Projective representations of Heisenberg groups over the rings of order 𝑝2

Sumana Hatui; E. K. Narayanan; Pooja Singla

doi:10.1515/jgth-2022-0033

Article Publicly Available

Projective representations of Heisenberg groups over the rings of order 𝑝²

Sumana Hatui , E. K. Narayanan and Pooja Singla

Published/Copyright: June 27, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Journal of Group Theory Volume 27 Issue 2

Abstract

We describe the 2-cocycles, Schur multiplier and representation group of discrete Heisenberg groups over the unital rings of order p2. We also describe all projective representations of Heisenberg groups with entries from the rings Z/p2⁢Z and Fp⁢[t]/(t2) for odd primes 𝑝 and obtain a classification of their degenerate and non-degenerate 2-cocycles.

1 Introduction

The theory of projective representations of finite groups was first studied by Schur in a series of papers [16, 17, 18]. A projective representation of a group 𝐺 is a homomorphism from 𝐺 to the projective general linear group PGL⁢(V), where 𝑉 is a complex vector space. A projective representation is thus a map ρ:G→GL⁢(V) such that ρ⁢(1)=IdV and there is a 2-cocycle α:G×G→C× satisfying

ρ⁢(x⁢y)=α⁢(x,y)⁢ρ⁢(x)⁢ρ⁢(y),x,y∈G.

In this case, we say 𝜌 is an 𝛼-representation. An 𝛼-representation is also frequently called a projective representation with factor set 𝛼.

For cyclic groups, the irreducible projective representations are the same as the ordinary representations (up to equivalence) and hence one-dimensional. However, this is not true in general for abelian groups. Several authors have investigated the case of abelian groups; see [6, 14, 15].

Recently, the first- and third-named authors studied the projective representations of the discrete Heisenberg group over cyclic rings in [5]. This paper may be considered as a continuation of [5], where we study the projective representations of the discrete Heisenberg groups over unital rings 𝑅 of order p2, where 𝑝 is an odd prime. The discrete Heisenberg group of rank one over 𝑅, denoted by H2⁢n+1⁢(R), is the set R×Rn×Rn with the multiplication given by

(c1,b1,…,bn,a1,…,an)⁢(c1′,b1′,b2′,…,bn′,a1′,a2′,…,an′)=(c1+c1′+∑i=1nai*bi′,a1+b1′,…,bn+bn′,b1+a1′,…,an+an′),

where a*b denotes the multiplication in 𝑅.

From now onwards, we consider 𝑅 to be a ring of order p2 with unity. It is easy to prove that any such ring 𝑅 is commutative and is isomorphic to one of the following:

Z/p2⁢Z,
Z/p⁢Z×Z/p⁢Z,
Fp⁢[t]/(t2),
Fp2.

We note that the results in [5] account for the case H2⁢n+1⁢(R) when n>1. Indeed, for n>1, every irreducible projective representation of H2⁢n+1⁢(R) (where 𝑅 is any commutative ring) is obtained via inflation from an irreducible projective representation of the abelian group R2⁢n. As a result, we will focus on the case n=1, i.e. on H3⁢(R), in this article.

In the study of projective representations of a finite group 𝐺, the main ingredients are to describe the Schur multiplier of 𝐺, to determine a representation group G⋆ of 𝐺 and then describe the ordinary representations of G⋆. We refer the reader to [12, 11] for any unexplained terms or notation in this article. Following [2], we say a 2-cocycle α∈Z2⁢(G,C×) is non-degenerate if the twisted group algebra Cα⁢[G] is a simple algebra. If a non-degenerate 2-cocycle exists, the group 𝐺 is said to be of central type. These groups play an important role in classifying the semisimple triangular complex Hopf algebras; see [4]. Now let 𝑅 be one of the aforementioned rings of order p2. The main results obtained in this article can be summarized as follows:

description of the Schur multiplier H2⁢(H3⁢(R),C×),
explicit description of the 2-cocycles of H3⁢(R),
construction of a representation group G⋆ for H3⁢(R) and its irreducible representations,
description of the non-degenerate 2-cocycles of H3⁢(Fp⁢[t]/(t2)), H3⁢(Z/p2⁢Z).

We now provide statements of our results. Our first result gives

H2⁢(H3⁢(Fp⁢[t]/(tr)),C×)

for an odd prime 𝑝 and r≥2. For r=1, this result is already known in the literature; see [12, Theorem 3.3.6].

Theorem 1.1

Assume that 𝑝 is an odd prime and r∈N. Then

H2⁢(H3⁢(Fp⁢[t]/(tr)),C×)≅(Z/p⁢Z)2⁢r2.

We next describe H2⁢(H3⁢(R),C×), where 𝑅 is a unital ring of order p2 and 𝑝 is an odd prime. For R=Z/p2⁢Z, H2⁢(H3⁢(R),C×)≅(Z/p2⁢Z)2 by [10, Theorem 1.1]. For R=Z/p⁢Z×Z/p⁢Z, we use H2⁢(H3⁢(Z/p⁢Z),C×)≅(Z/p⁢Z)2 (see [12, Theorem 3.3.6]) and the following result due to Schur (see [12, Theorem 2.2.10]) to describe H2⁢(H3⁢(R),C×).

For finite groups G1 and G2,

H2⁢(G1×G2,C×)≅H2⁢(G1,C×)×H2⁢(G2,C×)×Hom⁢(G1G1′⊗G2G2′,C×),

where G1′ and G2′ denote the commutator subgroups of G1 and G2 respectively. Therefore, H2⁢(H3⁢(Z/p⁢Z×Z/p⁢Z),C×)≅(Z/p⁢Z)8. The following result describes H2⁢(H3⁢(R),C×) for the remaining cases of 𝑅.

Theorem 1.2

Suppose 𝑝 is an odd prime. Then

H2⁢(H3⁢(Fp⁢[t]/(t2)),C×)≅H2⁢(H3⁢(Fp2),C×)≅(Z/p⁢Z)8.

The proofs of the above results are given in Section 3 and mainly use the results of Blackburn and Evens [3]. Since H3⁢(Fp⁢[t]/(tr)) and H3⁢(Fp2) are 𝑝-groups of nilpotency class 2 with the property that their abelianization is elementary abelian, we can apply the main results from [3]. The hypothesis of p≠2 is used in our proofs because (a) this gives Fp2≅Fp⁢[t]/(t2−k) for some k∈Fp2∖Fp and (b) the groups H3⁢(R) are of exponent 𝑝, so we have easy presentations to work with. As a different treatment is required for the p=2 case, we assume p≠2 throughout this article. From now onwards, we will fix 𝑘 such that Fp2≅Fp⁢[t]/(t2−k) and use this wherever required.

Our next result is the description of the 2-cocycles of the group H3⁢(R). We prove the following for H3⁢(Fp⁢[t]/(t2)).

