INFINITE TWO-GENERATOR GROUPS OF CLASS TWO AND THEIR NON-ABELIAN TENSOR SQUARES

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PII. S0161171202112294 http://ijmms.hindawi.com

INFINITE TWO-GENERATOR GROUPS OF CLASS TWO AND THEIR NON-ABELIAN TENSOR SQUARES

NOR HANIZA SARMIN Received 24 December 2001

We classify all infinite 2-generator groups of nilpotency class two and determine their non-abelian tensor squares.

2000 Mathematics Subject Classification: 20F05, 20J99, 20F99.

1. Introduction. For a groupG, the non-abelian tensor squareG⊗Gof a groupG is generated by the symbolsg⊗h,g,h∈G, subject to the relations

gg⊗h=_g

g⊗^gh

(g⊗h), g⊗hh=(g⊗h)_h g⊗^hh

, (1.1)

for allg,g,h,h∈G, where^gg=ggg⁻¹. The non-abelian tensor square is a special case of the non-abelian tensor product which has its origins in homotopy theory and was introduced by Brown and Loday in [4,5].

In [3], Brown et al. started the investigation of non-abelian tensor squares as group theoretical objects. One of their main goals is the explicit computation of non-abelian tensor squares. The topic of this paper is the classification of infinite 2-generator groups of nilpotency class two and the determination of their non-abelian tensor squares. In [1, 7], this was done for 2-generator p-groups of class two, for p an odd prime orp=2, respectively. Thus this paper completes the classification of 2- generator groups of class two and determination of their non-abelian tensor squares.

For an overview of non-abelian tensor squares which have been determined, we re- fer to [6] and also to [2], where infinite metacyclic groups were classified and their non-abelian tensor squares determined.

2. The classification. In this section, we classify the infinite 2-generator groups of nilpotency class two up to isomorphism. As a preliminary step, we classify the above groups which are split extensions of ap-group by an infinite cyclic group.

Proposition 2.1. Let G= a,bbe a 2-generator group of nilpotency class less than or equal to 2 of the formG=P b, wherebis an infinite cyclic group, and P= [a,b]ais ap-group. ThenGis isomorphic to exactly one group of the following types:

G

a×cb, (2.1)

where[a,b]=c,[a,c]=[b,c]=1,|a| =p^α,|c| =p^γ,α≥γ≥1;

G ab, (2.2)

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where[a,b]=a^p^α−γ,|a| =p^α,α≥2γ≥2;

G

a×c

b, (2.3)

where[a,b]=a^p^α−γc,[c,b]=a^−p^2(α−γ)c^−p^α−γ,|a| =p^α,|c| =p^σ,γ > σ≥1,α+σ≥ 2γ;

G a×b, (2.4)

where[a,b]=1,|a| =p^α.

The groups in the above list have nilpotency class two precisely for (2.1), (2.2), and (2.3), and are abelian for (2.4).

Proof. LetGbe a group as in the hypothesis. SinceG is nilpotent of class less than or equal to 2,Pis abelian. Suppose first[a,b]=1. Obviously,Gis of type (2.4).

Now, let [a,b]≠1. First, let [a,b] ∩ a = 1. Set [a,b]=c. This implies that P= c×a. ThusGis the group of type (2.1). Next, let[a,b]∩a≠1. If[a,b] ⊆ a, then evidently,G is the group of type (2.2). Finally, suppose that[a,b] ∩ a is a proper subgroup of[a,b]. We have 1=[a,b]^p^γ =[a,b^p^γ]. This implies that b^p^γ∈Z(G), thusb^p^γG. Therefore,G/b^p^γis isomorphic toH=(u×v)w where[v,w]=v^p^α−γu,[u,w]=v^−p^2(α−γ)·u^−p^α−γ,|v| =p^α,|w| =p^β,|[v,w]| =p^γ, α,β,γ,σ∈N,γ > σ ≥1,α+σ ≥2γ,β≥γ, as follows from [1, Theorem 2.4] forp odd and [7, Theorem 2.5] forp=2, respectively.

It is easy to show that the nilpotency class of each group of type (2.1), (2.2), and (2.3) is two and they are all pairwise nonisomorphic.

Now, we are ready to state and prove the main theorem of this section. Note that F2/γ3(F2)is the free group of rank 2 and class 2, and is denoted byᏴ, known as the Heisenberg group.

