The Schur multiplier of groups of order 𝑝5

Sumana Hatui; Vipul Kakkar; Manoj K. Yadav

doi:10.1515/jgth-2018-0139

Publicly Available Published by De Gruyter January 20, 2019

The Schur multiplier of groups of order 𝑝⁵

Sumana Hatui , Vipul Kakkar and Manoj K. Yadav

From the journal Journal of Group Theory

https://doi.org/10.1515/jgth-2018-0139

Abstract

In this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian p-groups of order p5. As an application, we determine the capability of groups of order p5.

1 Introduction

The Schur multiplier M⁡(G) of a group G was invented by I. Schur in [42, 43] as an obstruction for a projective representation to become a linear representation, and is defined as the second integral homology group H2⁡(G,ℤ), where ℤ is a trivial G-module. It is well known that for a finite group G, the group M⁡(G) is isomorphic to the second cohomology group H2⁡(G,ℂ⋆), where ℂ⋆ is a trivial G-module. There has been a considerable interest in understanding the Schur multipliers of finite p-groups in recent past. Finite p-groups with trivial Schur multiplier have been investigated in [13, 34, 44]. Study of the Schur multiplier of finite p-groups of a given coclass is taken up in [12, 33]. Bounds on the order of the Schur multiplier of finite p-groups are obtained, and groups attaining these bounds are classified in [1, 9, 35, 36, 38, 39]. Bounds on the exponent of the Schur multiplier are obtained in [30, 19, 41].

In spite of all the efforts made, it has always been challenging to compute the Schur multiplier of an arbitrary finite p-group with a given presentation. Explicit computations of the Schur multipliers of certain classes of groups not only have applications in mathematics but also in physics. Interested readers may like to have a look at [16] for such an application in string theory. Some efforts have been made to explicitly compute the Schur multiplier for some special classes of groups, e.g., extraspecial p-groups [5], metacyclic group [2, 4], groups of order p4 [37], groups of order p2⁢q and p2⁢q⁢r [24]. Another related topic of current interest is the computation of non-abelian tensor (and exterior) squares of groups (see Section 2 for the definition), which is extremely useful for computing Schur multipliers. Many mathematicians have studied this topic in the recent past, and explicit computations of non-abelian tensor squares of various classes of finite groups have been taken up; see, e.g., [8, 6, 22, 23, 24, 25]. For a detailed survey on this topic, see [7].

It seems that explicit computations for more classes of finite p-groups will help in understanding the Schur multiplier. With this viewpoint, in this paper, we compute the structure of the Schur multiplier of groups of order p5. On the way, we also compute the non-abelian tensor square and non-abelian exterior square of these groups. We sometimes compute the Schur multiplier of a group and use this to compute the non-abelian tensor square, and sometimes the other way round. The technique of the proof depends strongly on the given group, and therefore, the article uses a blend of almost all known techniques. As an application, we categorise capable and non-capable groups of order p5. A group G is said to be capable if there exists a group H such that G≅H/Z⁡(H), where Z⁡(H) denotes the center of H.

The layout of the article is as follows. In Section 2, we present preliminary results including the Schur multiplier and tensor square of groups of order p3 and p4, p≥5. In the following sections, the Schur multiplier, exterior square and tensor square of groups G of order p5, p≥5, are computed. Section 3 deals with special p-groups of order p5. In Section 4, we consider groups of maximal class. In Section 5, all remaining groups of order p5 are dealt with, and for p≥5, our main results are presented in Table 2. The results on the capability of these groups are presented in Table 3. Similar results for the groups of order 25 and 35 can be easily obtained using HAP [15] of GAP [47]. A complete list of these results can be provided on request. Since our proofs depend heavily on the presentations of the groups of order p5 from the paper of James [26], the reader is advised to keep this paper handy.

The set of identities, obtained by computation in the proofs, is usually sequential, i.e., the first might have been used for obtaining the second, and the first and second for the third, and so on. The presentation of the article is a bit unusual. We only prove lemmas, in each of which we compute required information for specific classes of groups sharing the technique of proof. Our main result is a combination of all the lemmas.

We conclude this section by fixing some notation. The commutator and Frattini subroup of a finite group G are denoted by G′ and Φ⁢(G), respectively. By d⁢(G) we denote the cardinality of a minimal generating set of a finitely generated group G. By ℤp(k) we denote ℤp×ℤp×⋯×ℤp (k times). For a group G, γi⁢(G) denotes the i-th term of the lower central series of G, and Ga⁢b denotes the quotient group G/γ2⁢(G). Notice that γ2⁢(G)=G′. The epicenter of a group G, denoted by Z*⁢(G), is defined to be the smallest central subgroup K of G such that G/K is capable.

There are 10 isoclinism classes Φi, 1≤i≤10, of groups of order p5 as per the classification by James [26]. The groups in Φ1 are abelian groups. We recall some notation from [26]. For an element αi+1 of a finite p-group G, by αi+1(p), we mean

αi+1p⁢αi+2(p2)⁢⋯⁢αi+k(pk)⁢⋯⁢αi+p,

where αi+2,…,αi+p are suitably defined elements of G. Observe that, for the groups under consideration in this article, αi(p)=αip for p>3. By ν we denote the smallest positive integer which is a non-quadratic residue (mod p), and by ζ we denote the smallest positive integer which is a primitive root (mod p). Relations of the form [α,β]=1 for generators α and β are omitted in the presentations of the groups.

2 Preliminaries

The non-abelian tensor productG⊗H of two groups G and H, acting on each other and satisfying certain compatibility conditions, was defined by Brown and Loday [11] as a generalization of the abelian tensor product. In particular, when a group G acts on itself by conjugation then G⊗G is called the non-abelian tensor square which is defined as follows. Let G act on itself by conjugation, i.e., hg=g-1⁢h⁢g for all h,g∈G. Then the non-abelian tensor square G⊗G of G is the group generated by the symbols g⊗h for all g,h∈G, subject to the relations

g⁢g′⊗h=(gg′⊗hg′)⁢(g′⊗h),

g⊗h⁢h′=(g⊗h′)⁢(gh′⊗hh′)

for all g,g′,h,h′∈G. The non-abelian exterior square of G, denoted by G∧G, is the quotient group of G⊗G by ∇⁡(G), where ∇⁡(G) is the normal subgroup generated by the elements g⊗g for all g∈G. It follows from the definition that the map f:G⊗G→G′, defined on the generators by f⁢(g⊗h)=[g,h], is an epimorphism, where [g,h]=g-1⁢h-1⁢g⁢h. The epimorphism f then induces an epimorphism f′:G∧G→G′. The kernel of f′ is isomorphic to the Schur multiplier M⁡(G) of G (see [11]).

We now present a different description of G⊗G, introduced in [40], which, sometimes, is more useful for evaluating the tensor square of a group G. By Gϕ we denote the isomorphic image of a group G via the isomorphism ϕ. Consider the group

ν(G):=〈G,Gϕ∣ℜ,ℜϕ,[g1,g2ϕ]g=[g1g,(g2g)ϕ]=[g1,g2ϕ]gϕfor allg,g1,g2∈G〉,

where ℜ,ℜϕ are the defining relations of G and Gϕ, respectively.

Recall that the commutator subgroup of G and Gϕ in ν⁢(G) is defined as

[G,Gϕ]=〈[g,hϕ]∣g,h∈G〉.

Then the map Φ:G⊗G→[G,Gϕ], defined by

Φ⁢(g⊗h)=[g,hϕ],g,h∈G,

is an isomorphism [40]. Let π:[G,Gϕ]→[G,Gϕ]/〈[g,gϕ]∣g∈G〉 be the natural projection. Then we have an isomorphism between G∧G and

[G,Gϕ]/〈[g,gϕ]∣g∈G〉.

In the rest of this section, we list some known results, which we will use in the upcoming sections.

Theorem 2.1 ([43]).

For two groups H and K,

M⁡(H×K)≅M⁡(H)×M⁡(K)×(H/H′⊗K/K′).

Theorem 2.2 ([45]).

If a group G is a semidirect product of a normal subgroup N and a subgroup T, and M is a G-module with trivial G-action, then the following sequence is exact:

1→H1⁡(T,Hom⁡(N,M))→H2(G,M)2→H2(N,M)T→H2(T,Hom(N,M)),

where

H2(G,M)2=Ker(resTG:H2(G,M)→H2(T,M)),

and H2(N,M)T is the T-stable subgroup of H2⁡(N,M).

Theorem 2.3 ([27, Theorem 3.1]).

Let G be a finite group and K any normal subgroup of G. Set H=G/K. Then

|M⁡(H)| divides |M⁡(G)|⁢|G′∩K|,
d⁢(M⁡(H))≤d⁢(M⁡(G))+d⁢(G′∩K).

Theorem 2.4 ([27, Theorem 4.1]).

Let G be a finite group and K a central subgroup of G. Set A=G/K. Then

|M⁡(G)|⁢|G′∩K| 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 |M⁡(A)|⁢|M⁡(K)|⁢|Aa⁢b⊗K|.

Theorem 2.5 ([28, Theorem 3.1]).

Let G be a finite group and N any normal subgroup such that G/N is cyclic. Then

|M⁡(G)| divides |M⁡(N)|⁢|N/N′|,
d⁢(M⁡(G))≤d⁢(M⁡(N))+d⁢(N/N′).

Theorem 2.6 ([17]).

Let Z be a central subgroup of a finite group G. Then the following sequence is exact

G/G′⊗Z→𝜆M⁡(G)→𝜇M⁡(G/Z)→G′∩Z→1,

where λ is the Ganea map.

Theorem 2.7 ([3]).

Let Z be a central subgroup of a finite group G. Consider the Ganea map λ:G/G′⊗Z→M⁡(G). Then Z⊆Z*⁢(G) if and only if G/G′⊗Z is contained in the kernel of λ.

Lemma 2.8 ([32, Lemma 9]).

If a group G has nilpotency class ≤5, then

[xn,y]=[x,y]n⁢[x,y,x](n2)⁢[x,y,x,x](n3)⁢[x,y,x,x,x](n4)⁢[x,y,x,[x,y]]σ⁢(n)

for x,y∈G and any positive integer n, where σ⁢(n)=n⁢(n-1)⁢(2⁢n-1)/6.

The subsequent results follow from [8] and [40].

Lemma 2.9.

For a group G, the following properties hold in ν⁢(G).