Theorem 1.3

Every 2-cocycle of H3⁢(Fp⁢[t]/(t2)) is cohomologous to the following cocycles:

α⁢(x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2,x1a1′⁢x2a2′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′)=μ1a2⁢a1′⁢μ2b2⁢a1′⁢μ3b2⁢a2′⁢μ4b2⁢b1′⁢μ5b1⁢(a1′2)+a1′⁢c1μ6b2⁢(a1′2)+a1′⁢c2+a2′⁢c1⁢μ7a1′⁢(b12)−b1⁢c1′⁢μ8a2′⁢(b12)−b2⁢c1′−b1⁢c2′,

where μi∈C× such that μip=1. Furthermore, any two distinct cocycles of the above form are not cohomologous to each other.

Here

x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2

denotes a general element in the group H3⁢(Fp⁢[t]/(t2)). See Section 2.2 for more details. A description of the 2-cocycles of H3⁢(Z/p2⁢Z) already appeared in [5]. For other groups, a description of the 2-cocycles is given in Section 5.

As is well known, the projective representations of a group are obtained from ordinary representations of its representation group. To study the projective representations of H3⁢(R), we construct a representation group of H3⁢(R). In this direction, we have the following result for H3⁢(Fp⁢[t]/(t2)). For a group 𝐺 and x,y∈G, the commutator x−1⁢y−1⁢x⁢y is denoted by [x,y]. Whenever we write a presentation of a group, we assume all the commutators [x,y] for generators x,y, which are not explicitly stated in the presentation, are trivial.

Theorem 1.4

A representation group of G=H3⁢(Fp⁢[t]/(t2)) is given by

G⋆=⟨xi,yi,i=1,2∣[y1,x1]=z1,[y1,x2]=z2,[x2,x1]=w1,[y2,x1]⁢z2−1=w2,[y2,x2]=w3,[y2,y1]=w4,[z1,xi]=vi,[z1,yi]=ui,[z2,x1]=[z1,x2],[z2,y1]=[z1,y2],xip=yip=zip=wjp=1, 1≤j≤4⟩.

The parallel results for H3⁢(Z/p2⁢Z) have already appeared in [5]. For other groups, a description of their representation group is included in Section 4. In Section 6, we give a construction of all projective representations of H3⁢(Fp⁢[t]/(t2)) and H3⁢(Z/p2⁢Z). We use our construction of projective representations to classify the degenerate and non-degenerate cocycles of H3⁢(Fp⁢[t]/(t2)) and H3⁢(Z/p2⁢Z). More specifically, we prove the following result.

Theorem 1.5

For a finite group 𝐺, let

X⁢(G)={[α]∈H2⁢(G,C×)∣[α]⁢is non-degenerate}.

Then

|X⁢(G)|={p2⁢(p−1)2,G=H3⁢(Z/p2⁢Z),p5⁢(p−1)2⁢(p+1),G=H3⁢(Fp⁢[t]/(t2)).

For a proof of the above, see Section 6. We obtain the above from a construction of the irreducible representations of G⋆; see Sections 6.2 and 6.4.

As an application of this result, we compute certain 1-cocycles of H3⁢(Fp). Consider G=H3⁢(Fp⁢[t]/(t2)) and the following exact sequence obtained via the natural projection map from 𝐺 onto H3⁢(Fp):

1→⟨x2,y2,z2⟩≅(Fp)3→G→H3⁢(Fp)→1.

Let Q=H3⁢(Fp) and A=⟨x2,y2,z2⟩. Then 𝐺 is a semidirect product of 𝑄 by 𝐴. The following result describes H1⁢(Q,A^) and enumerates its bijective classes. The elements of H1⁢(Q,A^) can also be described explicitly; see Remark 6.2.

Corollary 1.6

With the above notation, the following hold.

H1⁢(Q,A^)≅(Z/p⁢Z)5.
The number of bijective classes in H1⁢(Q,A^) is p2⁢(p−1)2⁢(p+1).

Combining the results of [2] with a description of the elements of

H2⁢(H3⁢(Fp⁢[t]/(t2)),C×)

yields a proof of the above; see Section 6.

2 Preliminaries

In this section, we fix the notation and recall a few results which will be used in the upcoming sections. The center and commutator subgroup of 𝐺 are denoted by Z⁢(G) and G′ respectively.

A central extension

(2.1)1→A→G→G/A→1

is called a stem extension if A⊆Z⁢(G)∩G′. For a central extension (2.1), the map tra:Hom⁢(A,C×)→H2⁢(G/A,C×) given by f↦[tra⁢(f)],

tra⁢(f)⁢(x¯,y¯)=f⁢(μ⁢(x¯)⁢μ⁢(y¯)⁢μ⁢(x⁢y¯)−1),x¯,y¯∈G/A,

for a section μ:G/A→G, is a group homorphism and is called the transgression homomorphism. The map

inf:H2⁢(G/A,C×)→H2⁢(G,C×)

given by [α]↦[inf⁢(α)], where inf⁢(α)⁢(x,y)=α⁢(x⁢A,y⁢A), is a group homomorphism and is called the inflation homomorphism. By [7, Theorem 2, p. 129] and [9, Proposition 1.1], the spectral sequence for cohomology of groups yields the following exact sequence:

(2.2)1→Hom⁢(A,C×)→traH2⁢(G/A,C×)→infH2⁢(G,C×)→(res,χ)H2⁢(A,C×)⊕Hom⁢(G/G′⊗A,C×),

where the map 𝜒, defined by Iwahori and Matsumoto [9], is given by

χ⁢([α])⁢(g⁢G′⊗a)=α⁢(g,a)⁢α⁢(a,g)−1for⁢g∈G,a∈A.

Lemma 2.1

Lemma 2.1 (Hall–Witt identity)

Let 𝐺 be a finite group of nilpotency class 3. For x,y,z∈G, we have [x,y−1,z]⁢[y,z−1,x]⁢[z,x−1,y]=1.

2.1 Projective representations of a finite group

In this section, we include the results that we require regarding the projective representations of a finite group. We will use these in Section 6.

Let 𝐺 be a finite group. We use Z2⁢(G,C×) to denote the set of all 2-cocycles of 𝐺. For α∈Z2⁢(G,C×), we use Irrα⁢(G) to denote the set of equivalence classes of irreducible 𝛼-representations of 𝐺. For α=1, we use Irr⁢(G) instead of Irrα⁢(G) and call this the set of ordinary irreducible characters of 𝐺. Let G⋆ be a representation group of 𝐺 with A≅H2⁢(G,C×) such that

(2.3)1→A→G⋆→G→1

is a stem extension. The existence of such G⋆ follows from [12, Theorem 2.1.4]. Let Irr⁢(G|χ) denote the set of inequivalent irreducible representations of 𝐺 lying above 𝜒, that is ρ∈Irr⁢(G|χ) if and only if HomN⁢(ρ|N,χ) is non-trivial. The following well-known result relates the projective representations of 𝐺 and the ordinary ones of G⋆.