Theorem2.2. LetGbe an infinite non-abelian2-generator group of nilpotency class two. ThenGis isomorphic to exactly one group of the following types:

G

a×cb, (2.5)

where[a,b]=c,[a,c]=[b,c]=1,|a| = ∞,|b| = ∞,|c| ≤ ∞; G

P1×P2×···×Pi×···×Pn

b, n≥1, (2.6)

where, fori=1,...,n, the componentP_iis ap_i-group,p_i≠p_jfori≠j,|b| = ∞, and Pibis of type (2.1), (2.2), (2.3), and (2.4) ofProposition 2.1.

Proof. LetGbe an infinite non-abelian 2-generator group of nilpotency class two.

Then,G=AbwhereA= ca,c=[a,b],Ais abelian and normal inG, andG= a,b. Furthermore,G= [a,b]. ConsiderG/Z(G). Then eitherG/Z(G)is torsion free orG/Z(G)has a nontrivial element of finite order. In caseG/Z(G)is torsion free, we can show that|G| = ∞. SoGᏴand is of type (2.5) with|c| = ∞.

Now, supposeG/Z(G)has a nontrivial element of finite order. It follows thatG/Z(G) is finite which implies|G|<∞. Then eitherG/Gis torsion free or not. In the first

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INFINITE TWO-GENERATOR GROUPS OF CLASS TWO 617 case,G/GZ×ZandGis a group of type (2.5) with|c|<∞, and in the second case, G/GZm×Z. Without loss of generality, assume that|b| = ∞and|a|<∞, oth- erwise, relabel the generators so thatG= a,b, with|b| = ∞andA= a,c,|A|<∞. Then|b| = ∞implies thatA∩ b = 1, and we haveG=Ab.

Now, sinceA= c,ais a finite abelian group,A=P1×···×Pi×···×Pn, wherePiis a Sylowp_i-subgroup ofA. Leta = a1×···×a_i×···×a_n,c = c1×···×c_n, withai,ci∈Piandci=[ai,b]. We havePi= ai,ciandPiis normal inG. Thus, there are subgroupsHi=PibofGsuch thatHi= ai,b. IfHiis an abelian group, then it follows thatP_i= a_i, |a_i| =p^αⁱ, andH_i=P_i× b. IfH_i is a non-abelian group, thenH_iis one of the groups of type (2.1), (2.2), and (2.3) ofProposition 2.1. Thus,G is the group of type (2.6).

3. The tensor squares of groups of class two. In [1, Proposition 3.3], it was shown that the tensor square for a group of class two is abelian. This fact helps us in using the concept of crossed pairing in our computations. We define it here in the case relevant for non-abelian tensor squares.

Definition3.1. LetGandLbe groups. A functionφ:G×G→Lis called a crossed pairing if

φ(gg,h)=φ_gg,^ghφ(g,h),

φ(g,hh)=φ(g,h)φ_hg,^hh, (3.1) for allg,g,h,h∈G.

Crossed pairings allow us to determine homomorphic images ofG⊗Gas follows.

Proposition3.2[3]. A crossed pairingφ:G×G→Ldetermines a unique homo- morphism of groupsφ:G⊗G→Lsuch thatφ(g⊗h)=φ(g,h)for allg,h∈G.

In this section, we also include two results that will be used in the next section. First, suppose we are given groupsH,G,K, andL, whereGandLare homomorphic images ofHandK, respectively. The following proposition enables us to find a crossed pairing fromG×GtoLgiven a crossed pairing fromH×HtoK, provided certain conditions are met.

Proposition3.3[7]. LetG,H,K, andLbe groups withπ:H→Gan epimorphism, ϕ:K→La homomorphism, andΓ :H×H→Ka crossed pairing. IfΓ(kerπ,H)and Γ(H,kerπ)are contained inkerϕ, then there exists a crossed pairing∆:G×G→Lfor which the following diagram commutes:

H×H

(π,π)

Γ K

ϕ

G×G ^∆ L

(3.2)

The second result is a lemma on finitely generated abelian groups. The proof is easy and thus omitted here. Observe that we say that a nontrivial element in an infinite cyclic group has order zero.

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Lemma3.4. LetA= a1,...,anbe a finitely generated abelian group, and letB= b1 × ··· × bnbe a direct sum ofncyclic groups, such that the order ofaidivides the order ofb_ifori=1,...,n. Ifφ:A→Bis a homomorphism such thatφ(a_i)=b_i, thenφis an isomorphism.

4. Computation of the tensor squares. In this section, we determine the tensor squares of the groups classified inSection 2beginning with the squares of groups of type (2.5).