If G is nilpotent of class c, then ν⁢(G) is nilpotent of class at most c+1.
If G is a p-group, then ν⁢(G) is a p-group.
[g1ϕ,g2,g3]=[g1,g2ϕ,g3]=[g1,g2,g3ϕ]=[g1ϕ,g2ϕ,g3]=[g1ϕ,g2,g3ϕ]=[g1,g2ϕ,g3ϕ] for all g1,g2,g3∈G.
If either g∈G′ or h∈G′, then [g,hϕ]=[h,gϕ]-1.
[g,gϕ]=1 for all g∈G′.
[[g1,g2ϕ],[h1,h2ϕ]]=[[g1,g2],[h1,h2]ϕ] for all g1,g2,h1,h2∈G.
[[g1,g2ϕ],[g2,g1ϕ]]=1 for all g1,g2∈G.
If g,g1,g2∈G such that [g,g1]=1=[g,g2], then [g1,g2,gϕ]=1.
[g,gϕ] is central in ν⁢(G) for all g∈G.

The next result follows from the proof of [40, Lemma 2.1 (iv)].

Lemma 2.10.

For all g1,g2∈G, [g1,g2ϕ]=[g2,g1ϕ]-1 in G∧G, i.e., modulo ∇⁡(G).

Proposition 2.11 ([8, Proposition 20]).

Let G be a polycyclic group with a polycyclic generating sequence g1,…,gk. Then G∧G is generated by

{[gi,gjϕ],i>j}.

The generating sets for the groups G, given in [26], form polycyclic generating sequences. Then Proposition 2.11 provides a generating set for G∧G. This information will be used several times throughout the article without any further reference.

By items (vi) and (viii) of Lemma 2.9, we get the following lemma.

Lemma 2.12.

If G is of nilpotency class 2, then G⊗G is abelian.

Let Γ denote Whitehead’s quadratic functor (see [46]). For finite abelian groups G and H, we get

Γ⁢(G×H)≅Γ⁢(G)×Γ⁢(H)×(G⊗H),

Γ⁢(ℤn)={ℤn,n⁢odd,ℤ2⁢n,n⁢even.

Theorem 2.13 ([6, Theorem 1.3]).

Let Ga⁢b be finitely generated and have no element of order 2. Then G⊗G≅Γ⁢(Ga⁢b)×(G∧G). In particular, if G is a finite p-group, p odd, then G⊗G≅Γ⁢(Ga⁢b)×(G∧G).

Proposition 2.14 ([10, Proposition 11]).

For the groups G and H, we have

(G×H)⊗(G×H)=(G⊗G)×(G⊗H)×(H⊗G)×(H⊗H).

The non-abelian tensor square of groups of order p4 has already been computed in [18]. The purpose of the following result is to compute the generators of G∧G for the groups G of order p3 and p4, p≥5.

Theorem 2.15.

We have Table 1 for non-abelian groups of order p3 and p4, p≥5.

Table 1

Groups of order p3 and p4, p≥5.

G	Ga⁢b	Γ⁢(Ga⁢b)	M⁡(G)	G∧G	G⊗G	Generators of G∧G
Φ2⁢(21)	ℤp(2)	ℤp(3)	{1}	ℤp	ℤp(4)	[α1,αϕ]
Φ2⁢(111)	ℤp(2)	ℤp(3)	ℤp(2)	ℤp(3)	ℤp(6)	[α1,αϕ], [α2,αϕ], [α2,α1ϕ]
Φ2⁢(211)⁢a	ℤp(3)	ℤp(6)	ℤp(2)	ℤp(3)	ℤp(9)	[α1,αϕ], [α3,αϕ], [α3,α1ϕ]
Φ2⁢(14)	ℤp(3)	ℤp(6)	ℤp(4)	ℤp(5)	ℤp(11)	[α1,αϕ], [α2,αϕ], [α2,α1ϕ], [α3,αϕ], [α3,α1ϕ]
Φ2⁢(31)	ℤp2×ℤp	ℤp2×ℤp(2)	{1}	ℤp	ℤp2×ℤp(3)	[α1,αϕ]
Φ2⁢(22)	ℤp2×ℤp	ℤp2×ℤp(2)	ℤp	ℤp2	ℤp2(2)×ℤp(2)	[α1,αϕ]
Φ2⁢(211)⁢b	ℤp(3)	ℤp(6)	ℤp(2)	ℤp(3)	ℤp(9)	[α1,αϕ], [γ,αϕ], [γ,α1ϕ]
Φ2⁢(211)⁢c	ℤp2×ℤp	ℤp2×ℤp(2)	ℤp(2)	ℤp(3)	ℤp2×ℤp(5)	[α1,αϕ], [α2,αϕ], [α2,α1ϕ]
Φ3⁢(211)⁢a	ℤp(2)	ℤp(3)	ℤp	ℤp(3)	ℤp(6)	[α1,αϕ], [α2,αϕ], [α2,α1ϕ]
Φ3⁢(211)⁢br	ℤp(2)	ℤp(3)	ℤp	ℤp(3)	ℤp(6)	[α1,αϕ], [α2,αϕ], [α2,α1ϕ]
Φ3⁢(14)	ℤp(2)	ℤp(3)	ℤp(2)	ℤp(4)	ℤp(7)	[α1,αϕ], [α2,αϕ], [α2,α1ϕ], [α3,αϕ]

Proof.

Schur multipliers of groups of order p4 are taken from [37] for |G′|=p and from [14, p. 4177] for |G′|=p2. The Schur multipliers of groups of order p3 are well known. So we mainly work for computing the exterior squares. Tensor squares will then follow easily by Theorem 2.13. We work in the group ν⁢(G).

In the class Φ2, we only work out the exterior square of Φ2⁢(211)⁢b. The other cases go on the same lines.

Consider the group G:=Φ2(211)b. Since |M⁡(G)|=p2 and |γ2⁢(G)|=p, it follows that |G∧G|=p3. By Lemma 2.9 (viii), we have

[α2,γϕ]=[α1,α,γϕ]=1,as⁢γ∈Z⁡(G).

By Lemma 2.8, the following identities hold:

[α2,αϕ]=[γp,αϕ]=[γ,αϕ]p=[γ,(αp)ϕ]=1,

[α2,α1ϕ]=[γp,α1ϕ]=[γ,α1ϕ]p=[γ,(α1p)ϕ]=1,

[α1,αϕ]p=[α1p,αϕ]=1.

These identities, along with Proposition 2.11 and Lemma 2.12, imply that G∧G is generated by [α1,αϕ], [γ,αϕ], [γ,α1ϕ], all of which are of order p, and hence G∧G≅ℤp(3).

Now we work out the exterior square of G=Φ3⁢(14). Since |M⁡(G)|=p2, |G∧G|=p4. By Proposition 2.11, G∧G is generated by the set

{[α1,αϕ],[α2,αϕ],[α2,α1ϕ],[α3,αϕ],[α3,α1ϕ],[α2,α3ϕ]}.

By Lemma 2.9 (viii), we have [α2,α3ϕ]=[α1,α,α3ϕ]=1 as α3∈Z⁡(G). The Hall–Witt identity yields

1=[α2,α,α1ϕ]α-1⁢[α-1,α1-1,α2ϕ]α1=[α3,α1ϕ]⁢[α1⁢α⁢α2-1⁢α-1⁢α1-1,α2ϕ]α1=[α3,α1ϕ]⁢[α⁢α2-1⁢α-1,α2ϕ]=[α3,α1ϕ]⁢[α3⁢α2-1,α2ϕ]=[α3,α1ϕ]⁢[α3,α2ϕ]⁢[α2,α2ϕ]-1=[α3,α1ϕ]⁢[α2,α2ϕ]-1.

Consequently, [α3,α1ϕ]=[α2,α2ϕ]=1 in G∧G. Also, by Lemma 2.8, we get

[α3,αϕ]p=[α3p,αϕ]=1=[α2p,α1ϕ]=[α2,α1ϕ]p,

=p[α2p,αϕ]=1=[α1p,αϕ]=[α1,αϕ]p.

Hence, in view of Lemma 2.9 (vi),

G∧G≅〈[α1,αϕ],[α2,αϕ],[α2,α1ϕ],[α3,αϕ]〉≅ℤp(4).

The remaining cases go on the same lines. ∎

3 Classes Φ4,Φ5: Groups of class 2 with G/G′ elementary abelian

Throughout the article, we make calculations in the subgroup [G,Gϕ] of ν⁢(G) modulo ∇⁡(G); i.e., we work in G∧G. For commutator and power calculations, we use Lemma 2.8 without any further reference.

First we consider extra-special p-groups of order p5, for which we have the following result.

Lemma 3.1.

If G is one of the groups Φ5⁢(2111) or Φ5⁢(15), then M⁡(G) is isomorphic to Zp(5) and G∧G is isomorphic to Zp(6).

Proof.

The groups G are extra-special groups. So it follows from [29, Theorem 3.3.6 (i)] that M⁡(G) is an elementary abelian p-group of order p5. By [31, Corollary 2.3], we have G⊗G≅ℤp(16). Now, by Theorem 2.13, G∧G≅ℤp(6), which completes the proof. ∎

Now let G be a finite p-group of class 2 such that G/G′ and G′ are elementary abelian of order p3 and p2, respectively. We consider G/G′ and G′ as vector spaces over 𝔽p, which we denote by V,W, respectively. The bilinear map (-,-):V×V→W is defined by (v1,v2)=[g1,g2] for v1,v2∈V such that vi=gi⁢G′,i∈{1,2}. The following construction is from [5]. Let X1 be the subspace of V⊗W spanned by all

v1⊗(v2,v3)+v2⊗(v3,v1)+v3⊗(v1,v2)

for v1,v2,v3∈V. Consider a map f:V→W given by f⁢(g⁢G′)=gp for g∈G. We denote by X2 the subspace spanned by all v⊗f⁢(v), v∈V, and take

X:=X1+X2.

Now consider a homomorphism σ:V∧V→(V⊗W)/X given by

σ⁢(v1∧v2)=(v1⊗f⁢(v2)+(p2)⁢v2⊗(v1,v2))+X.

Then there exists an abelian group M* admitting a subgroup N, isomorphic to (V⊗W)/X, such that

1→N→M*→𝜉V∧V→1

is exact. Now we consider a homomorphism ρ:V∧V→W given by

ρ⁢(v1∧v2)=(v1,v2) for all⁢v1,v2∈V.

Notice that ρ is an epimorphism. Denote by M the subgroup of M* containing N such that M/N≅Ker⁡ρ. Then it follows that |M/N|=|V∧V|/|W|, which will be used for calculating the order of M without any further reference.

With the above setting, we have the following theorem.

Theorem 3.2 ([5, Theorem 3.1]).

M⁡(G)≅M.

The following information will be used throughout the rest of this section without further reference.

