Some applications of Two-Generator p-Groups of Nilpotency class two

Some Applications of Two-Generator

p-Groups of Nilpotency Class Two

Nor Haniza Sarmin

Jabatan Matematik, Fakulti Sains

Universiti Teknologi Malaysia

81310 UTM Skudai, Johor

[email protected]

Idariyana Mohd Seran

Sekolah Sains Kuantitatif

Universiti Utara Malaysia

06010 UUM Sintok, Kedah

[email protected]

Abstract The classification of all two-generator groups of nilpotency class two have been done by Nor Haniza Sarmin in 2000. Earlier research on the finite case was introduced by Kappeet. alin 1999 and Bacon and Kappe in 1993. Later, in 2000, James Beuerle and Kappe focus their classification to infinite metacyclic groups and some of their applications. This paper will focus on the applications of two-generatorp-groups of nilpotency class two including the nonabelian tensor squares and the exterior squares. We will also determine which of these groups are capable.

Keywords Nonabelian tensor squares, two-groups, two-generator, nilpotency class two.

Abstrak Klasifikasi bagi semua kumpulan dengan dua penjana yang mempun-yai kelas nilpoten dua telah diperolehi oleh Nor Haniza Sarmin dalam tahun 2000. Kajian yang terdahulu bagi kumpulan terhingga telah dilakukan oleh Kappeet. aldalam tahun 1999 dan Bacon dan Kappe dalam tahun 1993. Kemu-dian, dalam tahun 2000, James Beuerle dan Kappe telah menumpukan klasifikasi mereka kepada kumpulan metakitaran tak terhingga dan beberapa aplikasinya. Penyelidikan ini ditumpukan kepada beberapa aplikasi bagi kumpulan-p den-gan dua penjana termasuk kuasa dua tensor yang tak abelan dan kuasa dua peluaran. Kajian ini juga akan menentukan kumpulan berupaya dari kumpulan tersebut.

1

Introduction

The classification of all two-generator groups of nilpotency class two up to isomorphism have been done by Nor Haniza Sarmin [8]. Earlier research on the classification of two-generator

p-groups of nilpotency class two was introduced by Bacon and Kappe [1] for the casepodd, and Kappe,et. al[7] for the casep= 2.

Recent work of Beuerle and Kappe in [3] focus the classification on infinite metacylic groups and some of their applications including the nonabelian tensor squares, exterior squares, the symmetric products and the second homology groups. They also showed that the only capable group of the nonabelian infinite metacylic group is the infinite dihedral group.

In this paper, we will combine the paper by Bacon and Kappe [1] and Kappe,et. al[7] to give the classification of two-generatorp-groups of nilpotency class two, up to isomorphisms in Theorem 2.1. We will compute some applications of two-generatorp-groups of nilpotency class two including their nonabelian tensor squares and exterior squares. Then, we will use the result of these applications to determine which of these groups are capable.

In 1938, Baer [2] initiated a systematic investigation to determine the conditions of a groupGmust fulfill in order to be the group of inner automorphisms of some groupE(that isG∼=E/Z(E)). Such group is called a capable group. Baer already determined all finitely generated abelian capable groups which are direct sums of cyclic groups. Baer stated that such group are capable if and only if their two highest torsion coefficients agree. Ellis in [5] investigated the capability of other classes of groups, concentrating to some extent on nilpotent group of class two. He used the method that involves the third integral homology of a group. He showed that the quaternion group

Q2n=ha, b|an=b2, b−1ab=a−1i

of order 4n is not capable. He also showed that any finitely generated (but non-cyclic) nilpotent group of class≤2 with prime exponent is capable. These results will be used to determine which of the two-generatorp-groups of nilpotency class two are capable.

2

The Classification of Two-Generator

p

-Groups of

Nilpotency Class Two

In this section, we bring the results of the classification of two-generatorp-groups of nilpo-tency class two up to isomorphism from the papers by Bacon and Kappe [1] and Kappe,et. al [7]. The proofs of the classification are omitted and the details can be referred from the papers stated.

Theorem 2.1 states the classification of two-generatorp-groups of nilpotency class two up to isomorphism. Groups of type (2.1.1) - (2.1.3) is the classification for the case p6= 2.