Theorem 2.2

Let 𝛼 be a 2-cocycle of 𝐺. Suppose that χ∈Hom⁢(A,C×) is such that tra⁢(χ)=[α]. Then there is a bijective correspondence between

Irrα⁢(G)↔Irr⁢(G⋆|χ)

obtained via lifting a projective representation of 𝐺 to an ordinary representation of G⋆. In particular, we obtain the following:

⋃[α]∈H2⁢(G,C×)Irrα⁢(G)↔Irr⁢(G⋆).

A proof of this follows from the proof of [11, Chapter 3, Section 3]; see also [5, Theorem 3.2]. To determine the projective representations of H3⁢(Fp⁢[t]/(t2)) and H3⁢(Z/p2⁢Z), it therefore suffices to determine the ordinary representations of their representation groups. We now discuss a method that will work in our situation.

In this direction, we first recall the results of Clifford theory regarding the ordinary characters of a finite group. For proofs, see [8, Theorem 6.11, Corollary 6.17].

Theorem 2.3

Let 𝐺 be a finite group and 𝑁 a normal subgroup. For any irreducible representation 𝜌 of 𝑁, let IG⁢(ρ)={g∈G∣ρg≅ρ} denote the stabilizer of 𝜌 in 𝐺. Then the following hold.

The map
θ↦IndIG⁢(ρ)G⁢(θ)
is a bijection of Irr⁢(IG⁢(ρ)∣ρ) onto Irr⁢(G∣ρ).
Let 𝐻 be a subgroup of 𝐺 containing 𝑁, and suppose that 𝜌 is an irreducible representation of 𝑁 which has an extension ρ~ to 𝐻 (i.e. ρ~|N=ρ). Then the representations δ⊗ρ~ for δ∈Irr⁢(H/N) are irreducible, distinct for distinct 𝛿 and
IndNH⁢(ρ)=⨁χ∈Irr⁢(H/N)χ⊗ρ~.

Let 𝐺 be a finite group with an abelian normal subgroup 𝑁 such that G/N is abelian. Let χ:N→C× be a one-dimensional representation of 𝑁 and let

IG⁢(χ)={g∈G∣χg=χ}

be the inertia group of 𝜒 in 𝐺. By Theorem 2.3, the problem of determining Irr⁢(G|χ) reduces to that of determining Irr⁢(IG⁢(χ)|χ). We now mention a method that helps us to determine Irr⁢(IG⁢(χ)|χ) for every 𝜒.

Let 𝐾 be a finite group with an abelian normal subgroup 𝑁 such that K/N is abelian. Let χ:N→C× be a one-dimensional representation of 𝑁 such that χk=χ for all k∈K, that is IK⁢(χ)=K. Let 𝑇 be a fixed set of left coset representatives of 𝑁 in 𝐾. Define a map χ′:K→C× by χ′⁢(k⁢n)=χ⁢(n) for all k∈T and n∈N. Following [8, Chapter 11], let α∈Z2⁢(K,C×) be a 2-cocycle of 𝐾 associated to χ′ and let β∈Z2⁢(K/N,C×) be defined by β⁢(g⁢N,h⁢N)=α⁢(g,h) for g,h∈K.

Lemma 2.4

The following are equivalent.

The character 𝜒 extends to 𝐾.
[K,K]⊆Ker⁢(χ).
[β]=1.

Proof

The equivalence of (1) and (2) follows from the fact that [K,K]⊆N and N/[K,K] is an abelian group. The equivalence of (1) and (3) follows from [8, Theorem 11.7]. ∎

With the above notation, the following result can be obtained from [11, Theorem 4.2, Chapter 6].

Lemma 2.5

There is a dimension-preserving bijection between the sets Irr⁢(K|χ) and Irrβ−1⁢(K/N).

Hence it boils down to understanding the projective representations of the quotient group K/N. For our case, this quotient group will turn out to be an abelian group. The projective representations of abelian groups are well studied, and we use some of these results. In particular, we will use the following lemma without further reference.

Lemma 2.6

Let 𝐺 be a finite abelian group. Then the following hold.

For any fixed α∈Z2⁢(G,C×), all irreducible representations in Irrα⁢(G) have equal dimension.
For |G|∈{p2,p3} and 1≠[α]∈H2⁢(G,C×), every ρ∈Irrα⁢(G) has dimension 𝑝.
For |G|=p4 and 1≠[α]∈H2⁢(G,C×), every ρ∈Irrα⁢(G) has dimension either 𝑝 or p2.

Proof

Here (1) follows from [1, Theorem 1]. Assertions (2) and (3) follow from the fact that any ρ∈Irrβ⁢(G) for [β]≠1 satisfies

1<dim(ρ)≤|G|

and dim(ρ) divides the order of 𝐺. ∎

At this point, we mention the ways to identify the non-degenerate cocycles of a group 𝐺 by using the ordinary representations of its representation group G⋆. Consider the stem extension (2.3). Let 𝛼 be a 2-cocycle of 𝐺 and χ∈Hom⁢(A,C×) such that tra⁢(χ)=[α].

Lemma 2.7

The following conditions are equivalent.

The 2-cocycle 𝛼 is non-degenerate.
|Irrα⁢(G)|=1.
There exists ρ∈Irrα⁢(G) such that dim(ρ)=|G|.
There exists ρ∈Irr⁢(G⋆|χ) such that dim(ρ)=|G|.
|Irr⁢(G⋆|χ)|=1.

Proof

By definition, a 2-cocycle 𝛼 is non-degenerate if and only if 𝐺 has a unique irreducible 𝛼-character if and only if the group algebra Cα⁢[G] is simple. The equivalence of (1), (2) and (3) follows from this. Equivalence of (2) and (5) as well as of (3) and (4) follows from Theorem 2.2. ∎

We use this result to classify the non-degenerate cocycles of H3⁢(Fp⁢[t]/(t2)) and H3⁢(Z/p2⁢Z) in Section 6.

2.2 Presentation and matrix form of groups

In this section, we give a presentation of H3⁢(Fp⁢[t]/(tr)) and H3⁢(Fp⁢[t]/(t2−k)) that we use throughout this article.

The groups H3⁢(Fp⁢[t]/(tr)) for r≥1 are of nilpotency class 2 and have the following presentation:

H3⁢(Fp⁢[t](tr))=⟨xm,ym, 1≤m≤r∣[yj,xi−j+1]=zi,xmp=ymp=1,1≤i≤r, 1≤j≤i⟩.

As mentioned earlier, Fp2≅Fp⁢[t]/(t2−k) for p≠2 and k∈Fp2∖Fp. Therefore,

H3⁢(Fp2)≅H3⁢(Fp⁢[t](t2−k))for⁢p≠2.

We have the following presentation of H3⁢(Fp⁢[t]/(t2−k)):

H3⁢(Fp⁢[t](t2−k))=⟨x1,x2,y1,y2∣[y1,x1]=z1,[y2,x2]=z1k,[y1,x2]=[y2,x1]=z2,xip=yip=1,i=1,2⟩.