Remark 3.1.

Let G be any group in the isoclinism class Φ4. Consider the natural epimorphism

[G,Gϕ]→[G/Z⁡(G),(G/Z⁡(G))ϕ].

Since G/Z⁡(G)∧G/Z⁡(G) is elementary abelian of order p3, it follows that the elements [α1,αϕ], [α2,αϕ], [α2,α1ϕ] are non-trivial and independent in G∧G. Furthermore, by Lemma 2.12, G⊗G is abelian.

Lemma 3.3.

If G is one of the groups

Φ4⁢(221)⁢a,Φ4⁢(221)⁢b,Φ4⁢(221)⁢c,Φ4⁢(221)⁢d12⁢(p-1),Φ4(221)dr(r≠12(p-1)),Φ4(221)e,Φ4(221)f0 𝑜𝑟 Φ4(221)fr,

then M⁡(G) is isomorphic to Zp, Zp×Zp, Zp, Zp2, Zp, Zp, Zp2 or Zp, respectively, and G∧G is isomorphic to Zp(3), Zp2×Zp×Zp, Zp(3), Zp2×Zp×Zp, Zp(3), Zp(3), Zp2×Zp×Zp or Zp(3), respectively.

Proof.

For the group G=Φ4⁢(221)⁢a, notice that X1 is spanned by

α1⁢G′⊗αp-α2⁢G′⊗α1p,

and dim⁡X1=1. Observe that α⁢G′⊗αp, α1⁢G′⊗α1p∈X2 and (α⁢G′+u⁢G′)⊗αp, (α1⁢G′+u⁢G′)⊗α1p∈X2 for u⁢G′∈Ker⁡f. So we have u⁢G′⊗αp, u⁢G′⊗α1p∈X2 for all u⁢G′∈Ker⁡f. This implies α2⁢G′⊗αp, α2⁢G′⊗α1p∈X2. Now a general element of X2 is of the form

(p1⁢α⁢G′+p2⁢α1⁢G′+p3⁢α2⁢G′)⊗(p1⁢αp+p2⁢α1p)=p12⁢α⁢G′⊗αp+p22⁢α1⁢G′⊗α1p+p1⁢p2⁢(α⁢G′⊗α1p+α1⁢G′⊗αp)+p3⁢p1⁢(α2⁢G′⊗αp)+p3⁢p2⁢(α2⁢G′⊗α1p).

This shows that X2 is spanned by the set

{α⁢G′⊗αp,α1⁢G′⊗α1p,α2⁢G′⊗αp,α2⁢G′⊗α1p,(α⁢G′⊗α1p+α1⁢G′⊗αp)}.

Hence dim⁡X2=5. Observe that (α1⁢G′⊗αp-α2⁢G′⊗α1p) is not contained in X2. Thus dim⁡X=6, and consequently, |N|=1, |M|=p. Now, by Theorem 3.2, we have M⁡(G)≅ℤp, which gives |G∧G|=p3. Hence

G∧G=〈[α1,αϕ],[α2,αϕ],[α1,α2ϕ]〉≅ℤp(3).

For the group G=Φ4⁢(221)⁢b, X1 is spanned by (α1⁢G′⊗αp-α2⁢G′⊗α2p), and dim⁡X1=1. As described in the preceding case, the subspace X2 is spanned by the set

{α⁢G′⊗αp,α2⁢G′⊗α2p,α1⁢G′⊗αp,α1⁢G′⊗α2p,(α⁢G′⊗α2p+α2⁢G′⊗αp)},

and dim⁡X2=5. Observe that X1⊂X2, so dim⁡X=5 and |N|=p. By Theorem 3.2, |M|=|M⁡(G)|=p2. Hence |G∧G|=p4.

By Lemma 2.9 (viii), [β1,β2ϕ]=[α1,α,β2ϕ]=1. For i∈{1,2}, x∈{α,α1,α2}, we have (by Lemma 2.8)

[βi,xϕ]p=[βip,x]=1,[β2,xϕ]=[αp,xϕ]=[α,xϕ]p,

[β1,xϕ]=[α2p,xϕ]=[α2,xϕ]p,

[α,α1ϕ]p=[α,(α1p)ϕ]=1=[α2,(α1p)ϕ]=[α2,α1ϕ]p,

[α2,αϕ]p2=[α2p2,αϕ]=1.

Hence it follows that G∧G=〈[α2,αϕ],[α1,αϕ],[α1,α2ϕ]〉≅ℤp2×ℤp×ℤp and

M⁡(G)≅〈[α2,αϕ]p,[α1,α2ϕ]〉≅ℤp×ℤp.

For the group G=Φ4⁢(221)⁢c, X1 is spanned by (α1⁢G′⊗α2p-α2⁢G′⊗α1p), and dim⁡X1=1. It follows that X2 is spanned by the set

{α1⁢G′⊗α1p,α2⁢G′⊗α2p,α⁢G′⊗α1p,α⁢G′⊗α2p,(α1⁢G′⊗α2p+α2⁢G′⊗α1p)},

and dim⁡X2=5. So dim⁡X=6 and |N|=1, |M|=p. Hence

M⁡(G)≅ℤp and G∧G≅〈[α1,αϕ],[α2,αϕ],[α1,α2ϕ]〉≅ℤp(3).

For the group G=Φ4⁢(221)⁢dr, X1 is spanned by (α1⁢G′⊗α2p-α2⁢G′⊗β1), and dim⁡X1=1. The space X2 is spanned by the set

{α1⁢G′⊗β1k,α2⁢G′⊗α2p,α⁢G′⊗β1k,α⁢G′⊗α2p,(α1⁢G′⊗α2p+α2⁢G′⊗β1k)},

and dim⁡X2=5.

For r=12⁢(p-1), it follows that in the presentation of G, k≡-1(modp). So we have X1⊂X2. Hence dim⁡X=5, and therefore, |N|=p, |M⁡(G)|=p2 and |G∧G|=p4. By Lemma 2.9 (viii), [β1,β2ϕ]=[α1,α,β2ϕ]=1. For i∈{1,2}, x∈{α,α1,α2}, we have

[βi,xϕ]p=[βip,x]=1,[β2,xϕ]=[α2p,xϕ]=[α2,xϕ]p,

[β1,xϕ]=[α1-p,xϕ]=[α1-1,xϕ]p=[α1,xϕ]-p,

[α,α1ϕ]p=[αp,α1ϕ]=1=[α2,(αp)ϕ]=[α2,αϕ]p,

[α1,α2ϕ]p2=[α1p2,α2ϕ]=1.

Hence it follows that

G∧G=〈[α1,α2ϕ],[α1,αϕ],[α2,αϕ]〉≅ℤp2×ℤp×ℤp,

and since [α1,α2]=1,

M⁡(G)≅〈[α1,α2ϕ]〉≅ℤp2.

For r≠12⁢(p-1), X1∩X2=∅, so dim⁡X=6, and therefore, |N|=1. Hence

M⁡(G)≅ℤp and G∧G=〈[α1,αϕ],[α2,αϕ],[α1,α2ϕ]〉≅ℤp(3).

For the group G=Φ4⁢(221)⁢e, X1 is spanned by (α1⁢G′⊗β2-α2⁢G′⊗β1), and dim⁡X1=1. The subspace X2 is spanned by the set

{α1⁢G′⊗β2-14,(α2⁢G′⊗β1+α2⁢G′⊗β2),α⁢G′⊗β2-14,(α⁢G′⊗β1+α⁢G′⊗β2),(α1G′⊗β1+α1G′⊗β2+α2G′⊗β2-14)},

and dim⁡X2=5. So dim⁡X=6. Therefore, |N|=1 and |M|=|M⁡(G)|=p. Hence

M⁡(G)≅ℤp⁢and⁢G∧G=〈[α1,αϕ],[α2,αϕ],[α1,α2ϕ]〉≅ℤp(3).

For the group G=Φ4⁢(221)⁢f0, X1 is spanned by (α1⁢G′⊗β2-α2⁢G′⊗β1), and dim⁡X1=1. The subspace X2 is spanned by the set

{α1⁢G′⊗β2,α2⁢G′⊗β1ν,α⁢G′⊗β2,α⁢G′⊗β1ν,(α1⁢G′⊗β1ν+α2⁢G′⊗β2)},

and dim⁡X2=5. Observe that X1⊂X2, so dim⁡X=5. Therefore,

|N|=p,|M|=|M⁡(G)|=p2.

By Lemma 2.9 (viii), [β1,β2ϕ]=[α1,α,β2ϕ]=1. For i∈{1,2}, x∈{α,α1,α2}, we have

[βi,xϕ]p=[βip,xϕ]=1,

[β2,xϕ]=[α1p,xϕ]=[α1,xϕ]p,

[β1,xϕ]=[α2p⁢ν-1,xϕ]=[α2,xϕ]p⁢ν-1,

[α,α1ϕ]p=[αp,α1ϕ]=1=[α2,(αp)ϕ]=[α2,αϕ]p.

Hence

G∧G=〈[α1,α2ϕ],[α2,αϕ],[α1,αϕ]〉≅ℤp2×ℤp×ℤp,

and since [α1,α2]=1, we have M⁡(G)≅〈[α1,α2ϕ]〉≅ℤp2.

On the same lines, we can prove that

M⁡(Φ4⁢(221)⁢fr)≅ℤp,

Φ4⁢(221)⁢fr∧Φ4⁢(221)⁢fr=〈[α1,αϕ],[α2,αϕ],[α1,α2ϕ]〉≅ℤp(3).

This completes the proof of the lemma. ∎

Lemma 3.4.

If G is one of the groups Φ4⁢(2111)⁢a, Φ4⁢(2111)⁢b, Φ4⁢(2111)⁢c or Φ4⁢(15), then M⁡(G) is isomorphic to Zp(3), Zp(3), Zp(3) or Zp(6), respectively, and G∧G is isomorphic to Zp(5), Zp(5), Zp(5) or Zp(8), respectively.

Proof.

For the group G=Φ4⁢(2111)⁢a, X1 is spanned by the element

(α1⁢G′⊗αp-α2⁢G′⊗β1),

and dim⁡X1=1. The space X2 is spanned by {α⁢G′⊗αp,α1⁢G′⊗αp,α2⁢G′⊗αp}, and dim⁡X2=3. So dim⁡X=4. Therefore |N|=p2, |M⁡(G)|=|M|=p3 and |G∧G|=p5. By Lemma 2.9 (viii), [β1,β2ϕ]=[α1,α,β2ϕ]=1.