Theorem 2.1 LetGbe a finite nonabelian two-generatorp-group of nilpotency class two. Then Gis isomorphic to exactly one group of the following four types:

(2.1.1) G∼= (hci × hai)ohbi, where[a, b] =c,[a, c] = [b, c] = 1,|a|=pα,|b|=pβ,|c|=pγ, α, β, γ∈ N, α≥β≥γ;

(2.1.2) G∼=haiohbi, where[a, b] =apα−γ_,|a|=pα_,|b|=pβ_{,|[a, b]|=p}γ_{, α, β, γ ∈ N,}

α≥2γ, β≥γ, and ifp= 2 thenα+β >3;

(2.1.3) G∼= (hci × hai)ohbi, where[a, b] =apα−γ_{c,[c, b] =a}−pp(α−γ)_c−pα−γ_,|a|=pα_,

|b|=pβ_,|c|=pσ_{,|[a, b]|=p}γ_{, α, β, γ, σ ∈ N, β≥γ > σ, α+σ≥2γ;}

(2.1.4) G∼= (hci × hai)hbi, where|a|=|b|= 2γ+1,|[a, b]|= 2γ,|c|= 2γ−1,[a, b] =a2c,

[c, b] =a−4c−2, a2γ=b2γ, γ∈ N.

The groups in the above list are pairwise nonisomorphic and have nilpotency class two precisely.

3

Some Preparatory Results

In this section we bring some results that will be used in Section 4. These results are important to compute the exterior square of a group and to determine which groups are capable. The proofs of some theorems and lemmas are omitted.

Definition 3.1 [3] Exterior Square

LetG be a group. Then we define the exterior square of the group as

G∧G= (G⊗G)/hx⊗x|x∈Gi.

Theorem 3.1 [3]The exterior center of a group Gis

Z∧(G) ={g∈G|g∧x= 1∧, ∀x∈G}.

Definition 3.2 Epicenter

LetG be a group. ThenZ∗(G) ={z∈Z(G)|x∧z= 1∧, ∀x∈G}is called the epicenter

of G.

Definition 3.3, Theorem 3.2, Theorem 3.3 and Proposition 3.1 below can be used to determine which of the groups in the classification are capable.

Definition 3.3 [4]Capable Group

A groupGis said to be capable if it is isomorphic to the inner automorphism group,Inn(H)

of some group H, that is

Theorem 3.2 [3]A groupGis capable if and only if its exterior center is trivial, that is

Z∧(G) ={1}.

Theorem 3.3 [6]A groupGis capable if and only if its epicenter is trivial, denoted as

Z∗(G) ={1}.

Proposition 3.1 [5]Any finitely generated (but non-cyclic) nilpotent group of class ≤ 2

and prime exponent is capable.

In the lemma below, we prove that hx⊗x|x ∈ Gi is generated by a⊗a, b⊗b and (a⊗b)(b⊗a). This result will be used to compute the exterior squares of two-generator

p-groups of nilpotency class two.

Lemma 3.1 LetG=ha, bibe a two-generator p-group of class two. Then

hx⊗x;x∈Gi=ha⊗a, b⊗b,(a⊗b)(b⊗a)i.

Proof Letx=am_bn_zl_{, z= [a, b]. Then by Proposition 3.5 in [1], withm=m}0_{, n=n}0_{, l=}

l0_{, we have}

x⊗x= (a⊗a)α1_(b⊗b)α2_(a⊗b)α3_(b⊗a)α4_(a⊗z)α5_(b⊗z)α6_{, (1)}

where

α1 = m2, α2=n2, α3=mn, α4=mn,

α5 = n(m2)−n(

m

2) + (n−n)m

2_{+ml−ml= 0,}

α6 = m(n2)−m(n2) +nl−nl= 0.

Hence,

x⊗x= (a⊗a)m2(b⊗b)n2((a⊗b)(b⊗a))nm. (2) Thushx⊗x;x ∈Gi ⊆ ha⊗a, b⊗b,(a⊗b)(b⊗a)i. Letm= 1, n= 0 orm= 0, n= 1, thena⊗a, b⊗b∈ hx⊗x;x∈Gi.Forn=m= 1 we have (a⊗a)(b⊗b)(a⊗b)(b⊗a)

∈ hx⊗x;x∈Gi.This, together with the above, yields

ha⊗a, b⊗b,(a⊗b)(b⊗a)i ⊆ hx⊗x;x∈Gi.

