be a consistent polycyclic presentation for the polycyclic group G in the generators g1,…,gn, relators r1,…,rl and index set I.
Let Fr/U be a consistent polycyclic presentation for G∧qG in the generators w1,…,wr and relators u1,…,us, as found in Section 4.
We determine the image of the q-biderivation λ:G×G×G→G∧qG: (g,h,1)↦(g∧h) and (1,1,k)↦k^ in the consistent polycyclic presentation obtained for G∧qG and construct the natural action of G on that presentation found for G∧qG (as defined before: (g∧h)x=gx∧hx, (k^)x=kx^, according to Remark 4.3).
Theorem 7.6.
Let G be a polycyclic group given as above.
If G is q-perfect, then the group νq(G) is the group generated by g1,…,gn, g1φ,…,gnφ, w1,…,wr, subject to the defining relations
ri(g1,…,gn)=1 for 1≤i≤l,
ri(g1φ,…,gnφ)=1 for 1≤i≤l,
ui(w1,…,wr)=1 for 1≤i≤s,
gi-1gjφgi=gjφ{(gi,gj,1)λ}-1 for 1≤i,j≤n,
gigjφgi-1=gjφ{(gi-1,gj,1)λ}-1 for 1≤i,j≤n,
i∉I
gj-1wigj=wigj for 1≤i≤r,
1≤j≤n,
gjwigj-1=wigj-1 for 1≤i≤r,
1≤j≤n,
j∉I,
gj-φwigjφ=wigj for 1≤i≤r,
1≤j≤n,
gjφwigj-φ=wigj-1 for 1≤i≤r,
1≤j≤n,
j∉I.
Proof.
In effect, according to Definition 6.1, the above presentation is the same as that of τq(G).
By Theorem 6.2, τq(G)≅νq(G)/Δq(G).
If G is q-perfect, then we have Δq(G)=1 and so τq(G)≅νq(G).
Consequently, the given presentation is a presentation of νq(G).
∎
7.1 A polycyclic presentation for the q-tensor square of a polycyclic group
By all we have seen, a method for determining a consistent polycyclic presentation for the q-tensor square G⊗qG from a given consistent polycyclic presentation of G consists of the following steps.
Algorithm 7.7.
Determine a consistent polycyclic presentation for
the subgroup Υq(G) of νq(G).
Step (a) is a direct application of the method for computing a central extension in Section 3.
If G=Fn/R is a consistent polycyclic presentation of G, then we can determine a consistent polycyclic presentation for Eq(G)=Fn/[Fn,Rq]Rq, and we get G∧qG as the subgroup (Eq(G))′(Eq(G))q.
Step (b) is thus a direct application of the method developed in Section 4.
Step (c) is obtained by another application of the method for computing a central extension in Section 3, in order to compute 𝔈q(τq(G)) which, by Theorem 7.1, is isomorphic to νq(G).
Finally, step (d) is an application of the standard method for computing presentations of subgroups of polycyclically presented groups.
and