For i∈{1,2}, x∈{α,α1,α2}, we have

[βi,xϕ]p=[βip,x]=1,

[α2,xϕ]p=[α2p,xϕ]=1=[α1p,αϕ]=[α1,αϕ]p.

Therefore, every generator of G∧G is of order at most p. Hence

G∧G≅ℤp(5) and M⁡(G)≅ℤp(3).

In the remaining three cases, the proof goes on the same lines. ∎

4 Groups of maximal class

Lemma 4.1.

If G is one of the groups Φ9⁢(2111)⁢a,Φ9⁢(2111)⁢br,Φ10⁢(2111)⁢ar or Φ10⁢(2111)⁢br, then M⁡(G) is isomorphic to Zp and G∧G is isomorphic to Zp(4).

Proof.

For the groups G under consideration, taking K=Z⁡(G) in Theorem 2.3 (i), we get p≤|M⁡(G)|. Thus |G∧G|≥p4.

If G is either Φ9⁢(2111)⁢a or Φ9⁢(2111)⁢br, then, by Lemma 2.9 (viii),

[α2,α4ϕ]=[α1,α,α4ϕ]=1=[α2,α,α4ϕ]=[α3,α4ϕ],

as α4∈Z⁡(G).

Now we have

[α-1,α1-1,α2ϕ]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α2ϕ]α1=[α⁢α2-1⁢α-1,α2ϕ]=[α⁢α3⁢α-1⁢α2-1,α2ϕ]=[α3⁢α4-1⁢α2-1,α2ϕ]=[α3,α2ϕ]⁢[α4,α2ϕ]-1=[α3,α2ϕ],

[α-1,α1-1,α3ϕ]α1=[α-1,α1-1,α3ϕ]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α3ϕ]α1=[α⁢α2-1⁢α-1,α3ϕ]=[α3⁢α4-1⁢α2-1,α3ϕ]=[α4,α3ϕ]-1⁢[α2,α3ϕ]-1=[α2,α3ϕ]-1.

By the Hall–Witt identity, we have

1=[α2,α,α1ϕ]α-1⁢[α-1,α1-1,α2ϕ]α1⁢[α1,α2-1,(α-1)ϕ]α2=[α3,α1ϕ]⁢[α3,α2ϕ],

1=[α3,α,α1ϕ]α-1⁢[α-1,α1-1,α3ϕ]α1⁢[α1,α3-1,(α-1)ϕ]α3=[α4,α1ϕ]⁢[α2,α3ϕ]-1.

This implies that [α3,α1ϕ]=[α2,α3ϕ]=[α4,α1ϕ].

Now consider the group G≅Φ9⁢(2111)⁢br. We have the following identities:

[α4,αϕ]p=[α4p,αϕ]=1=[α4p,α1ϕ]=[α4,α1ϕ]p,

[α3,αϕ]p=[α3p,αϕ]=1=[α3p,α1ϕ]=[α3,α1ϕ]p,

[α2,αϕ]p=[α2p,αϕ]=1=[α2p,α1ϕ]=[α2,α1ϕ]p,

[α4,αϕ]=[α1p⁢k-1,αϕ]=[α1,αϕ]p⁢k-1=[α1,(αp⁢k-1)ϕ]=1,

[α4,α1ϕ]=[α1p⁢k-1,α1ϕ]=[α1,α1ϕ]p⁢k-1=1,

[α1,αϕ]p=[α1,(αp)ϕ]=1.

Thus, by Proposition 2.11, G∧G is generated by

{[α1,αϕ],[α2,αϕ],[α3,αϕ],[α2,α1ϕ]}.

Since, by Lemma 2.9 (vi), [[α2,αϕ],[α1,αϕ]]=[α3,α2ϕ]=1, it follows that G∧G is elementary abelian of order p4. Hence G∧G≅ℤp(4), and consequently,

M⁡(G)≅〈[α2,α1ϕ]〉≅ℤp.

Similarly, we can compute

Φ9⁢(2111)⁢a∧Φ9⁢(2111)⁢a≅ℤp(4) and M⁡(Φ9⁢(2111)⁢a)≅ℤp.

If G is either Φ10⁢(2111)⁢ar or Φ10⁢(2111)⁢br, then, by Lemma 2.9 (viii),

[α2,α4ϕ]=[α1,α,α4ϕ]=1=[α2,α,α4ϕ]=[α3,α4ϕ],

[α-1,α1-1,α2ϕ]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α2ϕ]α1=[α⁢α2-1⁢α-1,(α1-1⁢α2⁢α1)ϕ]=[α⁢α3⁢α-1⁢α2-1,(α2⁢α4-1)ϕ]=[α3⁢α4-1⁢α2-1,(α2⁢α4-1)ϕ]=[α3,α2ϕ]⁢[α4,α2ϕ]-1=[α3,α2ϕ],

[α-1,α1-1,α3ϕ]α1=[α-1,α1-1,α3ϕ]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α3ϕ]α1=[α⁢α2-1⁢α-1,α3ϕ]=[α⁢α3⁢α-1⁢α2-1,α3ϕ]=[α3⁢α4-1⁢α2-1,α3ϕ]=[α4-1,α3ϕ]⁢[α2,α3ϕ]-1=[α2,α3ϕ]-1.

By the Hall–Witt identity, we have

1=[α2,α,α1ϕ]α-1⁢[α-1,α1-1,α2ϕ]α1⁢[α1,α2-1,(α-1)ϕ]α2=[α3,α1ϕ]⁢[α3,α2ϕ]⁢[α4-1,(α-1)ϕ]α2=[α3,α1ϕ]⁢[α3,α2ϕ]⁢[α4,αϕ],

1=[α3,α,α1ϕ]α-1⁢[α-1,α1-1,α3ϕ]α1⁢[α1,α3-1,(α-1)ϕ]α3=[α4,α1ϕ]⁢[α2,α3ϕ]-1.

This implies that [α2,α3ϕ]=[α4,α1ϕ]=[α4,αϕ]⁢[α3,α1ϕ].

Consider the group G≅Φ10⁢(2111)⁢ar. We have the following identities:

[α4,αϕ]p=[α4p,αϕ]=1=[α4p,α1ϕ]=[α4,α1ϕ]p,

[α3,αϕ]p=[α3p,αϕ]=1=[α3p,α1ϕ]=[α3,α1ϕ]p,

[α2,αϕ]p=[α2p,αϕ]=1=[α2p,α1ϕ]=[α2,α1ϕ]p,

[α4,αϕ]=[αp⁢k-1,αϕ]=[α,αϕ]p⁢k-1=1,

[α4,α1ϕ]=[αp⁢k-1,α1ϕ]=[α,α1ϕ]p⁢k-1=[α,(α1p⁢k-1)ϕ]=1,

[α1,αϕ]p=[α1p,αϕ]=1.

Thus G∧G is generated by {[α1,αϕ],[α2,αϕ],[α3,αϕ],[α2,α1ϕ]}. By Lemma 2.9 (vi), G∧G is elementary abelian of order p4. Hence G∧G≅ℤp(4), and consequently, M⁡(G)≅〈[α2,α1ϕ]⁢[α3,αϕ]〉≅ℤp.

Similarly, we can compute

Φ10⁢(2111)⁢br∧Φ10⁢(2111)⁢br≅ℤp(4) and M⁡(Φ10⁢(2111)⁢br)≅ℤp.

This completes the proof. ∎

We remark that the order of G⊗G has already been computed in [23, Theorem 3.1] for G≅Φ9⁢(15),Φ10⁢(15), which is p9 in both cases. Consequently, the orders of the Schur multiplier and exterior square of these groups are p3 and p6, respectively. Our motive here is to compute the structure of the Schur multiplier, exterior square and tensor square of these groups. One could use [23, Theorem 3.1] to shorten the proof of the following result, but, since our method of proof is different, we have chosen to present a complete proof.

Lemma 4.2.

If G is one of the groups Φ9⁢(15) or Φ10⁢(15), then M⁡(G) is isomorphic to Zp(3), and G∧G is isomorphic to Φ2⁢(111)×Zp(3).

Proof.

Consider the group G≅Φ9⁢(15). Let F be the free group generated by {α,α1}. Define αi+1=[αi,α], 1≤i≤3. Set β1=[α1,α2], β2=[α4,α] and β3=[α1,α4]. Then, modulo γ6⁢(F), we have

[α-1,α1-1,α3]α1=[α-1,α1-1,α3]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α3]α1=[α⁢α2-1⁢α-1,α3⁢[α3,α1]]=[α⁢α3⁢α-1⁢α2-1,α3⁢[α3,α1]]=[α3⁢α⁢α4-1⁢α-1⁢α2-1,α3⁢[α3,α1]]=[α3⁢α⁢β2⁢α-1⁢α4-1⁢α2-1,α3⁢[α3,α1]]=[α3⁢β2⁢α4-1⁢α2-1,α3⁢[α3,α1]]=[α2-1,α3]=[α2,α3]-1,

[α-1,α1-1,α2]α1=[α-1,α1-1,α2]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α2]α1=[α3⁢β2⁢α4-1⁢α2-1,α2⁢β1-1]=[α3,α2⁢β1-1]⁢[α2-1,α2⁢β1-1]=[α3,α2]⁢[α2,β1].

By the Hall–Witt identity, we have the following identities modulo γ6⁢(F):

(4.1)1=[α3,α,α1]α-1⁢[α-1,α1-1,α3]α1⁢[α1,α3-1,α-1]α3=[α4,α1]α-1⁢[α2,α3]-1⁢[α3,α1,α3-1⁢α-1⁢α3]=[α4,α1]⁢[α2,α3]-1⁢[α3,α1,α4⁢α-1]=[α4,α1]⁢[α2,α3]-1⁢[α3,α1,α-1],

(4.2)1=[α2,α,α1]α-1⁢[α-1,α1-1,α2]α1⁢[α1,α2-1,α-1]α2=[α3,α1]α-1⁢[α3,α2]⁢[α2,β1]⁢[β1-1,α2-1⁢α-1⁢α2]=[α3,α1]⁢[α3,α1,α-1]⁢[α3,α2]⁢[α2,β1]⁢[β1-1,α3⁢α-1]=[α3,α1]⁢[α3,α1,α]-1⁢[α3,α2]⁢[α2,β1]⁢[β1-1,α-1]=[α3,α1]⁢[α3,α1,α]-1⁢[α3,α2]⁢[α2,β1]⁢[β1,α].

Now consider

H1=F/〈γ6⁢(F),Fp,β1,β3,[α3,α1,α],[α3,α1,α1]〉.