Theorem4.1. LetGbe a group of type (2.5). Then

G⊗G





Z⁶, for|c| = ∞,

Z⁴×Z²k, for|c| =k. (4.1) Proof. LetGbe a group of type (2.5). If|c| = ∞, thenGᏴ, and the result follows from [1, Corollary 3.8].

If|c| =k, then by [1, Proposition 3.5] it follows thatG⊗Gis generated bya⊗a, b⊗b,a⊗b,b⊗a,a⊗c, andb⊗c.

We now establish order bounds for some of the generators ofG⊗G. Observing that c∈Z(G), we have 1_⊗=(a⊗c)^k,and 1_⊗=(b⊗c)^k. Since|a| = |b| = ∞, the first four generators ofG⊗Gdo not necessarily have finite order, and indeed, it will be shown that they have an infinite order.

Letg,h∈Gwithg=a^mbⁿc^landh=a^mbⁿc^l, wherem,m,n,n∈Zandl,lare integers modulok. LetL=Z⁴×Z²_k, and denote withz1,z2,z3,z4the components of the four factors of the formZ, and withz5,z6the components of the two factors of the formZk. Define the mappingθ:G×G→Lcomponentwise by

θ(g,h)=z1(g,h),z2(g,h),z3(g,h),z4(g,h),z5(g,h),z6(g,h), (4.2) where

z1(g,h)=mm, z2(g,h)=nn, z3(g,h)=mn, z4(g,h)=nm, z5(g,h)≡n

m 2

−n m

2

+(n−n)mm+ml−mlmodk,

z6(g,h)≡m n

2

−m n

2

+nl−nlmodk.

(4.3)

Sincem,m,n,n are unique integers andl,l are unique modulo k, it follows that θis well defined. As in [1],θis a crossed pairing. ByProposition 3.2, the mappingθ defined above lifts to a homomorphismθ ofG⊗GontoLsuch thatθ(g⊗ h)= θ(g,h). In particular, θ(a⊗a)=(1,0,0,0,0,0),θ(b⊗b)=(0,1,0,0,0,0), θ(a⊗ b)=(0,0,1,0,0,0),θ(b⊗a)=(0,0,0,1,0,0),θ(a⊗c)=(0,0,0,0,1,0), andθ(b⊗ c)=(0,0,0,0,0,1). Thus the generators ofG⊗Gmap to the generators ofL. Further- more, by the order estimates previously established, the order of a generator ofG⊗G divides the order of the corresponding generator ofL. Thus, byLemma 3.4,θ is an isomorphism and it follows thatG⊗GZ⁴×Z²kas claimed.

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INFINITE TWO-GENERATOR GROUPS OF CLASS TWO 619 In order to determine the tensor squares of groups of type (2.6), we first determine the tensor squares of the groups inProposition 2.1. We start with the groups of type (2.1), dealing with the casepodd andp=2 in two separate propositions.

Proposition4.2. LetGbe a group of type (2.1) withp≠2. Then

G⊗GZ³p^α×Z²p^γ×Z. (4.4) Proof. LetGbe a group of type (2.1) withpprime,p≠2. It follows thatG⊗Gis generated bya⊗a,b⊗b,a⊗b,b⊗a,a⊗c, andb⊗c. We give now order estimates for 5 of the 6 generators ofG⊗G. Using [1, Lemma 3.4], we obtain 1_⊗=(a⊗c)^p^γ, 1_⊗= b⊗c^p^γ=(b⊗c)^p^γ, 1_⊗=a⊗a^p^α=(a⊗a)^p^α, and 1_⊗=a^p^α⊗b=(a⊗b)^p^α(a⊗c)(^pα²). Sincep≠2 andα≥γ, we havep^γ|_pα

2 . Thus(a⊗c)(^pα²)=1_⊗. We conclude that 1_⊗=(a⊗b)^p^α. Similarly, it follows that 1_⊗ =(b⊗a)^p^α. Since|b| = ∞,b⊗b is not necessarily finite.

Letg,h∈G with g=a^mbⁿc^land h=a^mbⁿc^l, wheren,n∈Z, m,mare integers modulop^α, andl,lare integers modulop^γ. LetL=Zp^α×Z×Z²p^α×Z²p^γ, and denote withz1,z3,z4the components of the three factors of the formZp^α, withz2

the component ofZ, and withz5,z6the components of the two factors of the form Zp^γ.