It is helpful to think of H3⁢(R) in its matrix form, that is as a group of 3×3 matrices with entries from the ring 𝑅. In the above two presentations, the element

x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2

corresponds to

(1(b1,b2)(c1,c2)01(a1,a2)001)

in the matrix form of H3⁢(R).

2.3 Schur multiplier of certain 𝑝-groups

Let 𝐺 be a 𝑝-group of nilpotency class 2 with G/G′ elementary abelian. In this section, we recall the theory given in [3, Section 3] to compute the Schur multiplier of 𝐺.

Consider G/G′ and G′ as vector spaces over Fp, denoted by 𝑉 and 𝑊 respectively. For g1,g2,g3∈G, let U1 be the subspace of V⊗W spanned by the elements of the form

g¯1⊗[g2,g3]+g¯2⊗[g3,g1]+g¯3⊗[g1,g2],

where g¯i=gi⁢G′∈V for i=1,2,3. Let U2 be the subspace of V⊗W spanned by all g¯1⊗g1p. Now consider U=U1+U2.

We have the following result from [3, Theorem 3.1].

Proposition 2.8

Let 𝐺 be a 𝑝-group of nilpotency class 2 such that G/G′ is elementary abelian. Then

|H2⁢(G,C×)|=|V∧VW|⁢|V⊗WU|.

We will continue to use this result in the next section.

Lemma 2.9

Let 𝑝 be an odd prime and 𝐺 a 𝑝-group of exponent 𝑝, of nilpotency class 2. Then H2⁢(G,C×) is an elementary abelian 𝑝-group.

Proof

If 𝐺 has a free presentation F/R, then

H2⁢(G,C×)≅H2⁢(G,Z)≅F′∩R[F,R]

from [12, Theorem 2.4.6]. For x,y∈F and for odd 𝑝,

1≡[xp,y]≡[x,y]p⁢[[x,y],x](p2)≡[x,y]p(mod[F,R])

implies that (F′)p⊂[F,R]. Therefore, for any x∈F′∩R, xp∈[F,R]. Since

H2⁢(G,C×)≅(F′∩R)[F,R],

H2⁢(G,C×) is an elementary abelian 𝑝-group.∎

The following result will help us to describe the 2-cocycles of H3⁢(Fp⁢[t]/(t2)).

Proposition 2.10

Let 𝑝 be an odd prime, 𝐺 a 𝑝-group of exponent 𝑝 and of nilpotency class 2 such that there is a central subgroup Z⊆Z⁢(G)∩G′ with the property

|H2⁢(G,C×)|=|H2⁢(G/Z,C×)Hom⁢(Z,C×)|⁢|Hom⁢(G/G′⊗Z,C×)|.

Then

H2⁢(G,C×)≅H2⁢(G/Z,C×)Hom⁢(Z,C×)×Hom⁢(G/G′⊗Z,C×).

Proof

By (2.2), we have the following exact sequence:

1→H2⁢(G/Z,C×)Hom⁢(Z,C×)→infH2⁢(G,C×)→χHom⁢(G/G′⊗Z,C×)→1.

The group H2⁢(G,C×) is an elementary abelian 𝑝-group by Lemma 2.9. Therefore, the result follows. ∎

3 Schur multiplier of H3⁢(Fp⁢[t]/(tr)) and H3⁢(R)

In this section, we prove Theorem 1.1 and Theorem 1.2.

Proof of Theorem 1.1

By Lemma 2.9, H2⁢(G,C×) is an elementary abelian 𝑝-group for an odd prime 𝑝. Let N=⟨zr⟩. Then

G/N≅H3⁢(Fp⁢[t](tr−1))×(Z/p⁢Z)2.

For r=1, we have H2⁢(H3⁢(Fp),C×)≅(Z/p⁢Z)2 from [12, Theorem 3.3.6]. Now we use induction on 𝑟 to prove our result. Let 𝐺 have a free presentation F/R, where 𝐹 is the free group generated by the symbols ⟨xm,ym, 1≤m≤r⟩. Viewing H2⁢(G,C×) as (F′∩R)/[F,R], consider the exact sequence from [12, Theorem 2.5.6],

G/G′⊗N→λH2⁢(G,C×)→H2⁢(G/N,C×)→N→1,

where 𝜆 is the Ganea map defined by

λ⁢(xm⁢F′⁢R⊗xs⁢R)=[xm,xs]⁢[F,R].

Our claim is that Im⁡(λ)≅(Z/p⁢Z)2.

From [3, Remark, p. 110], it follows that Ker⁢λ=(G/G′⊗N)∩U. Observe that here U2=1 so that U=U1. For 2≤m≤r, by the Hall–Witt identity in 𝐹 (mod⁢[F,R]), we have

[xm,zr]=[xm,[yr,x1]]=[[x1,xm],yr]⁢[[xm,yr],x1]=1,[ym,zr]=[ym,[y1,xr]]=[[xr,ym],y1]⁢[[ym,y1],xr]=1,

which says that

x¯m⊗[yr,x1]=x¯m⊗zr∈Ker⁢λ,y¯m⊗[y1,xr]=y¯m⊗zr∈Ker⁢λ.

Now, for 1<j<r, by the Hall–Witt identity in 𝐹 (mod⁢[F,R]), we have the following:

[x1,zr]=[x1,[yj,xr−j+1]]=[[xr−j+1,x1],yj]⁢[[x1,yj],xr−j+1]=[zj−1,xr−j+1],

[y1,zr]=[y1,[yj,xr−j+1]]=[[xr−j+1,y1],yj]⁢[[y1,yj],xr−j+1]=[zr−j+1−1,yj],

which says that

x¯1⊗[yj,xr−j+1]+x¯r−j+1⊗[x1,yj]=x¯1⊗zr+x¯r−j+1⊗zj−1∈U∖Ker⁢λ,

y¯1⊗[yj,xr−j+1]+y¯j⊗[xr−j+1,y1]=y¯1⊗zr+y¯j⊗zr−j+1−1∈U∖Ker⁢λ.

Therefore,

Ker⁢λ=⟨x¯m⊗zr,y¯m⊗zr, 2≤m≤r⟩≅(Z/p⁢Z)2⁢r−2.

|Im⁢(λ)|=|(G/G′⊗N)p2⁢r−2|=p2andIm⁢(λ)≅(Z/p⁢Z)2.

Hence, for r≥2,

H2⁢(H3⁢(Fp⁢[t](tr)),C×)≅H2⁢(H3⁢(Fp⁢[t](tr−1))×(Z/p⁢Z)2,C×)×Z/p⁢Z≅(Z/p⁢Z)2⁢r2.∎

Proof of Theorem 1.2.

For G=H3⁢(Fp⁢[t]/(t2)), by Theorem 1.1,

H2⁢(G,C×)≅(Z/p⁢Z)8.