Using (4.1) and (4.2), we have [α4,α1]=[α2,α3]=[α3,α1]=1 in H1. So

H1≅〈α,α1,α2,α3,α4,β2∣[αi,α]=αi+1,[α4,α]=β2,αp=α1p=αi+1p=β2p=1,i=1,2,3〉,

which is the group Φ35⁢(16) of order p6 in [26]. Now consider

H2=F/〈γ6⁢(F),Fp,β1,[α3,α1,α],[α3,α1,α1]〉.

Using (4.1) and (4.2), we have [α4,α1]=[α2,α3]=[α3,α1] in H2. Observe that

H2≅〈α,αi,α4,β2,β3∣[αi,α]=αi+1,[α4,α]=β2,[α1,α4]=[α1,α3]=[α3,α2]=β3,αp=αip=α4p=β2p=β3p=1, 1≤i≤3〉.

Since H2/〈β3〉≅H1, it follows that H2 is of order p7. Now consider

H=F/〈γ6⁢(F),Fp,[β1,α],[β1,α1],[α3,α1,α],[α3,α1,α1]〉.

Observe that

H≅〈α,αj,βi∣[αi,α]=αi+1,[α1,α2]=β1,[α4,α]=β2,[α1,α4]=[α1,α3]=[α3,α2]=β3,αp=αjp=βip=1, 1≤i≤3, 1≤j≤4〉.

Then H is a group of order p8 and H/〈β1,β2,β3〉≅G.

Take Z=〈β1,β2,β3〉. Now the image of the transgression homomorphism tra:Hom⁡(Z,ℂ*)→M⁡(H/Z) is H′∩Z≅Z≅ℤp(3). Hence ℤp(3) is contained in M⁡(G). By Theorem 2.4, |M⁡(G)|≤p3. Hence

M⁡(G)≅ℤp(3).

As |G′|=p3, we get |G∧G|=p6. Since α4∈Z⁡(G), by Lemma 2.9 (viii),

[α2,α4ϕ]=[α3,α4ϕ]=1.

Thus G∧G is generated by the set

{[α1,αϕ],[α2,αϕ],[α3,αϕ],[α4,αϕ],[α2,α1ϕ],[α3,α1ϕ],[α4,α1ϕ],[α3,α2ϕ]}.

Now we have

[α-1,α1-1,α2ϕ]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α2ϕ]α1=[α⁢α2-1⁢α-1,α2ϕ]=[α⁢α3⁢α-1⁢α2-1,α2ϕ]=[α3⁢α4-1⁢α2-1,α2ϕ]=[α3,α2ϕ]⁢[α4,α2ϕ]-1=[α3,α2ϕ],

[α-1,α1-1,α3ϕ]α1=[α-1,α1-1,α3ϕ]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α3ϕ]α1=[α⁢α2-1⁢α-1,α3ϕ]=[α⁢α3⁢α-1⁢α2-1,α3ϕ]=[α3⁢α4-1⁢α2-1,α3ϕ]=[α2,α3ϕ]-1.

By the Hall–Witt identity, we have

1=[α2,α,α1ϕ]α-1⁢[α-1,α1-1,α2ϕ]α1⁢[α1,α2-1,(α-1)ϕ]α2=[α3,α1ϕ]⁢[α3,α2ϕ],

1=[α3,α,α1ϕ]α-1⁢[α-1,α1-1,α3ϕ]α1⁢[α1,α3-1,(α-1)ϕ]α3=[α4,α1ϕ]⁢[α2,α3ϕ]-1.

This implies that [α4,α1ϕ]=[α2,α3ϕ]=[α3,α1ϕ]. By Lemma 2.9 (vi),

[[α1,αϕ],[α2,αϕ]]=[α2,α3ϕ].

By Lemma 2.8, we have

[α4,αϕ]p=[α4p,αϕ]=1=[α3p,αϕ]=[α3,αϕ]p,

[α3,α2ϕ]p=[α3p,α2ϕ]=1=[α2p,α1ϕ]=[α2,α1ϕ]p,

[α2,αϕ]p=[α2p,αϕ]=1=[α1p,αϕ]=[α1,αϕ]p.

Hence

G∧G≅〈[α1,αϕ],[α2,αϕ],[α2,α3ϕ]〉×〈[α2,α1ϕ]〉×〈[α3,αϕ]〉×〈[α4,αϕ]〉≅Φ2⁢(111)×ℤp(3).

Now consider the group G≅Φ10⁢(15). Let F be the free group generated by {α,α1}. Define αi+1=[αi,α], i=1,2,3. Set

β=[α1,α2],β1=[α4,α],β2=[α1,α4] and β3=α4-1⁢[α1,α2].

Reading modulo γ6⁢(F), we have

[α-1,α1-1,α3]α1=[α-1,α1-1,α3]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α3]α1=[α⁢α2-1⁢α-1,α3⁢[α3,α1]]=[α⁢α3⁢α-1⁢α2-1,α3⁢[α3,α1]]=[α3⁢α⁢α4-1⁢α-1⁢α2-1,α3⁢[α3,α1]]=[α3⁢α⁢β1⁢α-1⁢α4-1⁢α2-1,α3⁢[α3,α1]]=[α3⁢β1⁢α4-1⁢α2-1,α3⁢[α3,α1]]=[α2-1,α3]=[α2,α3]-1,

[α-1,α1-1,α2]α1=[α-1,α1-1,α2]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α2]α1=[α3⁢β1⁢α4-1⁢α2-1,α2⁢β3-1⁢α4-1]=[α3,α2⁢β3-1⁢α4-1]⁢[α2-1,α2⁢β3-1⁢α4-1]=[α3,α2]⁢[α2,β3].

By the Hall–Witt identity, we have the following identities in F modulo γ6⁢(F):

(4.3)1=[α3,α,α1]α-1⁢[α-1,α1-1,α3]α1⁢[α1,α3-1,α-1]α3=[α4,α1]α-1⁢[α2,α3]-1⁢[α3,α1,α3-1⁢α-1⁢α3]=[α4,α1]⁢[α2,α3]-1⁢[α3,α1,α4⁢α-1]=[α4,α1]⁢[α2,α3]-1⁢[α3,α1,α-1],

(4.4)1=[α2,α,α1]α-1⁢[α-1,α1-1,α2]α1⁢[α1,α2-1,α-1]α2=[α3,α1]α-1⁢[α3,α2]⁢[α2,β3]⁢[β3-1⁢α4-1,α2-1⁢α-1⁢α2]=[α3,α1]⁢[α3,α1,α-1]⁢[α3,α2]⁢[α2,β3]⁢[β3-1⁢α4-1,α3⁢α-1]=[α3,α1]⁢[α3,α1,α]-1⁢[α3,α2]⁢[α2,β3]⁢[β3,α]⁢[α4,α].

Consider

H1=F/〈γ6⁢(F),Fp,β1,β3,[α3,α1,α],[α3,α1,α1]〉.

By (4.3) and (4.4), respectively, we have

[α4,α1]=[α2,α3] and [α1,α3]=[α4,α]⁢[α3,α2] in⁢H1.

Observe that

H1≅〈α,α1,α2,α3,α4,β2∣[αi,α]=αi+1,[α1,α2]=α4,[α1,α4]=[α3,α2]=[α1,α3]=β2,αp=α1p=αi+1p=β2p=1,i=1,2,3〉.

Then H1 is the group Φ39⁢(16) of order p6 in [26]. Now define

H2=F/〈γ6⁢(F),Fp,β3,[α3,α1,α],[α3,α1,α1]〉.

Again, by (4.3) and (4.4), we have

[α4,α1]=[α2,α3] and [α1,α3]=[α4,α]⁢[α3,α2] in⁢H2.

Hence

H2≅〈α,αi,α4,β1,β2∣[αi,α]=αi+1,[α1,α2]=α4,[α4,α]=β1,[α1,α4]=[α3,α2]=β2,[α1,α3]=β1⁢β2,αp=αi+1p=β1p=β2p=1, 1≤i≤3〉.

Notice that |H2|=p7. Finally, consider

H=F/〈γ6⁢(F),Fp,[α3,α1,α],[α3,α1,α1],[β3,α],[β3,α1]〉.

It follows that

H≅〈α,αj,βi∣[αi,α]=αi+1,[α1,α2]=α4⁢β3,[α4,α]=β1,[α1,α4]=[α3,α2]=β2,[α1,α3]=β1⁢β2,αp=αi+1p=βip=1, 1≤i≤3, 1≤j≤4〉.

Notice that |H|=p8 and H/〈β1,β2,β3〉≅G. Then, as in the preceding case, we have M⁡(G)≅ℤp(3). As |G′|=p3, we get |G∧G|=p6. Since α4∈Z⁡(G), by Lemma 2.9 (viii),

[α2,α4ϕ]=[α1,α,α4ϕ]=1=[α2,α,α4ϕ]=[α3,α4ϕ].

Hence G∧G is generated by the set

{[α1,αϕ],[α2,αϕ],[α3,αϕ],[α4,αϕ],[α2,α1ϕ],[α3,α1ϕ],[α4,α1ϕ],[α3,α2ϕ]}.

We have

[α-1,α1-1,α2ϕ]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α2ϕ]α1=[α⁢α2-1⁢α-1,(α1-1⁢α2⁢α1)ϕ]=[α⁢α3⁢α-1⁢α2-1,(α2⁢α4)ϕ]=[α3⁢α4-1⁢α2-1,(α2⁢α4)ϕ]=[α4,α2ϕ]-1⁢[α3,α2ϕ]=[α3,α2ϕ],

[α-1,α1-1,α3ϕ]α1=[α-1,α1-1,α3ϕ]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α3ϕ]α1=[α⁢α2-1⁢α-1,α3ϕ]=[α⁢α3⁢α-1⁢α2-1,α3ϕ]=[α3⁢α4⁢α2-1,α3ϕ]=[α2,α3ϕ]-1.

By the Hall–Witt identity, we have

1=[α2,α,α1ϕ]α-1⁢[α-1,α1-1,α2ϕ]α1⁢[α1,α2-1,(α-1)ϕ]α2=[α3,α1ϕ]⁢[α3,α2ϕ]⁢[α4-1,(α-1)ϕ]α2=[α3,α1ϕ]⁢[α3,α2ϕ]⁢[α4,αϕ],

1=[α3,α,α1ϕ]α-1⁢[α-1,α1-1,α3ϕ]α1⁢[α1,α3-1,(α-1)ϕ]α3=[α4,α1ϕ]⁢[α2,α3ϕ]-1.