Define the mappingθ:G×G→Lcomponentwise by

θ(g,h)=z1(g,h),z2(g,h),z3(g,h),z4(g,h),z5(g,h),z6(g,h), (4.5) where

z1(g,h)≡mmmodp^α, z2(g,h)=nn, z3(g,h)≡mnmodp^α, z4(g,h)≡nmmodp^α, z5(g,h)≡n

m 2

−n m

2

+(n−n)mm+ml−mlmodp^γ,

z6(g,h)≡m n

2

−m n

2

+nl−nlmodp^γ.

(4.6)

Sincem,m are unique modulo p^α, n,n are unique integers, and l,l are unique modulop^γ, and in additionα≥γ≥1, it follows thatθis well defined.

Next, we show that the mappingθ is a crossed pairing. By [1, Proposition 3.7], the mappingψ:Ᏼ×Ᏼ→Z⁶, defined asθ, butm,n,l,m,n,ljust being integers, is a crossed pairing. Now, equations (3.1) hold componentwise forψas identities in integers. It follows that they hold as congruences modulo any integer. Since, at the same time, the modules given forz1,z3,z4,z5,andz6are the largest for whichθis well defined, we conclude thatθis a crossed pairing. ByProposition 3.2, the mapping defined above lifts to a unique homomorphismθ:G⊗G→Lsuch thatθ(g⊗h)=θ(g,h), where the generators of G⊗G, as given above, map to the corresponding genera- tors of L. Furthermore, by the order estimates established before, the order of a generator ofG⊗G divides the order of the corresponding generator ofL. Thus, by Lemma 3.4, it follows thatθis an isomorphism andG⊗GZ³p^α×Z²p^γ×Zas claimed.

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Proposition4.3. LetGbe a group of type (2.1) withp=2. Then

G⊗G







Z³2^α×Z²2^γ×Z, ifα > γ,

Z³2^γ×Z2^γ+1×Z2^γ−1×Z, ifα=γ. (4.7)

Proof. LetGbe a group of type (2.1) withp=2. We have two cases, namely,α > γ andα=γ. The first case follows from the proof ofProposition 4.2, settingp=2 and observingα > γ. So we omit the details.

Now, letα=γ. From [1, Proposition 3.5], it follows thatG⊗Gcan be generated by a⊗a,b⊗b,a⊗b,(a⊗b)(b⊗a),(a⊗b)²(a⊗c), and(a⊗b)²(b⊗c).

We now establish order bounds for the generators ofG⊗G with the exception of b⊗b. First, notice that the expansion formula as before yields 1_⊗=(a⊗c)²^γ and 1_⊗=(b⊗c)²^γ. Furthermore, 1_⊗=(a⊗a)²^γ, 1_⊗=(a²^γ⊗b)²=(a⊗b)²^γ+1(a⊗c)²^γ= (a⊗b)²^γ+1, 1_⊗=(a²^γ⊗b)(b⊗a²^γ)=((a⊗b)(b⊗a))²^γ, 1_⊗=((a⊗b)²(a⊗c))²^γ−1, and 1_⊗=(a²^γ⊗b)²(b⊗c²^γ)=(a⊗b)²^γ+1(b⊗c)²^γ =((a⊗b)²(b⊗c))²^γ. Let Ᏼ= x,y. Defineπ:Ᏼ→Gbyπ(h)=a^mbⁿc^lforh∈Ᏼ^{, where}^h=x^myⁿz^l,m,n,l∈Z,z= [x,y]. It follows thatπis a homomorphism ontoG. Next, setL=Z2^α×Z×Z2^γ+1×Z2^γ× Z2^α−1×Z2^γ andZ⁶= x1×···×x6. Defineϕ:Z⁶→Lbyϕ=µ◦λ, whereλ:Z⁶→Z⁶ is given byλ(x_i)=x_ifori=1,2,4,5,6,λ(x3)=x3x₄⁻¹x₅⁻²x⁻²₆ , andµ:Z⁶→Lreduces the generators ofZ⁶modulo the appropriate powers. Clearly,λis an automorphism and µ is a homomorphism, soϕ is a homomorphism withϕ(xi)=i, i=1,...,6, where1=(1,0,0,0,0,0),..., 6=(0,0,0,0,0,1)∈L.