We now proceed to prove the result for G=H3⁢(Fp2). By Lemma 2.9 and Proposition 2.8, H2⁢(G,C×) is an elementary abelian group and

|H2⁢(G,C×)|=p12|U|,

where

U=⟨(y¯1⊗z1k+y¯2⊗z2−1),(y¯2⊗z1+y¯1⊗z2−1),(x¯1⊗z1k+x¯2⊗z2−1),(x¯2⊗z1+x¯1⊗z2−1)⟩.

Therefore, |U|=p4 and we have

H2⁢(H3⁢(Fp⁢[t](t2−k)),C×)≅(Z/p⁢Z)8.∎

4 Representation group of H3⁢(R)

4.1 G=H3⁢(Fp⁢[t]/(t2))

In this section, we prove Theorem 1.4.

Proof of Theorem 1.4.

Recall that G⋆ is given by the following:

G⋆=⟨xi,yi,i=1,2∣[y1,x1]=z1,[y1,x2]=z2,[x2,x1]=w1,[y2,x1]⁢z2−1=w2,[y2,x2]=w3,[y2,y1]=w4,[z1,xi]=vi,[z1,yi]=ui,[z2,x1]=[z1,x2],[z2,y1]=[z1,y2],xip=yip=zip=wjp=1, 1≤j≤4⟩.

If G⋆ is a group of order p14, then we have the stem extension

1→Z→G⋆→G→1

for

Z=⟨ui,vi,wj, 1≤i≤2, 1≤j≤4⟩≅(Z/pZ)8.

Since H2⁢(G,C×)≅(Z/p⁢Z)8, G⋆ is a representation group of 𝐺. We now proceed to prove that G⋆ is of order p14.

Consider the free group 𝐹 generated by four elements x1,x2,y1,y2. The basic commutators [x,y] form a basis for the free abelian group γ2⁢(F)/γ3⁢(F). We fix the following notation:

[y1,x1]=z1,[y1,x2]=z2,[y2,x1]⁢z2−1=w2,[x2,x1]=w1,[y2,x2]=w3,[y2,y1]=w4,

and

[z1,xi]=vi,[z1,yi]=ui,i=1,2.

To prove our result, we show that G⋆ is isomorphic to a certain quotient group of 𝐹.

Let H1=F/⟨γ3⁢(F),Fp,w1,w2,w3,w4⟩. Then

H1≅⟨y1,y2,x1,x2∣[y1,x1]=z1,[y1,x2]=[y2,x1]=z2,xip=yip=1⟩≅H3⁢(Fp⁢[t]/(t2))

is a group of order p6.

In 𝐹 modulo ⟨γ4⁢(F),[wj,xi],[wj,yi],i=1,2, 1≤j≤4⟩, we have the following identities:

[x1−1,y1−1,x2]=[z1,x2]−1,[x1−1,y1−1,y2]=[z1,y2]−1,[x1−1,y2−1,x2]=[z2,x2]−1,[x2−1,y1−1,y2]=[z2,y2]−1,[y1,x2−1,x1−1]=[z2,x1].

Observe that, using the Hall–Witt identity, we have the following relations in 𝐹 modulo ⟨γ4⁢(F),[wj,xi],[wj,yi],i=1,2, 1≤j≤4⟩:

(4.1)[z2,x1]=[z1,x2],[z2,y1]=[z1,y2],[z2,y2]=[z2,x2]=1.

Now consider the group

H2=F/⟨γ4(F),Fp,w1,w2,w3,w4,[z1,x2],[z1,yi],i=1,2⟩.

By (4.1), we have

H2≅⟨x1,x2,y1,y2∣[y1,x1]=z1,[y1,x2]=[y2,x1]=z2,[z1,x1]=v1,xip=1⟩

and H2/⟨v1⟩≅H1. Hence H2 is of order p7. We proceed further step by step considering

H3=F/⟨γ4(F),Fp,w1,w2,w3,w4,[z1,yi],i=1,2⟩,H4=F/⟨γ4⁢(F),Fp,w1,w2,w3,w4,[z1,y2]⟩,H5=F/⟨γ4⁢(F),Fp,w1,w2,w3,w4⟩.

Using the identities in (4.1), we see that

H5≅⟨x1,x2,y1,y2∣[y1,x1]=z1,[y1,x2]=[y2,x1]=z2,[z1,x1]=v1,[z1,y1]=u1,[z2,x1]=[z1,x2]=v2,[z2,y1]=[z1,y2]=u2,xip=1⟩.

and H5/⟨u2⟩≅H4,H4/⟨u1⟩≅H3,H3/⟨v2⟩≅H2. Hence H5 is of order p10.

Now consider the group

K1=F/⟨γ4(F),Fp,w1,w3,w4,[w2,xi],[w2,yi],i=1,2⟩.

By (4.1), we have

[z2,x1]=[z1,x2]=v2,[z2,y1]=[z1,y2]=u2,[z2,y2]=[z2,x2]=1.

K1≅⟨y1,y2,x1,x2∣[y1,x1]=z1,[y1,x2]=z2,[y2,x1]⁢z2−1=w2,[z1,x1]=v1,[z1,y1]=u1,[z2,x1]=[z1,x2]=v2,[z2,y1]=[z1,y2]=u2,xip=1⟩.

Therefore, K1/⟨w2⟩≅H5. Hence K1 is of order p11. In a similar way, step by step, we consider the groups

K2=F/⟨γ4(F),Fp,w3,w4,[wj,xi],[wj,yi],i=1,2,j=1,2⟩,K3=F/⟨γ4(F),Fp,w4,[wj,xi],[wj,yi],i=1,2,j=1,2,3⟩,K4=F/⟨γ4(F),Fp,[wj,xi],[wj,yi],i=1,2,j=1,2,3,4⟩,

and we see that K2/⟨w1⟩≅K1, K3/⟨w3⟩≅K2, K4/⟨w4⟩≅K3. Finally, we see that K4≅G⋆. Hence G⋆ is of order p14. ∎

In the following corollary, we point out the cardinality of a specific group that appeared in the above proof. We require it for later.

Corollary 4.1

Consider the following nilpotent class three group:

(4.2)G~=⟨xi,yi,i=1,2∣[y1,x1]=z1,[y1,x2]=[y2,x1]=z2,[z1,xi]=vi,[z1,yi]=ui,[z2,x1]=[z1,x2],[z2,y1]=[z1,y2],xip=yip=zip=1⟩.

The group G~ is of order p10.

Proof

Let H5 be the group that appeared in the proof of Theorem 1.4, where we also proved that |H5|=p10. The result follows because G~≅H5. ∎

4.2 H3⁢(Fp⁢[t]/(t2−k))

In this section, we give a representation group of H3⁢(Fp⁢[t]/(t2−k)).