This implies that [α4,α1ϕ]=[α2,α3ϕ]=[α4,αϕ]⁢[α3,α1ϕ]. By Lemma 2.9 (vi), [[α1,αϕ],[α2,αϕ]]=[α2,α3ϕ]. We have the following identities:

[α4,αϕ]p=[α4p,αϕ]=1=[α3p,αϕ]=[α3,αϕ]p,

=p[α3p,α2ϕ]=1=[α2p,α1ϕ]=[α2,α1ϕ]p,

=p[α2p,αϕ]=1=[α1p,αϕ]=[α1,αϕ]p.

Hence

G∧G=〈[α1,αϕ],[α2,αϕ],[α2,α3ϕ]〉×〈[α2,α1ϕ]〉×〈[α3,αϕ]〉×〈[α4,αϕ]〉≅Φ2⁢(111)×ℤp(3).

The proof is now complete. ∎

5 Remaining groups of order p5

We start with the groups which occur as direct product of groups of smaller orders. For such groups we compute the Schur multiplier in the following lemma, whose proof follows using Theorem 2.1 and Theorem 2.15.

Lemma 5.1.

The following assertions hold:

M⁡(Φ2⁢(311)⁢a)≅M⁡(Φ2⁢(221)⁢b)≅ℤp×ℤp,
M⁡(Φ2⁢(221)⁢a)≅ℤp(3),
M⁡(Φ2⁢(2111)⁢a)≅M⁡(Φ2⁢(2111)⁢b)≅ℤp(5),
M⁡(Φ2⁢(2111)⁢c)≅M⁡(Φ2⁢(2111)⁢d)≅ℤp(4),
M⁡(Φ2⁢(15))≅ℤp(7),
M⁡(Φ3⁢(2111)⁢a)≅M⁡(Φ3⁢(2111)⁢br)≅ℤp(3),
M⁡(Φ3⁢(15))≅ℤp(4).

The proof of the following lemma is a direct consequence of Proposition 2.14 and Theorem 2.15.

Lemma 5.2.

The following assertions hold:

Φ2⁢(311)⁢a⊗Φ2⁢(311)⁢a≅Φ2⁢(221)⁢b⊗Φ2⁢(221)⁢b≅ℤp2×ℤp(8),
Φ2⁢(221)⁢a⊗Φ2⁢(221)⁢a≅ℤp2(2)×ℤp(7),
Φ2⁢(2111)⁢a⊗Φ2⁢(2111)⁢a≅Φ2⁢(2111)⁢b⊗Φ2⁢(2111)⁢b≅ℤp(16),
Φ2⁢(2111)⁢c⊗Φ2⁢(2111)⁢c≅Φ2⁢(2111)⁢d⊗Φ2⁢(2111)⁢d≅ℤp2×ℤp(10),
Φ2⁢(15)⊗Φ2⁢(15)≅ℤp(18),
Φ3⁢(2111)⁢a⊗Φ3⁢(2111)⁢a≅Φ3⁢(2111)⁢br⊗Φ3⁢(2111)⁢br≅ℤp(11),
Φ3⁢(15)⊗Φ3⁢(15)≅ℤp(12).

Lemma 5.3.

If G is one of the groups Φ2⁢(41),Φ2⁢(32)⁢a1, Φ2⁢(32)⁢a2 or Φ8⁢(32), then M⁡(G) is isomorphic to {1}, Zp, Zp or {1}, respectively, and G∧G is isomorphic to Zp, Zp2, Zp2 or Zp2, respectively.

Proof.

Since these groups are metacyclic, the assertion about the Schur multipliers follows from [29, Theorem 2.11.3]. Now we compute the exterior square. Since the Schur multiplier of the groups G isomorphic to Φ2⁢(41) or Φ8⁢(32) is trivial, we have G∧G≅G′. Hence

Φ2⁢(41)∧Φ2⁢(41)≅ℤp and Φ8⁢(32)∧Φ8⁢(32)≅ℤp2.

Now we consider G=Φ2⁢(32)⁢a1. We have the following identities:

[α2,αϕ]=[αp2,αϕ]=[α,αϕ]p2=1=[α2p,α1ϕ]=[α2,α1ϕ]p,

[α2,α1ϕ]=[αp2,α1ϕ]=[α,α1ϕ]p2=[α,(α1p2)ϕ]=1.

Hence G∧G is a cyclic group generated by [α1,αϕ]. Since both M⁡(G) and γ2⁢(G) are of order p, it follows that G∧G≅〈[α1,αϕ]〉≅ℤp2.

Similarly, for G=Φ2⁢(32)⁢a2, we have

[α2,α1ϕ]=[α1p,α1ϕ]=[α1,α1ϕ]p=1,[α2,αϕ]=[α1p,αϕ]=[α1,αϕ]p.

Hence G∧G is generated by [α1,αϕ]. As above, G∧G≅〈[α1,αϕ]〉≅ℤp2, and the proof is complete. ∎

Lemma 5.4.

If G is one of the groups Φ2⁢(311)⁢b or Φ2⁢(311)⁢c, then M⁡(G) is isomorphic to Zp×Zp, and G∧G is isomorphic to Zp(3).

Proof.

For the group G=Φ2⁢(311)⁢b, taking K=Z⁡(G) in Theorem 2.4, we have |M⁡(G)|≤p2. On the other hand, taking K=Φ⁢(G) in Theorem 2.3 (ii), we get d⁢(M⁡(G))≥2. Hence

M⁡(G)≅ℤp×ℤp.

The group G∧G is generated by the set

{[α1,αϕ],[α2,αϕ],[α2,α1ϕ],[γ,αϕ],[γ,α1ϕ],[γ,α2ϕ]}.

By Lemma 2.9 (viii), [α2,γϕ]=[α1,α,γϕ]=1. We get the following set of identities:

[γ,αϕ]p=[γ,(αp)ϕ]=1=[γ,(α1p)ϕ]=[γ,α1ϕ]p,

[α2,αϕ]=[γp2,αϕ]=[γ,αϕ]p2=1,

[α2,α1ϕ]=[γp2,α1ϕ]=[γ,α1ϕ]p2=1,

[α,α1ϕ]p=[αp,α1ϕ]=1.

Hence G∧G is generated by {[α1,αϕ],[γ,αϕ],[γ,α1ϕ]}. Since the nilpotency class of G is 2, by Lemma 2.12, G∧G is abelian. Hence G∧G≅ℤp(3).

Now consider the group G=Φ2⁢(311)⁢c. Again, using Theorem 2.4 (K=G′), we get |M⁡(G)|≤p2. By Theorem 2.3 (ii) with K=〈αp〉, we have d⁢(M⁡(G))≥2. Hence M⁡(G)≅ℤp×ℤp. Since |G′|=p, it follows that G∧G is of order p3. Notice that

[α2,α1ϕ]p=[α2p,α1ϕ]=1,[α2,αϕ]p=[α2p,αϕ]=1,[α1,αϕ]p=[α1p,αϕ]=1.

Consequently, G∧G is generated by {[α1,αϕ],[α2,αϕ],[α2,α1ϕ]}, which shows that G∧G≅ℤp(3). The proof is now complete. ∎

Lemma 5.5.

If G is one of the groups Φ3⁢(311)⁢a, Φ3⁢(311)⁢br or Φ2⁢(221)⁢d, then M⁡(G) is isomorphic to Zp, Zp or Zp(3), respectively, and G∧G is isomorphic to Zp(3), Zp(3) or Zp2×Zp×Zp, respectively.

Proof.

Let G be one of the groups Φ3⁢(311)⁢a or Φ3⁢(311)⁢br. Taking K=Z⁡(G) in Theorem 2.3 (i), p divides |M⁡(G)|. Since |G′|=p2, it follows that |G∧G|≥p3.

For the group G=Φ3⁢(311)⁢a, by Lemma 2.9 (viii), [α2,α3ϕ]=[α1,α,α3ϕ]=1. It now follows by Proposition 2.11 that G∧G is generated by the set

{[α1,αϕ],[α2,αϕ],[α2,α1ϕ],[α3,αϕ],[α3,α1ϕ]}.

The following identities hold:

[α3,αϕ]=[αp2,αϕ]=[α,αϕ]p2=1,

[α3,α1ϕ]p=[α3p,α1ϕ]=1,

[α2,αϕ]p=[α2p,αϕ]=1=[α2p,α1ϕ]=[α2,α1ϕ]p,

[α1,αϕ]p=[α1p,αϕ]=1,

[α3,α1ϕ]=[αp2,α1ϕ]=[α,α1ϕ]p2=1.

Hence G∧G is generated by the set {[α1,αϕ],[α2,αϕ],[α2,α1ϕ]}. Since, in view of Lemma 2.9 (vi), G∧G is abelian, we have G∧G≅ℤp(3). Consequently,

M(G)≅[α1,α2ϕ]〉≅ℤp.

The computation for the group G=Φ3⁢(311)⁢br goes on the same lines for r=1,ν.

Finally, we consider the group G=Φ2⁢(221)⁢d. The group G∧G is generated by the set {[α1,αϕ],[α2,αϕ],[α1,α2ϕ]}. By Lemma 2.12, G∧G is abelian. We have the identities

[α2,α1ϕ]p=[α2p,α1ϕ]=1=[α2p,αϕ]=[α2,αϕ]p,

[α1,αϕ]p2=[α1p2,αϕ]=1.

Now the natural epimorphism [G,Gϕ]→[Ga⁢b,(Ga⁢b)ϕ] implies that the order of [α1,αϕ] is p2 in G∧G. Notice that G/〈α1p〉≅Φ2⁢(211)⁢c. Consider the natural epimorphism

[G,Gϕ]→[G/〈α1p〉,(G/〈α1p〉)ϕ]≅[Φ2⁢(211)⁢c,(Φ2⁢(211)⁢c)ϕ],

which induces an epimorphism G∧G→(Φ2⁢(211)⁢c)∧(Φ2⁢(211)⁢c). By Theorem 2.15, we know that Φ2⁢(211)⁢c∧Φ2⁢(211)⁢c≅ℤp(3), and hence the generators [α1,αϕ], [α2,αϕ], [α1,α2ϕ] of G∧G are non-trivial, independent, and [α2,αϕ], [α1,α2ϕ] have order p in G∧G. Consequently, we get G∧G≅ℤp2×ℤp×ℤp, which, by observing the fact that G′=〈[α1,α]〉, gives M⁡(G)≅ℤp×ℤp×ℤp. The proof is complete. ∎

Lemma 5.6.

If G is one of the groups Φ3⁢(221)⁢a or Φ2⁢(221)⁢c, then M⁡(G) is isomorphic to Zp or Zp2×Zp, respectively, and G∧G is isomorphic to Zp(3) or Zp2×Zp×Zp, respectively.