Now, letΓ=ψ be the crossed pairing of [1, Proposition 3.7] withψ:Ᏼ×Ᏼ→Z⁶, andπthe epimorphism ofᏴontoG. ByProposition 3.3, there exists a crossed pairing

∆:G×G→Lfor which the diagram commutes, provided

ϕ

ψ(kerπ,Ᏼ)

=ϕ

ψ(Ᏼ,kerπ)

=1. (4.8)

To establish (4.8), letl=(l1,...,l6)∈Landh,h∈Ᏼ^with^h=x^myⁿz^l,h=x^myⁿz^l, wherem,m,n,n,l,l∈Z. By [1, Proposition 3.7] and the definition ofϕ, we obtain ϕ ◦ ψ(h,h) = ϕ(x1(h,h),...,x6(h,h)) = (ϕ(x1(h,h)),...,ϕ(x6(h,h))) = (l1(h,h),...,l6(h,h)), where

l1(h,h)≡mmmod 2^γ, l2(h,h)=nn, l4(h,h)≡nmmod 2^γ, l3(h,h)≡mn−mn−2

n

m 2

−n m

2

+(n−n)mm+ml−ml

−2

m n

2

−m n

2

+nl−nl

mod 2^γ+1, l5(h,h)≡n

m 2

−n m

2

+(n−n)mm+ml−mlmod 2^γ−1, l6(h,h)≡m

n 2

−m n

2

+nl−nlmod 2^γ.

(4.9)

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INFINITE TWO-GENERATOR GROUPS OF CLASS TWO 621 Suppose now thath∈kerπ, thenm≡l≡0 mod 2^γandn=0. Thusl1,l2,l4,l5, and l6are obviously trivial inL. As forl3, the terms of the right side can be rearranged as

l3(h,h)≡2m(mn−mn−l)+2n

m+1 2

−l

+2l(m+n)−m²n+nn²mmod 2^γ+1.

(4.10)

Sinceα=γ, we havem≡l≡0 mod 2^γ andn=0. Therefore, 2m≡2l≡0 mod 2^γ+1, 2n=0, andm²≡0 mod 2^γ+1. Thusl3is trivial inL. Soϕ(ψ(kerπ,Ᏼ⁾⁾=1. In a similar mannerϕ(ψ(Ᏼ,kerπ))=1, hence, (4.8) is established.

Since the diagram ofProposition 3.3commutes for∆andϕ◦ψis onto, we conclude that∆is onto. Thus, byProposition 3.2,∆lifts to a homomorphism∆ofG⊗Gonto L, where the generators of G⊗G map to the generators of L. Furthermore, by the order estimates established before, the order of a generator ofG⊗Gdivides the order of the corresponding generator ofL. Thus, by Lemma 3.4, it follows that ∆ is an isomorphism andG⊗GZ³2^γ×Z2^γ+1×Z2^γ−1×Zas claimed.

The metacyclic groups of type (2.2) can be viewed as a special case of groups of type (2.3), for whichσ =0, that is, the torsion subgroup has rank one. Thus, the tensor squares of groups of type (2.2) are obtained together with those of type (2.3). Now, we determine the tensor square for a group of type (2.3) and (2.2), dealing with the cases p≠2 andp=2 in two separate propositions.

Proposition4.4. LetGbe a group of type (2.3) or (2.2) withp≠2. Then

G⊗GZ²_pα−γ+σ×Zp^α×Z²p^σ×Z. (4.11) Proof. LetG be a group of type (2.3) or (2.2) with p≠2. Setz=[a,b]. By [1, Lemma 3.5], it follows thatG⊗Gcan be generated bya⊗a,b⊗b,a⊗b,(a⊗b)(b⊗a), (a⊗a)^p^α−γ(a⊗z), and(a⊗b)^p^α−γ(b⊗z). Now, we establish order bounds for some of the generators ofG⊗G. If we setz=[a,b], thenc=za^−p^α−γ by (2.3). To obtain an order bound for(a⊗a)^p^α−γ(a⊗z), expandc^p^σ⊗ato obtain

1_⊗=c^p^σ⊗a=(a⊗a)^−p^α−γ+σ(z⊗a)^p^σ=(a⊗a)^p^α−γ(z⊗a)⁻¹−p^σ, (4.12) thus((a⊗a)^p^α−γ(a⊗z))^p^σ=1_⊗. Similarly,

1_⊗=c^p^σ⊗b=(a⊗b)^−p^α−γ+σ(a⊗z)(^−pα^−γ+σ² )(b⊗z)^−p^σ. (4.13) Since α+σ ≥ 2γ and p ≠ 2, this implies that (a⊗z)(^−pα^−γ+σ² ) = 1_⊗. Therefore, ((a⊗b)^p^α−γ(b⊗z))^p^σ=1_⊗.

Turning now to(a⊗b)(b⊗a), expansion ofb⊗c^p^σ leads to

1_⊗=(b⊗a)^−p^α−γ+σ(a⊗z)(^{−pα−γ+σ}² )(b⊗z)²^σ. (4.14) Multiplying (4.13) and (4.12) yields((a⊗b)(b⊗a))^p^α−γ+σ=1_⊗.