The next two lemmas will be used to determine the capability of two-generatorp-groups of nilpotency class two using Proposition 3.1.

Lemma 3.2 LetGbe a group of type (2.1.1) - (2.1.4) of Theorem 2.1 with p= 2. Then,

exp(G) =       

2γ+1_{, ifα=β=γ,}

2α_{, ifα=β > γ,}

2α_{, ifα > β,}

2β_{, ifβ > α.}

Proof To prove this lemma we need to consider that

(22δ) =

2δ(2δ−1)(2δ−2)! 2!(2δ_−2)!

= 2

δ

2(2

δ₋₁₎

= 2δ−1(2δ−1).

Let g ∈ G. Then we can write g =aibj[a, b]k, where i, j, k are integers. Thus, for some

δ∈N,

g2δ = (aibj[a, b]k)2δ = (aibj)2δ[a, b]k2δ = ai2δbj2δ[a, b]ij(2 δ

2 )_{[a, b]}k2

δ

= ai2δbj2δ[a, b]ij(2 δ

2 )+k2δ

= ai2δ[a, b]ij(2 δ

2 )+k2

δ

.

Then, there are three cases, according to the conditions of the classification : (i)α > β, (ii)β > α, and (iii)α=β,

where G=ha, bi,|a|= 2α_{,|b|= 2}β_{,|[a, b]|= 2}γ_{, and γ≥1.}

(i) Assume α > β. Then

g2α = ai2αbj2α[a, b]ij(2 α

2 )+k2

α

= (a2α)i(b2α)j[a, b]2α−1(2α−1)ij+k2α = 1,

sinceα > β≥γimpliesα > β andα > γ,anda2α−1

6

= 1.Thus exp(G) = 2α_.

(ii) Assume β > α. Then we obtain

g2β = ai2βbj2β[a, b]ij(2 β

2 )+k2

β

= (a2β)i(b2β)j[a, b]2β−1(2β−1)ij+k2β = 1,

sinceβ > α > γ andb2β−1 _{6= 1.Thus exp(G) = 2}β_.

(iii) Assumeα=β. Then we have two subcases : (a) α=β=γ. Then

g2α+1 = ai2α+1bj2α+1[a, b]ij(2 α+1 2 )+k2

α+1

= (a2α+1)i(b2α+1)j[a, b]2α(2α+1−1)ij+k2α+1 = 1,

sinceα=β=γ. On the contrary, (ab)2α =a2αb2α[a, b](2 α

2 )_{= [a, b]}(2

α

2 )_{= [a, b]}2α

−1₍₂α₋₁₎

6

(b)α=β > γ.We obtain

g2α = ai2αbj2β[a, b]ij(2 α

2 )+k2α

= (a2α)i(b2β)j[a, b]2α−1(2α−1)ij+k2α = 1,

sinceα=β > γanda2α−1 6= 1, b2α−1 6= 1.Thus exp(G) = 2α= 2β.

Lemma 3.3 LetGbe a group of type (2.1.1) - (2.1.3) of Theorem 2.1 with p6= 2.Then,

exp(G) =       

pγ_{, if α=β=γ,}

pα_{, if α=β > γ,}

pα_{, ifα > β,}

pβ_{, ifβ > α.}

(4)

Proof To proof this lemma we need to consider that

(p₂δ) = p

δ_(pδ_−1)(pδ_−2)!

2!(pδ_−2)!

= p

δ

2(p

δ_−1).

For the case p odd, we can see that pδ _{−1 is an even, so that it can be divided by 2.}

Then, we can factorize the power of the variables involved. Let g∈G. Then we can write

g=aibj[a, b]k, wherei, j, kare integers. Thus, for someδ∈ N,

gpδ = (aibj[a, b]k)pδ

= (aibj)pδ[a, b]kpδ

= aipδbjpδ[a, b]ij(pδ2 )_{[a, b]}kp

δ

= aipδbjpδ[a, b]ij(pδ2 )+kpδ

= aipδ[a, b]ij(pδ2 )+kpδ_.