Proof.

The group G=Φ3⁢(221)⁢a is a semidirect product of its subgroups

N=〈α2,α,α3〉≅Φ2⁢(21) and T=〈α1〉≅ℤp2,

where N is normal.

By Theorem 2.15, we know that M⁡(N)=1. Now, using Theorem 2.2, we get the exact sequence

1→H1⁡(T,Hom⁡(N,ℂ*))→H2⁡(G,ℂ*)→1.

Hence

M⁡(G)≅H1⁡(T,Hom⁡(N,ℂ*)).

Notice that Hom⁡(N,ℂ*)≅N/N′≅〈α⁢N′,α2⁢N′〉. Let ζ be a primitive p-th root of unity and Hom⁡(N,ℂ*)≅〈ϕ1,ϕ2〉, where ϕi:N→ℂ*, i=1,2, are defined by setting ϕ1⁢(α)=ζ, ϕ1⁢(α2)=1 and ϕ2⁢(α)=1, ϕ2⁢(α2)=ζ-1. Recall that T acts on Hom⁡(N,ℂ*) as follows. For ϕ1∈Hom(N,ℂ*), we set

ϕ1α1⁢(α)=ϕ1⁢(α1-1⁢α⁢α1)=ϕ1⁢(α) and ϕ1α1⁢(α2)=ϕ1⁢(α1-1⁢α2⁢α1)=ϕ1⁢(α2).

So ϕ1α1=ϕ1. Similarly, the action of α1 on ϕ2 is given by ϕ2α1=ϕ1⁢ϕ2.

Define the map Norm:Hom⁡(N,ℂ*)→Hom⁡(N,ℂ*) given by

Norm⁡(ϕ)=ϕ(1+α1+α12+…+α1p-1).

It is easy to check that Ker⁡(Norm)=Hom⁡(N,ℂ*). Define

β:Hom⁡(N,ℂ*)→Hom⁡(N,ℂ*)

by β⁢(ϕ)=ϕ(α1-1). Then Im⁡(β)=〈ϕ1〉. It is a general fact (see Step 3 in the proof of [21, Theorem 5.4]) that

H1⁡(T,Hom⁡(N,ℂ*))≅Ker⁡(Norm)Im⁡(β)≅ℤp.

Hence M⁡(G)≅ℤp.

We have the following identities:

[α2,α3ϕ]=[α1,α,α3ϕ]=1 by Lemma 2.9 (viii),

[α3,αϕ]=[αp,αϕ]=[α,αϕ]p=1,

[α2,αϕ]p=[α2p,αϕ]=1=[α2p,α1ϕ]=[α2,α1ϕ]p,

[α3,α1ϕ]=[αp,α1ϕ]=[α,α1ϕ]p.

Now we get

[α-1,α1-1,α2ϕ]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α2ϕ]α1=[α⁢α2-1⁢α-1,α2ϕ]=[α3⁢α2-1,α2ϕ]=[α3,α2ϕ]⁢[α2,α2ϕ]-1=1.

By the Hall–Witt identity,

1=[α2,α,α1ϕ]α-1⁢[α-1,α1-1,α2ϕ]α1=[α3,α1ϕ].

Hence G∧G is generated by {[α1,αϕ],[α2,αϕ],[α2,α1ϕ]}. By Lemma 2.9 (vi) G∧G is abelian, and therefore, G∧G≅ℤp(3).

The group G=Φ2⁢(221)⁢c is a semidirect product of its normal subgroup

N=〈α1,γ〉 and T=〈α〉.

Now using Theorem 2.2, we get the exact sequence

1→H1(T,Hom(N,ℂ*))→H2(G,ℂ*)→H2(N,ℂ*)T.

As above, H1⁡(T,Hom⁡(N,ℂ*))≅ℤp2, which embeds in M⁡(G). Now, by Theorem 2.3 (ii), taking K=〈γp〉, we have 2≤d⁢(M⁡(G)). By Theorem 2.4, taking K=〈γ〉, we have |M⁡(G)|≤p3. Hence M⁡(G)≅ℤp2×ℤp.

We have the following identities:

[α2,γϕ]=[α1,α,γϕ]=1 by Lemma 2.9 (viii),

[α2,α1ϕ]=[γp,α1ϕ]=[γ,α1ϕ]p=[γ,(α1p)ϕ]=1,

[α2,αϕ]=[γp,αϕ]=[γ,αϕ]p,[α1,αϕ]p=[α1p,αϕ]=1.

Hence G∧G is generated by the set {[α1,αϕ],[γ,αϕ],[γ,α1ϕ]}. Since the nilpotency class of G is 2, by Lemma 2.12, G∧G is abelian; hence

G∧G≅〈[γ,αϕ]〉×〈[γ,α1ϕ]〉×〈[α1,αϕ]〉≅ℤp2×ℤp×ℤp.

The proof is now complete. ∎

Lemma 5.7.

If G is one of the groups Φ3⁢(2111)⁢c, Φ3⁢(2111)⁢d or Φ3⁢(2111)⁢e, then M⁡(G) is isomorphic to Zp(3), Zp×Zp or Zp×Zp, respectively, and G∧G is isomorphic to Zp(5),Zp(4) or Zp(4), respectively.

Proof.

For the group G=Φ3⁢(2111)⁢c, by Theorem 2.4, taking K=Z⁡(G), we get |M⁡(G)|≤p3, and by Theorem 2.3 (i), taking K=〈γp〉, it follows that p3 divides |M⁡(G)|; hence |M⁡(G)|=p3.

We have the following identities:

[α2,γϕ]=[α1,α,γϕ]=1=[α2,α,γϕ]=[α3,γϕ] by Lemma 2.9 (viii),

[α3,αϕ]=[γp,αϕ]=[γ,αϕ]p=[γ,(αp)ϕ]=1,

[α3,αiϕ]=[γp,αiϕ]=[γ,αiϕ]p=[γ,(αip)ϕ]=1 for⁢i=1,2,

[α2,α1ϕ]p=[α2p,α1ϕ]=1=[α2p,αϕ]=[α2,αϕ]p,

[α1,αϕ]p=[α1p,αϕ]=1.

The group G∧G is generated by {[α1,αϕ],[α2,αϕ],[α2,α1ϕ],[γ,αϕ],[γ,α1ϕ]}, and every generator has order at most p. By Lemma 2.9 (vi), G∧G is abelian. Hence M⁡(G)≅ℤp(3). Since |G′|=p2, we get G∧G≅ℤp(5).

Consider the group G=Φ3⁢(2111)⁢d. By Theorem 2.3 (i), taking K=〈αp〉, p2≤|M⁡(G)|. Since |G′|=p2, we have |G∧G|≥p4. By Lemma 2.9 (viii),

[α2,α3ϕ]=[α1,α,α3ϕ]=1.

We have

[α-1,α1-1,α2ϕ]α1=[α1⁢α⁢α2-1⁢α-1⁢α1-1,α2ϕ]α1=[α⁢α2-1⁢α-1,α2ϕ]=[α3⁢α2-1,α2ϕ]=[α3,α2ϕ]⁢[α2,α2ϕ]-1=[α2,α2ϕ]-1=1.

By the Hall–Witt identity,

1=[α2,α,α1ϕ]α-1⁢[α-1,α1-1,α2ϕ]α1=[α3,α1ϕ].

We have the following identities:

[α3,αϕ]p=[α3p,αϕ]=1=[α2p,α1ϕ]=[α2,α1ϕ]p,

[α2,αϕ]p=[α2p,αϕ]=1=[α1p,αϕ]=[α1,αϕ]p.

So G∧G is generated by the set {[α1,αϕ],[α2,αϕ],[α2,α1ϕ],[α3,αϕ]}.

By Lemma 2.9 (vi), it follows that G∧G is an elementary abelian p-group, and hence |G∧G|≤p4. Thus

G∧G≅ℤp(4) and M⁡(G)≅ℤp×ℤp.

For the group G=Φ3⁢(2111)⁢e, by Theorem 2.3 (i), taking K=〈α1p〉, we get p2≤|M⁡(G)|. Since |G′|=p2, we have |G∧G|≥p4. As in the preceding case, observe that G∧G is an elementary abelian p-group of order p4. Hence

M⁡(G)≅ℤp×ℤp and G∧G≅ℤp(4).

The proof is now complete. ∎

Lemma 5.8.

If G is one of the groups Φ3⁢(221)⁢br, Φ6⁢(221)⁢b12⁢(p-1) or Φ6⁢(221)⁢d0, then M⁡(G) is isomorphic to Zp×Zp, Zp or Zp, respectively, and G∧G is isomorphic to Zp2×Zp×Zp in all the cases.

Proof.

From [26, Table 4.1], it follows that for the groups E of order p6 in the isoclinism classes Φ25, Φ26, Φ42 or Φ43, E/Z⁢(E) is isomorphic to Φ3⁢(221)⁢b1, Φ3⁢(221)⁢bν, Φ6⁢(221)⁢b12⁢(p-1) or Φ6⁢(221)⁢d0, respectively. Thus all the groups G under consideration are capable; hence Z*⁢(G)=1.

For the group G=Φ3⁢(221)⁢br, by Theorem 2.6, taking Z=〈αp〉, we have the exact sequence

G/G′⊗Z→𝜆M⁡(G)→ℤp→1.

By Theorem 2.7, it follows that α1⊗αp∉Ker⁡λ, as α⊗αp∈Ker⁡λ. So

|Im⁡(λ)|=p and |M⁡(G)|=p2.

Since |G′|=p2, |G∧G|=p4. By Lemma 2.9 (viii), [α2,α3ϕ]=[α1,α,α3ϕ]=1. The following identities hold:

[α3,αϕ]p=[α3p,αϕ]=1=[α2p,α1ϕ]=[α2,α1ϕ]p,

[α2,αϕ]p=[α2p,αϕ]=1,[α3,α1ϕ]=[α1p⁢r-1,α1ϕ]=[α1,α1ϕ]p⁢r-1=1,

[α3,αϕ]=[α1p⁢r-1,αϕ]=[α1,αϕ]p⁢r-1,[α1,αϕ]p2=[α1p2,αϕ]=1.

Observe that G∧G is generated by the set {[α1,αϕ],[α2,αϕ],[α1,α2ϕ]}.

By Lemma 2.9 (vi), G∧G is abelian, and therefore,

G∧G≅〈[α1,αϕ]〉×〈[α2,αϕ]〉×〈[α2,α1ϕ]〉≅ℤp2×ℤp×ℤp.

Hence

M⁡(G)≅〈[α1,αϕ]p〉×〈[α2,α1ϕ]〉≅ℤp×ℤp.