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Next, we establish an estimate for the order ofa⊗b. By observingα > γ and ex- pandinga^p^α⊗b, we obtain 1_⊗=(a⊗b)^p^α(a⊗z)(^pα²)=(a⊗b)^p^α.

Finally, we turn to the estimate for the order ofa⊗a. Here we have

1_⊗=a⊗c^p^σ=(a⊗a)^−p^α−γ+σ(a⊗z)^p^σ. (4.15)

Equating (4.12) and (4.15) yields

(a⊗z)^p^σ=(a⊗z)^−p^σ. (4.16)

Obviously, 1_⊗=a^p^α⊗z=(a⊗z)^p^α, soa⊗z hasp-power order. Hence, we obtain (a⊗z)^p^σ=1_⊗by (4.16). This, together with (4.12), implies(a⊗a)^p^α−γ+σ=1_⊗. LetᏴ= x,y. Defineπ:Ᏼ→Gbyπ(h)=a^m¹b^m²c^l, whereh=x^m¹y^m²v^lwithm1,m2,l∈ Z, v=x^−p^α−γ[x,y], and a,b,c ∈G as in the proposition. It follows that π is an epimorphism. Next, letL=Zp^α−γ+σ×Z×Zp^α×Zp^α−γ+σ×Z²p^σ, andZ⁶= x1×···×x6. Defineϕ:Z⁶→Lbyϕ=µ◦λ, whereλ:Z⁶→Z⁶is given byλ(x_i)=x_ifori=2,4,6, λ(x1)=x1x₅^−p^α−γ·x₆^−p^{2(α−γ)−1},λ(x3)=x3x⁻₄¹x₆^−p^α−γ, and λ(x5)=x5x^p₆^α−γ−1, while µ:Z⁶→Lreduces the generatorsλ(x_i)modulo the appropriatep-powers. Specifically, ϕ(xi)=i,i=1,...,6, where1=(1,0,0,0,0,0),...,6=(0,0,0,0,0,1)∈L. It follows thatλis an automorphism ofZ⁶andµis a homomorphism ofZ⁶ontoL.

Leth,h∈Ᏼ^with ^h=x^m¹y^m²v^l and h=x^m¹y^m²v^l, wherev=x^−p^α−γ[x,y]

as before. Setting u= [x,y], we obtain alternative presentations for h and h as h=x^myⁿu^kandh=x^myⁿu^k, wherem=m1−lp^α−γ,m=m₁−lp^α−γ,n=m2, n=m₂, k=l, and k=l. By [1, Proposition 3.7], there exists a crossed pairing ψ:Ᏼ×Ᏼ→Z⁶, where, in terms of the original presentation,

x1(h,h)=

m1−lp^α−γ

m₁−lp^α−γ

, x2(h,h)=m2m₂, x3(h,h)=m1−lp^α−γm₂, x4(h,h)=m₁−lp^α−γm2, x5(h,h)=m₂

m1−lp^α−γ 2

−m2

m₁−lp^α−γ 2

+

m1−lp^α−γ l

−

m₁−lp^α−γ l+

m₂−m2

m1−lp^α−γ

m₁−lp^α−γ , x6(h,h)=

m1−lp^α−γ m₂

2

−

m₁−lp^α−γ m2

2

+m2l−m₂l.

(4.17)

We apply nowProposition 3.3withGas given in (2.3),H=Ᏼ^,^K=Z⁶, andLas defined above. For the mappings, letϕ=µ◦λandΓ=ψ, all as given above, andπ:Ᏼ→ G. ByProposition 3.3, there exists a crossed pairing∆:G×G→Lsuch that the diagram (3.2) commutes, providedϕ(ψ(kerπ,Ᏼ⁾⁾=ϕ(ψ(Ᏼ^,ker^π))=1. Next, we show that this is the case. Supposeh,h∈Ᏼwhereh=x^m¹y^m²v^l andh=x^m¹y^m²v^l. Writingϕ◦ψ:Ᏼ×Ᏼ→Lcomponentwise asϕ◦ψ(h,h)=(l1(h,h),...,l6(h,h)), we

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INFINITE TWO-GENERATOR GROUPS OF CLASS TWO 623 obtainli(h,h)in terms ofxi(h,h)as given above,

l1(h,h)≡x1(h,h)−p^α−γx5(h,h)−p^{2(α−γ)−1}x6(h,h)modp^α−γ+σ; l2(h,h)=x2(h,h);

l3(h,h)≡x3(h,h)−x4(h,h)−p^α−γx6(h,h)modp^α; l4(h,h)≡x4(h,h)modp^α−γ+σ;

l5(h,h)≡x5(h,h)+p^α−γ−¹x6(h,h)modp^σ; l6(h,h)≡x6(h,h)modp^σ.