Then, there are three cases, according to the conditions of the classification : (i)α > β, (ii)β > α, and (iii)α=β,

where G=ha, bi,|a|=pα_,|b|=pβ_{,|[a, b]|=p}γ_{, andγ≥1.}

(i) Assume α > β. Then

gpα = aipαbjpα[a, b]ij(pα2 )+kp

α

= (apα)i(bpα)j[a, b]pα2 (p

α_−1)ij+kpα = (apα)i(bpα)j([a, b]pα)12(p

α_−1)ij+k = 1,

(ii) Assume β > α. Then we obtain

gpβ = aipβbjpβ[a, b]ij(pβ2 )+kpβ

= (apβ)i(bpβ)j[a, b]pβ2(p

β_−1)ij+kpβ

= (apβ)i(bpβ)j([a, b]pβ)12(p

β_−1)ij+k = 1,

sinceβ > α > γ andbpβ−1 6= 1.Thus exp(G) =pβ.

(iii) Assumeα=β. Then we have two subcases : (a) α=β=γ. Then

gpα = aipαbjpα[a, b]ij(pα2 )+kp

α

= (apα)i(bpα)j[a, b]pα2 (p

α_−1)ij+kpα = (apα)i(bpα)j([a, b]pα)12(p

α_−1)ij+k = 1,

sinceα=β=γ. Thus exp(G) =pα=pβ=pγ.

(b)α=β > γ.We obtain

gpα = aipαbjpβ[a, b]ij(pα2 )+kp

α

= (apα)i(bpβ)j[a, b]pα2 (p

α_−1)ij+kpα

= (apα)i(bpα)j([a, b]pα)12(p

α_−1)ij+k = 1,

sinceα=β > γandapα−1 6= 1, bpα−16= 1.Thus exp(G) =pα=pβ.

Now, we are ready to compute the exterior squares and determine which of the groups in Theorem 2.1 are capable.

4

The Exterior Squares And The Capability of Groups

In this section, we will arrive at our main objective to compute the exterior squares of some of the groups classified in Theorem 2.1 and to determine which of these groups are capable. Some results have been obtained by earlier papers. Baeyl [6] has already proved that an extra special p-group is capable if and only if it is either dihedral of order eight or of order

p3_{and exponentp >2. He also mentioned that all dihedral groups are capable. Beurle and}

Kappe [3], have computed the exterior square for the infinite metacyclic group. They have also determined that the only nonabelian infinite metacyclic group that is capable isZoZ₂, that is the infinite dihedral group. Specifically, if H =ZoZ4, thenZoZ2 ∼=H/Z(H).

Some of these results will be used to determine the capability of groups according to the classification in Theorem 2.1.

of type (2.1.1) for the case p 6= 2 and α= β = γ = 1 are capable. The computation of exterior squares and the determination of the capability of groups for the case p6= 2 and

p= 2 have to be done separately since they have different nonabelian tensor squares and conditions of parameters. At the end of this section, we will also show that D4 is capable

sinceD4∼=D8/Z(D8).

Theorem 4.1 below is needed to compute the exterior square of groups of type (2.1.2). For the complete proof of the theorem we refer to the paper by Bacon and Kappe [1].

Theorem 4.1 [1] Let G be a group of type (2.1.2) with p 6= 2, that is G ∼= hai o

hbi, whereα, β, γare integers with[a, b] =apα−γ

,|a|=pα_,|b|=pβ_{,|[a, b]|=p}γ_{, α≥2γ, β≥}

γ≥1.Then

G⊗G∼=Z_p2β × Zpα−γ× Z_min{pα−γ_,pβ},

generated by (a⊗a),(b⊗b),(a⊗b)(b⊗a) and(b⊗a).The orders of the generators are,

|(a⊗a)| = pα−γ,

|(b⊗b)| = pβ,

|(a⊗b)(b⊗a)| = pβ,

|(b⊗a)| = min{pα−γ, pβ}.

Next, we will compute the exterior square of groups of type (2.1.2) for the casep6= 2.