For the group G≅Φ6⁢(221)⁢d0, taking Z=〈β2〉, notice that G/Z≅Φ3⁢(211)⁢bν. By Theorem 2.15, M⁡(G/Z)≅ℤp. Then by Theorem 2.6, we have the exact sequence

G/G′⊗Z→𝜆M⁢(G)→𝜇ℤp→ℤp→1.

By Theorem 2.7, it follows that α2⁢G′⊗α1p∉Ker⁡λ, as α1⁢G′⊗α1p∈Ker⁡λ. Hence |Im⁡(λ)|=p, and therefore, M⁡(G)≅ℤp. Since |G′|=p3, we have |G∧G|=p4.

By Lemma 2.9 (viii), we have [β,β1ϕ]=[α1,α2,β1ϕ]=1. Similarly, [β,β2ϕ]=[β2,β1ϕ]=1. We have the following identities:

[x,yϕ]p=[xp,yϕ]=1 for⁢x∈{β,β1,β2},y∈{α1,α2},

[β1,α1ϕ]=[α2p⁢ν-1,α1ϕ]=[α2,α1ϕ]p⁢ν-1,

[β1,α2ϕ]=[α2p⁢ν-1,α2ϕ]=[α2,α2ϕ]p⁢ν-1=1,

[β2,α1ϕ]=[α1p,α1ϕ]=[α1,α1ϕ]p=1,

[β2,α2ϕ]=[α1p,α2ϕ]=[α1,α2ϕ]p,

[α1,α2ϕ]p2=[β2,α2ϕ]p=1.

Hence G∧G is generated by the set {[α1,α2ϕ],[β,α1ϕ],[β,α2ϕ]}, and therefore, G∧G≅ℤp2×ℤp×ℤp.

Finally, we consider the group G=Φ6⁢(221)⁢b12⁢(p-1). As in the preceding case, taking Z=〈β2〉, it follows that M⁡(G)≅ℤp. Recall that ζ is the smallest positive integer which is a primitive root (mod p). In this case, ζ12⁢(p-1)≡-1(modp). By Lemma 2.9 (viii), [β,β1ϕ]=[α1,α2,β1ϕ]=1. Similarly [β,β2ϕ]=[β2,β1ϕ]=1.

We have the following identities:

[x,yϕ]p=[xp,yϕ]=1 for⁢x∈{β,β1,β2},y∈{α1,α2},

[β1,α1ϕ]=[α1-p,α1ϕ]=[α1-1,α1ϕ]p=1,

[β1,α2ϕ]=[α1-p,α2ϕ]=[α1-1,α2ϕ]p=[α1,α2ϕ]-p,

[β2,α1ϕ]=[α2p,α1ϕ]=[α2,α1ϕ]p,

[β2,α2ϕ]=[α2p,α2ϕ]=[α2,α2ϕ]p=1,

[α2,α1ϕ]p2=[β2,α1ϕ]p=1.

Hence G∧G is generated by the set {[α1,α2ϕ],[β,α1ϕ],[β,α2ϕ]}, and therefore,

G∧G≅ℤp2×ℤp×ℤp,

which completes the proof. ∎

Lemma 5.9.

If G is one of the groups Φ6⁢(221)⁢a, Φ6⁢(221)⁢br (r≠12⁢(p-1)), Φ6⁢(221)⁢cr or Φ6⁢(221)⁢dr, then M⁡(G) is trivial and G∧G≅Zp(3).

Proof.

This follows from [44, Theorem 2.2]. ∎

Lemma 5.10.

If G is one of the groups Φ6⁢(2111)⁢a, Φ6⁢(2111)⁢br or Φ6⁢(15), then M⁡(G) is isomorphic to Zp, Zp, or Zp(3), respectively, and G∧G≅Zp(4), Zp(4) or Zp(6), respectively.

Proof.

For the group G≅Φ6⁢(2111)⁢a, by Theorem 2.3 (i), taking K=〈β1〉, we have p≤|M⁡(G)|, and so, since |G′|=p3, it follows that p4≤|G∧G|. By Lemma 2.9 (viii), [β,β1ϕ]=[β,β2ϕ]=[β2,β1ϕ]=1. Also, we have

[x,yϕ]p=[xp,yϕ]=1 for⁢x∈{β,β1,β2},y∈{α1,α2},

[β1,α1ϕ]=[α1p,α1ϕ]=[α1,α1ϕ]p=1,

[β1,α2ϕ]=[α1p,α2ϕ]=[α1,α2ϕ]p=[α1,(α2p)ϕ]=1.

Now we have

[α2-1,α1-1,βϕ]α1=[α2⁢α1⁢β-1⁢α1-1⁢α2-1,βϕ]α1=[α2⁢α1⁢β1⁢α1-1⁢β-1⁢α2-1,βϕ]α1=[α2⁢β1⁢β-1⁢α2-1,βϕ]α1=[α2⁢β1⁢β2⁢α2-1⁢β-1,βϕ]α1=[β1⁢β2⁢β-1,βϕ]α1=[β,βϕ]-1=1.

By the Hall–Witt identity, we get

1=[β,α2,α1ϕ]α2-1⁢[α2-1,α1-1,βϕ]α1⁢[α1,β-1,α2-ϕ]β=[β2,α1ϕ]α2-1⁢[β⁢β1⁢β-1,α2-ϕ]β=[β2,α1ϕ]⁢[β1,α2ϕ]-1.

Hence [β2,α1ϕ]=[β1,α2ϕ]=1, and consequently, G∧G is generated by the set {[α1,α2ϕ],[β,α1ϕ],[β,α2ϕ],[β2,α2ϕ]}, and is elementary abelian by Lemma 2.9 (vi). Hence G∧G≅ℤp(4), and therefore, M⁡(G)≅ℤp.

For the group G≅Φ6⁢(2111)⁢br, taking K=〈β1〉, the proof goes on the lines of the preceding case.

Now we consider the group G=Φ6⁢(15). By Theorem 2.4, taking K=〈β1〉, we have |M⁡(G)|≤p3. Since |G′|=p3, we get |G∧G|≤p6. As described above, [β,β1ϕ]=[β,β2ϕ]=[β2,β1ϕ]=1, and by the Hall–Witt identity, we get

[β2,α1ϕ]=[β1,α2ϕ].

Thus G∧G is generated by the set

{[α1,α2ϕ],[β,α1ϕ],[β,α2ϕ],[β1,α1ϕ],[β1,α2ϕ],[β2,α2ϕ]}.

It follows from Lemma 2.9 (vi) that G∧G is abelian. A straightforward calculation (as above) shows that each generator of G∧G is of order at most p. Consider the natural epimorphism

[G,Gϕ]→[G/〈β1〉,(G/〈β1〉)ϕ]≅[Φ3⁢(14),Φ3⁢(14)ϕ],

which shows that the elements [α1,α2ϕ], [β,α1ϕ], [β,α2ϕ], [β2,α2ϕ] are non-trivial and independent in G∧G. Now consider the natural epimorphism

[G,Gϕ]→[G/〈β2〉,(G/〈β2〉)ϕ]≅[Φ3⁢(14),Φ3⁢(14)ϕ],

which shows that the elements [α1,α2ϕ], [β,α1ϕ], [β,α2ϕ], [β1,α1ϕ] are non-trivial and independent in G∧G. We take the quotient group

G/〈β1⁢β2〉=〈α1,α2,β,β1∣[α1,α2]=β,[β,α1]=[α2,β]=β1,α1p=α2p=βp=β1p=1〉≅〈α1⁢α2,α1,β,β1∣[α1,α1⁢α2]=β,[β,α1]=β1,[β,α1⁢α2]=1,α1p=(α1α2)p=βp=β1p=1〉≅Φ3⁢(14).

In [G/〈β1⁢β2〉,(G/〈β1⁢β2〉)ϕ], by the Hall–Witt identity, we have

1=[β,α1,α2ϕ]α1-1⁢[α1-1,α2-1,βϕ]α2⁢[α2,β-1,α1-ϕ]β=[β1,α2ϕ]α1-1⁢[α1⁢α2⁢β⁢α2-1⁢α1-1,βϕ]α2⁢[β⁢β1-1⁢β-1,α1-ϕ]β=[β1,α2ϕ]⁢[α1⁢β⁢β1⁢α1-1,βϕ]α2⁢[β1-1,α1-ϕ]β=[β1,α2ϕ]⁢[α1⁢β⁢α1-1⁢β1,βϕ]α2⁢[β1,α1ϕ]=[β1,α2ϕ]⁢[β⁢α1⁢β1-1⁢α1-1⁢β1,βϕ]α2⁢[β1,α1ϕ]=[β1,α2ϕ]⁢[β,βϕ]⁢[β1,α1ϕ]=[β1,α2ϕ]⁢[β1,α1ϕ].

It follows from Theorem 2.15 that

G/〈β1⁢β2〉∧G/〈β1⁢β2〉≅〈[α1⁢α2,α1ϕ],[β-1,α1ϕ],[β-1,(α1⁢α2)ϕ],[β1-1,α1ϕ]〉.

A straightforward computation now shows that

G/〈β1⁢β2〉∧G/〈β1⁢β2〉≅〈[α1,α2ϕ],[β,α1ϕ],[β,α2ϕ],[β1,α2ϕ]〉≅ℤp(4).

Now consider the natural epimorphism

[G,Gϕ]→[G/〈β1⁢β2〉,(G/〈β1⁢β2〉)ϕ]≅[Φ3⁢(14),Φ3⁢(14)ϕ],

which shows that the generators [α1,α2ϕ],[β,α1ϕ],[β,α2ϕ],[β1,α2ϕ] of G∧G are independent. Thus it follows that all six generators of G∧G are non-trivial and independent, and so we have |G∧G|≥p6. Hence

G∧G≅ℤp(6) and M⁡(G)≅ℤp(3),

and the proof is complete. ∎

Lemma 5.11.

If G is one of the groups Φ7⁢(2111)⁢a, Φ7⁢(2111)⁢br, Φ7⁢(2111)⁢c or Φ7⁢(15), then M⁡(G) is isomorphic to Zp(3), Zp(3), Zp(3) or Zp(4), respectively, and G∧G is isomorphic to Zp(5), Zp(5), Zp(5) or Zp(6), respectively.

Proof.

For the groups G belonging to the isoclinism class Φ7, it follows from Theorem 2.3 (ii), taking K=Z⁡(G), that d⁢(M⁡(G))≥3.

For the group G=Φ7⁢(2111)⁢a, consider the normal subgroup

N=〈α,α1,α2,α3〉