(4.18)

Ifh∈kerπ, thenm1≡0 modp^α,m2=0 andl≡0 modp^σ. It follows directly from the definition ofxi(h,h)thatli(h,h)=0 fori=1,...,6. Thusϕ(ψ(kerπ,Ᏼ))=1, as claimed. Similarly, it can be shown thatϕ(ψ(Ᏼ,kerπ))=1. Thus ∆is a crossed pairing.

Since the diagram ofProposition 3.3commutes for∆andψ◦ϕis onto, we conclude that∆is onto. Thus, byProposition 3.2,∆lifts to a homomorphism∆ofG⊗GontoL, where the generators ofG⊗Gmap to the corresponding generators ofL. Furthermore, by the order estimates established before, the order of a generator ofG⊗Gdivides the order of the corresponding generator ofL. Thus, byLemma 3.4, it follows that∆ is an isomorphism andG⊗GZ²_pα−γ+σ×Zp^α×Z²p^σ×Zas claimed.

Proposition4.5. LetGbe a group of type (2.3) or (2.2) withp=2. Then

G⊗GZ2^α−γ+σ+1×Z2^α−γ+σ×Z2^α×Z²2^σ×Z. (4.19) Proof. LetG be a group of type (2.3) or (2.2) with p=2. Setz=[a,b]. By [1, Proposition 3.5], it follows thatG⊗G can be generated bya⊗a, b⊗b, a⊗b, (a⊗ b)(b⊗a),(a⊗a)²^α−γ(a⊗z), and(a⊗b)²^α−γ(a⊗z)⁻²^α−γ−1(b⊗z).

Now, we establish order bounds for some of the generators ofG⊗G. Notice that c=za⁻²^α−γ by (2.3). Following along the lines of the proof ofProposition 4.4, we get

1_⊗=c²^σ⊗a=(a⊗a)⁻²^α−γ+σ(a⊗z)⁻²^σ, (4.20) which gives((a⊗a)²^α−γ(a⊗z))²^σ=1_⊗. Similarly,

1_⊗=c²^σ⊗b=(a⊗b)⁻²^α−γ+σ(a⊗z)²^{α−γ+σ−1}(b⊗z)⁻²^σ. (4.21) Thus((a⊗b)²^α−γ(a⊗z)⁻²^α−γ−1(b⊗z))²^σ=1_⊗. Expansion ofb⊗c²^σ leads to

1_⊗=(b⊗a)⁻²^α−γ+σ(a⊗z)⁻²^{α−γ+σ−1}(b⊗z)²^σ. (4.22) Multiplying (4.21) and (4.22) yields((a⊗b)(b⊗a))²^α−γ+σ =1_⊗.

Next, we give an estimate for the order ofa⊗b. Sinceα > γ, expansion ofa²^α⊗b yields 1_⊗=(a⊗b)²^α(a⊗z)²^α−1=(a⊗b)²^α.

Finally, we estimate the order ofa⊗a. We obtain

1_⊗=a⊗c²^σ=(a⊗a)⁻²^α−γ+σ(a⊗z)²^σ. (4.23)

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Equating (4.21) and (4.23) yields(a⊗z)²^σ+1=1_⊗. Thus, squaring (4.20) leads to(a⊗ a)²^α−γ+σ+1=1_⊗.

The rest of the proof follows directly from the proof ofProposition 4.4, withp replaced by 2 and takingli(h,h)modulo the appropriate powers of 2. Thus

G⊗GZ2^α−γ+σ+1×Z2^α−γ+σ×Z2^α×Z²2^σ×Z, (4.24) as claimed.

For easier reference, we list the tensor squares of groups of type (2.2), the case σ=0, of Propositions4.4and4.5in the next corollary.

Corollary4.6. LetGbe a group of type (2.2). Then

G⊗G







Z²_pα−γ×Zp^α×Z, ifp≠2,

Z2^α−γ+1×Z2^α−γ×Z2^α×Z, ifp=2. (4.25) Our concluding theorem (Theorem 4.7) determines the tensor square of a group of type (2.6). To that end, we mention again that the non-abelian tensor square of an abelian group is just the standard abelian tensor. Thus, for groups of type (2.4), we haveG⊗GG⊗ZG.