Proposition 4.1 Let G be a group of type (2.1.2) with p 6= 2, that is G ∼= haio

hbi, where[a, b] =apα−γ

,|a|=pα_,|b|=pβ_{,|[a, b]|=p}γ_{, α, β, γ ∈ N, α≥2γ, β≥γ.Then,}

G∧G= (G⊗G)/hx⊗x|x∈Gi ∼=hb⊗ai

where hb⊗aihas the order of min(pα−γ_{, p}β_).

Proof By Theorem 4.1,G⊗Gis generated bya⊗a, b⊗b,(a⊗b)(b⊗a) andb⊗awhere their orders arepα−γ_{, p}β_{, p}β _{and min{p}α−γ_{, p}β_{},respectively. By Lemma 3.1, we have}

hx⊗x|x∈Gi=ha⊗a, b⊗b,(a⊗a)(b⊗a)i.Thus, we conclude that

G∧G∼=hb⊗ai

with the order of min{pα−γ, pβ}.

Now we state the tensor square for groups of type (2.1.2) for the casep= 2, which will be used to compute their exterior squares.

Theorem 4.2 [7] Let G be a group of type (2.1.2) with p = 2, that is G ∼= hai o

hbi, whereα, β, γare integers with[a, b] =a2α−γ,|a|= 2α,|b|= 2β,|[a, b]|= 2γ, α≥2γ, β≥ γ≥1, α+β >3.Then

G⊗G∼=Z₂α−γ+1× Z₂β× Z₂min{α−γ}× Z₂min{α,β},

generated by (a⊗a),(b⊗b),(a⊗b)(b⊗a) and(b⊗a).The orders of the generators are,

|(a⊗a)| = 2α−γ+1, |(b⊗b)| = 2β,

|(a⊗b)(b⊗a)| = 2min{α,β,}

In the next proposition, we will compute the exterior squares of groups of type (2.1.2) for the case p= 2.

Proposition 4.2 Let G be a group of type (2.1.2) with p = 2, that is G ∼= haio

hbi, where[a, b] =a2α−γ

,|a|= 2α_{,|b|= 2}β_{,|[a, b]|= 2}γ_{, α, β, γ∈ N, α≥2γ, β≥γ, α+β >}

3.Then,

G∧G= (G⊗G)/hx⊗x|x∈Gi ∼=hb⊗ai

where hb⊗aihas the order of 2min{α,β}_.

Proof By Theorem 4.1,G⊗Gis generated bya⊗a, b⊗b,(a⊗b)(b⊗a) andb⊗awhere their orders are 2α−γ+1_,2β_,2min{α−γ,β} _{and 2}min{α,β}_{,respectively. By Lemma 3.1, we have}

hx⊗x|x∈Gi=ha⊗a, b⊗b,(a⊗a)(b⊗a)i.Thus, we conclude that

G∧G∼=hb⊗ai

with the order of 2min{α,β}.

Now, we are ready to determine the capability of groups of type (2.1.2) using Theorem 3.3.

Proposition 4.3 LetG=ha, bi be a two-generatorp-group (p6= 2) of class two and type (2.1.2), that is G∼=haiohbi, where[a, b] =apα−γ_,|a|=pα_,|b|=pβ_{,|[a, b]|=p}γ_,

α, β, γ∈ N, α≥2γ, β≥γ.Then

Z∗(G)6= 1, and thusGis not capable.

Proof By Proposition 4.1,G∧Gis a cyclic group of order min(pα−γ_{, p}β_{). We will show}

that 16=apα−γ

= [a, b]∈Z∗_(G).Letg=am_bn_{[a, b]}k_{.We have}

g⊗[a, b] = ambn[a, b]k⊗[a, b]

= (ambn⊗[a, b])([a, b]k⊗[a, b])([ambn,[a, b]k]⊗[a, b])([a, b]k⊗[ambn,[a, b]]).

We already know that [a, b]k_{⊗[a, b] = [a}k_{, b]⊗[a, b] = 1}

⊗,and obviously [ambn,[a, b]] = 1.

Then

g⊗[a, b] = ambn⊗[a, b]

= (am⊗[a, b])(bn⊗[a, b])([am, bn]⊗[a, b])(bn⊗[am,[a, b]])

= (am⊗[a, b])(bn⊗[a, b])

= (am⊗apα−γ)(bn⊗apα−γ).