Theorem 4.2

For an odd prime 𝑝, a representation group of

G=H3⁢(Fp⁢[t](t2−k))

is the following group of nilpotency class three:

G⋆=⟨xi,yi,i=1,2∣[y1,x1]=z1,[y1,x2]=z2,[y2,x1]⁢z2−1=w2,[y2,x2]⁢z1−k=w3,[z1,yi]=ui,[z1,xi]=vi,[z2,x1]=[z1,x2],[z2,y1]=[z1,y2],[z2,y2]=[z1,y1]k,[z2,x2]=[z1,x1]k,[x2,x1]=w1,[y2,y1]=w4,xip=yip=wip=1⟩.

Proof

Consider the group

H=⟨xi,yi,i=1,2∣[y1,x1]=z1,[y2,x2]=z1k,[y1,x2]=[y2,x1]=z2,[z1,yi]=ui,[z1,xi]=vi,[z2,x1]=[z1,x2],[z2,y1]=[z1,y2],[z2,y2]=[z1,y1]k,[z2,x2]=[z1,x1]k,xip=yip=wip=1⟩.

By using a method similar to the proof of Theorem 1.4, we obtain that 𝐻 is of order p10 and G⋆ is of order p14. Hence it follows that G⋆ is a representation group of H3⁢(Fp⁢[t]/(t2−k)). ∎

4.3 H3⁢(Z/p⁢Z×Z/p⁢Z)

Theorem 4.3

A representation group of G=H3⁢(Z/p⁢Z×Z/p⁢Z) is the following nilpotency class three group:

G⋆=⟨x1,x2,y1,y2∣[x1,x2]=z1,[y1,y2]=z2,[z1,x1]=t1,[z1,x2]=t2,[z2,y1]=t3,[z2,y2]=t4,[x1,y1]=t5,[x1,y2]=t6,[x2,y1]=t7,[x2,y2]=t8,xip=yip=z1p=z2p=1,i=1,2⟩.

Proof

We have H2⁢(G,C×)≅(Z/p⁢Z)8 and

G⋆≅(⟨x1,x2∣[x1,x2]=z1,[z1,x1]=t1,[z1,x2]=t2⟩×⟨t5,t6,t7,t8⟩)⋊⟨y1,y2∣[y1,y2]=z2,[z2,y1]=t3,[z2,y2]=t4⟩≅(H′×(Z/p⁢Z)4)⋊K′,

where H′≅K′ is of order p5. Thus G⋆ is a semidirect product of a normal subgroup of order p9 and a subgroup of order p5. So G⋆ of order p14 such that the sequence

1→⟨ti⟩i=18≅H2⁢(G,C×)→G⋆→G→1

is exact. Hence the result follows. ∎

5 Description of 2-cocycles of H3⁢(R)

In this section, we describe the 2-cocycles of H3⁢(R).

5.1 H3⁢(Fp⁢[t]/(t2))

We start with a proof of Theorem 1.3.

Proof of Theorem 1.3

Let G=H3⁢(Fp⁢[t]/(t2)). We show that H2⁢(G,C×) is the direct product of two groups which we compute explicitly. In order to do this, consider the central subgroup Z=⟨z1⟩ of 𝐺. Then

H=G/Z≅⟨y1,y2,x1,x2∣[y1,x2]=[y2,x1]=z2,xip=yip=1⟩

is an extra-special 𝑝-group of order p5. From (2.2), we obtain the exact sequence

1→Hom⁢(Z,C×)→traH2⁢(H,C×)→infH2⁢(G,C×)→χHom⁢(G/G′⊗Z,C×).

Since H2⁢(G,C×)≅(Z/p⁢Z)8, H2⁢(H,C×)≅(Z/p⁢Z)5,

Im⁢(inf)≅H2⁢(H,C×)Im⁢(tra)andHom⁢(G/G′⊗Z,C×)≅(Z/p⁢Z)4,

we have, by Proposition 2.10,

(5.1)H2⁢(G,C×)≅Im⁢(inf)×μ⁢(Hom⁢(G/G′⊗Z,C×))

for a section μ:Hom⁢(G/G′⊗Z,C×)→H2⁢(G,C×) which is an injective homomorphism (𝜇 will be defined in Step 2 below). We describe

Im⁢(inf)andμ⁢(Hom⁢(G/G′⊗Z,C×))

in the following two steps. For simplification of notation, let

g=x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2,g′=x1a1′⁢x2a2′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′

such that 𝑔 and g′ are arbitrary elements of 𝐺 and let N=⟨Z,z2⟩ be the subgroup generated by 𝑍 and z2.

Step 1: To describe Im⁢(inf), we proceed as follows. Consider the exact sequence

1→Hom⁢(⟨z2⟩,C×)→traH2⁢(H/⟨z2⟩,C×)→infH2⁢(H,C×).

Since H2⁢(H,C×)≅(Z/p⁢Z)5 and H/⟨z2⟩≅(Z/p⁢Z)4, the map

inf:H2⁢(H/⟨z2⟩,C×)→H2⁢(H,C×)

is surjective. Hence every 2-cocycle 𝛽 of 𝐻 is of the form

[β]=inf⁢([δ]),δ∈H2⁢(H/⟨z2⟩,C×).

The 2-cocycles of H/⟨z2⟩, being an elementary abelian group, are well known. Therefore,

β⁢(g⁢Z,g′⁢Z)=δ⁢(g⁢N,g′⁢N)=μ1a2⁢a1′⁢λ1b1⁢a1′⁢λ2b2⁢a1′⁢λ3b1⁢a2′⁢μ3b2⁢a2′⁢μ4b2⁢b1′,

where μj, j∈{1,3,4}, and λℓ, ℓ∈{1,2,3}, are scalars whose 𝑝-th power is one.

Now recall (5.1) and define [α1]=inf⁢([β])∈H2⁢(G,C×) for β∈H2⁢(H,C×). Then

α1⁢(g,g′)=β⁢(g⁢Z,g′⁢Z)=μ1a2⁢a1′⁢λ1b1⁢a1′⁢λ2b2⁢a1′⁢λ3b1⁢a2′⁢μ3b2⁢a2′⁢μ4b2⁢b1′.

For i∈{1,2}, define maps fi:G→C× by

f1⁢(g)=λ1c1,f2⁢(g)=λ3c2for⁢g=x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2∈G.

This shows that λ1b1⁢a1′,λ3b2⁢a1′+b1⁢a2′ are coboundaries in 𝐺. Hence α1 is cohomologous to the following cocycle (again denoted by α1):

(5.2)α1⁢(g,g′)=μ1a2⁢a1′⁢(λ2⁢λ3−1)b2⁢a1′⁢μ3b2⁢a2′⁢μ4b2⁢b1′=μ1a2⁢a1′⁢μ2b2⁢a1′⁢μ3b2⁢a2′⁢μ4b2⁢b1′.

Therefore, in (5.1), Im⁢(inf) consists of [α1] such that α1 is a 2-cocycle of the above form.

Step 2: Consider the group G~ from (4.2). We have the central exact sequence

1→⟨u1,u2,v1,v2⟩→iG~→πG→1.