Theorem4.7. LetGbe a group of type (2.6), that is, G=

P1×P2×···×Pn

b, n≥1, (4.26)

where, fori=1,...,n, the componentPiis api-group,pi≠pjfori≠j,|b| = ∞, and Gi=Pibis of type (2.1), (2.2), (2.3), and (2.4) ofProposition 2.1. Then

G⊗GT G1⊗G1

×···×T

Gn⊗Gn

×Z, (4.27)

whereT (G_i⊗G_i)is the torsion subgroup ofG_i⊗G_i.

Proof. LetGbe a group of type (2.6). Observe thatT (G)=P1×···×Pnis abelian, and if(|g|,|h|)=1, theng⊗h=1_⊗.

We prove our claim by induction onn, the number of Sylowp-subgroups ofT (G).

Ifn=1, thenG=P1bandG⊗GT (G⊗G)×Zby Propositions4.2,4.3,4.4, and 4.5.

Supposen≥2. ThenG=(P×Q) bwithP=P1andQ=P2× ··· ×P_n. SetU= P bandW=Q b. Then, for anyg,h∈G, there existu,u∈P⊆U,w,w∈W such thatg=uwandh=uw. By expandingg⊗h,

g⊗h=uw⊗uw=X(g,h)·(w⊗w), (4.28) where

X(g,h)=(u⊗u)(u⊗w)(w⊗u)

[u,w]⊗u

[u,w]⊗w

w⊗[u,w]

. (4.29) For w,w ∈W, there exist v,v ∈ Q and integers s,t such that w =vb^s and w=vb^t. We substitutew=vb^s andw=vb^t into (4.29), expand, observing that

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INFINITE TWO-GENERATOR GROUPS OF CLASS TWO 625 elements inT (G)commute andg⊗h=1_⊗forg,h∈T (G)with(|g|,|h|)=1. We then arrive at

X(g,h)=(u⊗u)(u⊗b)^t

b⊗[u,b](^t2)(b⊗u)^s

b⊗[b,u](^s2)

[u,b]⊗us

·

[u,b]⊗bst

b⊗[u,b]t. (4.30)

We observeX(g,h)∈T (U⊗U), henceX(g,h)|g,h∈G ≤T (U⊗U). On the other hand,T (U⊗U)≤ u1⊗u2,u3⊗b,b⊗u4|u1,u2,u3,u4∈T (U). However, for suitable choices ofgandh, observe that all the above generators are in{X(g,h); g,h∈G}. We conclude thatT (U⊗U)= X(g,h)|g,h∈G. The above, together with (4.28), impliesG⊗G= T (U⊗U),W⊗W. Observing(|T (U⊗U)|,|T (W⊗W )|)=1, it follows thatG⊗G=T (U⊗U)×(W⊗W ). Since|Q| = |T (W )|has onlyn−1 distinct prime divisors, the claim follows by induction onn.

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[1] M. R. Bacon and L.-C. Kappe,The nonabelian tensor square of a2-generatorp-group of class2, Arch. Math. (Basel)61(1993), no. 6, 508–516.

[2] J. R. Beuerle and L.-C. Kappe,Infinite metacyclic groups and their non-Abelian tensor squares, Proc. Edinburgh Math. Soc. (2)43(2000), no. 3, 651–662.

[3] R. Brown, D. L. Johnson, and E. F. Robertson,Some computations of nonabelian tensor products of groups, J. Algebra111(1987), no. 1, 177–202.

[4] R. Brown and J.-L. Loday,Excision homotopique en basse dimension[Homotopical excision in low dimension], C. R. Acad. Sci. Paris Sér. I Math.298(1984), no. 15, 353–356 (French).

[5] ,Van Kampen theorems for diagrams of spaces, Topology26(1987), no. 3, 311–335.

[6] L.-C. Kappe,Nonabelian tensor products of groups: the commutator connection, Groups St. Andrews 1997 in Bath, II, London Mathematical Society Lecture Note Series, vol.

261, Cambridge University Press, Cambridge, 1999, pp. 447–454.

[7] L.-C. Kappe, M. P. Visscher, and N. H. Sarmin,Two-generator two-groups of class two and their nonabelian tensor squares, Glasgow Math. J.41(1999), no. 3, 417–430.

Nor Haniza Sarmin: Mathematics Department, Faculty of Science, Universiti Teknologi Malaysia (UTM),81310Skudai, Johor, Malaysia

E-mail address:[email protected]