So we obtain

g⊗[a, b] = (am⊗apα−γ)(bn⊗apα−γ)

= (am⊗apα−γ)(b⊗a)npα−γ(a⊗[b, a])n(pα

−γ

2 )_{(b⊗[b, a])}p

α−γ₍n

2)

= (am⊗apα−γ)(b⊗a)npα−γ(a⊗a−pα−γ)n(pα

−γ

2 )_{(b⊗[b, a])}p

α−γ₍n

Obviously,

am⊗apα−γ = (a⊗a)mpα−γ(a⊗[a, a])m(pα

−γ

2 )_{(a⊗[a, a])}p

α−γ₍m

2)

= (a⊗a)mpα−γ = 1⊗,

since by Theorem 4.1|a⊗a|=pα−γ _and

a⊗a−pα−γ = (a⊗a)−pα−γ(a⊗[a, a])(−pα

−γ

2 )_{(a⊗[a, a])}−p

α−γ₍1 2)

= (a⊗a)−pα−γ = ((a⊗a)pα−γ)−1 = 1⊗.

It follows that, g⊗[a, b] = (b⊗a)npα−γ(b⊗[b, a])pα−γ(n2) _{modhx⊗x|x∈Gi.By Theorem}

4.1, (b⊗a)npα−γ _{= 1}

⊗,and (b⊗[b, a])p

α−γ

=b⊗[b, a]pα−γ _{=b⊗1 = 1}

⊗,since [b, a]p

α−γ = (a−pα−γ₎pα−γ _{= 1.}

Thusg⊗[a, b]≡1⊗ modhx⊗x|x∈Gi.Therefore, the epicenter for this type of group

is not trivial. Hence, by Theorem 3.3Gis not capable.

In our next proposition, we will prove that groups of type (2.1.1) for the casep6= 2 and

α=β=γ= 1 is capable by Proposition 3.1.

Proposition 4.4 LetG=ha, bi be a two-generatorp-group (p6= 2) of class two and type (2.1.1) that is, G ∼= (hci × hai)ohbi, where [a, b] = c,[a, c] = [b, c] = 1,|a| = pα_{,|b| =}

pβ_,|c|=pγ_{, α, β, γ∈ N, α≥β≥γ.If α=β =γ= 1, thenGis capable.}

Proof By Lemma 3.3, we obtain

exp(G) =pγ=p,

sinceα=β=γ= 1.Thus, by Proposition 3.1,Gis capable.

As an example, the Burnside group, B(2,3), is capable using this proposition. The Burnside group is a group of type (2.1.1) in Theorem 2.1 with p= 3 andα=β =γ= 1, presented by

B(2,3) =ha, b|a3=b3= (ab)3= (a−1b)3= 1i.

Obviously, according to Lemma 3.2 and Lemma 3.3, exp(G) for other cases is a power ofpwhich is not a prime. Thus, we cannot use Proposition 3.1 to determine the capability of other types of groups classified in Theorem 2.1.

5

Conclusion

6

Acknowledgements

The authors would like to thank the Research and Management Center (RMC), UTM for the support in this work under Vot 71718.

References

[1] M. R. Bacon, L.-C. Kappe,The nonabelian tensor square of a 2-generator p-group of class 2.Arch. Math.61 (1993), 508-516.

[2] R. Baer, Groups with preassigned central and central quotient groups. Trans. Amer. Math Soc.44 (1938), 387-412.

[3] J. R. Beuerle, L.-C. Kappe, Infinite Metacylic Groups and Their Nonabelian Tensor Squares.Proc. Edinburgh Math. Soc.43 (2000), 651-662.

[4] G. Ellis,Capability, Homology, and Central Series of a Pair of Groups.J. of Algebra.

179 (1996), 31-46.

[5] G. Ellis, On the Capability of Groups. Proceedings Edinburgh Math. Soc. (2). 41

(1998), 487-495.

[6] U. Felgner, F.R Baeyl, P. Schmid, On Groups Occuring as Center Factor Groups. J. of Algebra.61 (1979), 161-177.

[7] L.-C. Kappe, N. Sarmin, M.P. Visscher,Two-generator Two-groups of Class Two and Their Nonabelian Tensor Squares. Glasgow Math. J.41(1999), 417-430.