Observe that, in G~, we have

[y1i,x1j]=z1i⁢j⁢v1i⁢(j2)⁢u1j⁢(i2),[y1i,x2j]=z2i⁢j⁢u2j⁢(i2),[y2i,x1j]=z2i⁢j⁢v2i⁢(j2).

For g=x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2, g′=x1a1′⁢x2a2′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′ in 𝐺,

g⁢g′=x1a1+a1′⁢x2a2+a2′⁢y1b1+b1′⁢y2b2+b2′⁢z1c1+c1′+b1⁢a1′⁢z2c2+c2′+b1⁢a2′+b2⁢a1′.

We denote the numbers

b1⁢(a1′2)+a1′⁢c1,b2⁢(a1′2)+a1′⁢c2+a2′⁢c1,a1′⁢(b12)−b1⁢c1′,a2′⁢(b12)−b2⁢c1′−b1⁢c2′

by k5,k6,k7,k8 respectively. Define a section s:G→G~ by

s⁢(x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2)=x1a1⁢x2a2⁢z1c1⁢z2c2⁢y1b1⁢y2b2∈G~.

Then

s⁢(g)⁢s⁢(g′)=(x1a1⁢x2a2⁢z1c1⁢z2c2⁢y1b1⁢y2b2)⁢(x1a1′⁢x2a2′⁢z1c1′⁢z2c2′⁢y1b1′⁢y2b2′)=x1a1+a1′⁢x2a2+a2′⁢z1c1+c1′+b1⁢a1′⁢z2c2+c2′+b1⁢a2′+b2⁢a1′y1b1+b1′⁢y2b2+b2′⁢v1k5⁢v2k6+b1⁢a1′⁢a2′u1k7−b12⁢a1′⁢u2k8−b1⁢(b1⁢a2′+b2⁢a1′),

s⁢(g)⁢s⁢(g′)⁢s⁢(g⁢g′)−1=v1k5⁢v2k6+b1⁢a1′⁢a2′⁢u1k7−b12⁢a1′⁢u2k8−b1⁢(b1⁢a2′+b2⁢a1′).

Consider the exact sequence

1→Hom⁢(⟨u1,u2,v1,v2⟩,C×)→traH2⁢(G,C×)→infH2⁢(G~,C×).

Let f∈Hom⁢(⟨u1,u2,v1,v2⟩,C×) and f⁢(v1), f⁢(v2), f⁢(u1), f⁢(u2) be denoted by μ5,μ6,μ7,μ8∈C× respectively. Then [α2]=tra⁢(f)∈H2⁢(G,C×) is defined by

α2⁢(g,g′)=f⁢(s⁢(g)⁢s⁢(g′)⁢s⁢(g⁢g′)−1)=f⁢(v1k5⁢v2k6+b1⁢a1′⁢a2′⁢u1k7−b12⁢a1′⁢u2k8−b1⁢(b1⁢a2′+b2⁢a1′))=μ5k5⁢μ6k6+b1⁢a1′⁢a2′⁢μ7k7−b12⁢a1′⁢μ8k8−b1⁢(b1⁢a2′+b2⁢a1′).

For i∈{1,2,3}, define the maps fi:G→C× by

f1⁢(g)=μ6c1,f2⁢(g)=μ7c1,f3⁢(g)=μ8c2for⁢g=x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2∈G.

This gives that μ6b1⁢a1′⁢a2′, μ7−b12⁢a1′, μ8(b1⁢a2′+b2⁢a1′) are coboundaries in 𝐺. Therefore, α2 is cohomologous to the following cocycle:

(5.3)α2⁢(g,g′)=μ5k5⁢μ6k6⁢μ7k7⁢μ8k8.

Consider the subgroup 𝑆 of H2⁢(G,C×) consisting of [α2] such that α2 is of the form (5.3). Now

χ⁢([α2])⁢(x1a1⁢x2a2⁢y1b1⁢y2b2⁢G′⊗z1c1)=α2⁢(x1a1⁢x2a2⁢y1b1⁢y2b2,z1c1)⁢α2⁢(z1c1,x1a1⁢x2a2⁢y1b1⁢y2b2)−1=μ5−a1⁢c1⁢μ6−a2⁢c1⁢μ7−b1⁢c1⁢μ8−b2⁢c1.

Define a section μ:Hom⁢(G/G′⊗Z,C×)→S by

μ⁢(f)⁢(g,g′)=f⁢(x1⊗z1)−k5⁢f⁢(x2⊗z1)−k6⁢f⁢(y1⊗z1)−k7⁢f⁢(y2⊗z1)−k8.

It is easy to check that 𝜇 is a homomorphism and χ|S∘μ=id|Hom⁢(G/G′⊗Z,C×) and μ∘χ|S=id|S. In particular,

μ⁢(Hom⁢(G/G′⊗Z,C×))≅S.

We are now in a position to complete the proof. Using (5.2) and (5.3), we define α=α1⁢α2, and so we have

α⁢(x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2,x1a1′⁢x2a2′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′)=μ1a2⁢a1′⁢μ2b2⁢a1′⁢μ3b2⁢a2′⁢μ4b2⁢b1′⁢μ5k5⁢μ6k6⁢μ7k7⁢μ8k8.

This gives that every cocycle of H3⁢(Fp⁢[t]/(t2)) is cohomologous to one of the above form. By the above description, there are at most p8 elements in

H2⁢(H3⁢(Fp⁢[t]/(t2)),C×).

By Theorem 1.2, we have H2⁢(H3⁢(Fp⁢[t]/(t2)),C×)≅(Z/p⁢Z)8. Therefore, all of the above cocycles must be not cohomologous to each other. This completes the proof of our result. ∎

5.2 H3⁢(Fp⁢[t]/(t2−k))

We denote the numbers

b1⁢(a1′2)+k⁢b1⁢(a2′2)+k⁢b2⁢a1′⁢a2′+a1′⁢c1+k⁢a2′⁢c2,b2⁢(a1′2)+k⁢b2⁢(a2′2)+b1⁢a1′⁢a2′+a1′⁢c2+a2′⁢c1,a1′⁢(b12)+k⁢a1′⁢(b22)−b12⁢a1′−b1⁢c1′−k⁢b2⁢c2′,a2′⁢(b12)+k⁢a2′⁢(b22)−k⁢b22⁢a2′−b2⁢c1′−b1⁢c2′

by q5, q6, q7 and q8 respectively.

Theorem 5.1

Every 2-cocycle of H3⁢(Fp⁢[t]/(t2−k)) is cohomologous to a cocycle of the following form:

α⁢(x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2,x1a1′⁢x2a2′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′)=μ1a2⁢a1′⁢μ2b2⁢a1′⁢μ3b2⁢a2′⁢μ4b2⁢b1′⁢μ5q5⁢μ6q6⁢μ7q7⁢μ8q8,

where μi∈C× such that μip=1.

Proof

The proof of this result goes along the same lines as the proof of Theorem 1.3, using the representation group of H3⁢(Fp⁢[t]/(t2−k)) that appeared in Section 4.2. ∎

5.3 H3⁢(Z/p⁢Z×Z/p⁢Z)

In the following result, the element x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2 denotes the element

(1(b1,b2)(c1,c2)01(a1,a2)001)∈H3⁢(Z/p⁢Z×Z/p⁢Z).

We use

H3⁢(Z/p⁢Z×Z/p⁢Z)≅H3⁢(Z/p⁢Z)×H3⁢(Z/p⁢Z)

and [13, Theorem 9.6] to describe elements of H3⁢(Z/p⁢Z×Z/p⁢Z). More precisely, every 2-cocycle 𝛼 of H3⁢(Z/p⁢Z×Z/p⁢Z) is cohomologous to a cocycle of the following form:

α⁢(x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2,x1a1′⁢x2a2′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′)=λ1c1′⁢a1+b1⁢(a1′2)+a1⁢b1⁢a1′⁢λ2c1′⁢b1+a1′⁢(b12)⁢λ3c2′⁢a2+b2⁢(a2′2)+a2⁢b2⁢a2′λ4c2′⁢b2+a2′⁢(b22)⁢λ5a1′⁢a2⁢λ6a1′⁢b2⁢λ7b1′⁢a2⁢λ8b1′⁢b2,

where λi∈C× such that λip=1.

5.4 H3⁢(Z/p2⁢Z)

In the following result, the element xa⁢yb⁢zc denotes the element

(1bc01a001)∈H3⁢(Z/p2⁢Z).

From [5, Lemma 2.2], every 2-cocycle of H3⁢(Z/p2⁢Z) is cohomologous to a cocycle of the following form:

α⁢(xa⁢yb⁢zc,xa′⁢yb′⁢zc′)=λ1c′⁢a+b⁢(a′2)+a⁢b⁢a′⁢λ2c′⁢b+a′⁢(b2),λ1p2=λ2p2=1.

6 Projective representations of H3⁢(R)

In this section, we give a construction of all projective irreducible representations of H3⁢(Fp⁢[t]/(t2)) and H3⁢(Z/p2⁢Z). We also classify their non-degenerate cocycles. In particular, Theorem 1.5 and Corollary 1.6 follow from this section.

6.1 Projective representations of H3⁢(Fp⁢[t]/(t2))

By Theorem 1.4, a representation group of G=H3⁢(Fp⁢[t]/(t2)) is given by

G⋆=⟨xi,yi,i=1,2∣[y1,x1]=z1,[y1,x2]=z2,[x2,x1]=w1,[y2,x1]⁢z2−1=w2,[y2,x2]=w3,[y2,y1]=w4,[z1,xi]=vi,[z1,yi]=ui,[z2,x1]=[z1,x2],[z2,y1]=[z1,y2],xip=yip=zip=wjp=1,1≤j≤4⟩.

Consider the normal subgroup

N=⟨zi,ui,vi,wj,i=1,2,1≤j≤4⟩≅(Z/pZ)10

of G⋆. Any one-dimensional ordinary representation, say 𝜒, of 𝑁 is given by

χ⁢(wj)=μj,1≤j≤4,χ⁢(u1)=μ7,χ⁢(u2)=μ6,χ⁢(v1)=μ5,χ⁢(v2)=μ8,χ⁢(z1)=μ9,χ⁢(z2)=μ10,

where μi∈C× such that μip=1 for all 𝑖. If μ6≠1, assume that μi=μ6ri for 1≤i≤8 with r6=1, and if μ8≠1, assume that μi=μ8ti for 1≤i≤7 with t8=1.

Our first step is to determine the stabilizer IG⋆⁢(χ) of the character 𝜒. An element g=x1a1⁢x2a2⁢y1b1⁢y2b2⁢n for n∈N satisfies g∈IG⋆⁢(χ) if and only if

χ⁢([x1a1⁢x2a2⁢y1b1⁢y2b2,z1])=1andχ⁢([x1a1⁢x2a2⁢y1b1⁢y2b2,z2])=1.

From the definition of G⋆, we observe that

(6.1)[x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2,x1a1′⁢x2a2′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′]=z1a1′⁢b1−a1⁢b1′⁢z2a2′⁢b1−a2⁢b1′+a1′⁢b2−a1⁢b2′⁢w1a1′⁢a2−a1⁢a2′w2a1′⁢b2−a1⁢b2′⁢w3a2′⁢b2−a2⁢b2′⁢w4b1′⁢b2−b1⁢b2′u1a1′⁢(b12)−a1⁢(b1′2)+b1⁢b1′⁢(a1′−a1)+(b1′⁢c1−b1⁢c1′)u2a2′⁢(b12)−a2⁢(b1′2)+b1⁢b1′⁢(a2′−a2)−a1⁢b1′⁢(b2′+b2)+b1⁢b2′⁢(a1′−a1)+a1′⁢b2⁢(b1+b1′)+(b1′⁢c2−b1⁢c2′)+(b2′⁢c1−b2⁢c1′)v1b1⁢(a1′2)−b1′⁢(a12)+a1′⁢c1−a1⁢c1′v2b2⁢(a1′2)−b2′⁢(a12)+(a1′⁢a2′⁢b1−a1⁢a2⁢b1′)+(a1′⁢c2−a1⁢c2′)+(a2′⁢c1−a2⁢c1′).

Therefore, g∈IG⋆⁢(χ) if and only if

(6.2)χ⁢(v1)a1⁢χ⁢(v2)a2⁢χ⁢(u1)b1⁢χ⁢(u2)b2=1andχ⁢(v2)a1⁢χ⁢(u2)b1=1.

Our next goal is to describe Irr⁢(IG⋆⁢(χ)|χ). We note that IG⋆⁢(χ)/N is abelian and |IG⋆⁢(χ)/N|≤p4. Either 𝜒 extends to IG⋆⁢(χ) or there exists a normal subgroup N′ of IG*⁢(χ) such that N′/N is cyclic, |IG⋆⁢(χ)/N′|≤p3 and 𝜒 extends to N′ by Lemma 2.4. In both cases, by Theorem 2.3, Lemma 2.4 and Lemma 2.6, all representations in Irr⁢(IG⋆⁢(χ)|χ) for a fixed 𝜒 will have the same dimension. We now consider various cases.

(i) Assume μ6=μ8=1. Then, by (6.2), we have g∈IG⋆⁢(χ) if and only if χ⁢(v1)a1⁢χ⁢(u1)b1=1. Hence |IG⋆⁢(χ)/N|∈{p3,p4}. We consider these cases separately.

Suppose |IG⋆⁢(χ)/N|=p4. For this case, IG⋆⁢(χ)=G⋆. By Lemma 2.5, representations are determined by certain projective representations of the abelian group G⋆/N. Since |G⋆/N|=p4, any irreducible representation of Irr⁢(G⋆|χ) will be of dimension either 𝑝 or p2.

Next suppose |IG⋆⁢(χ)/N|=p3. Again by Lemma 2.5, representations are determined by certain projective representative representations of the abelian group IG⋆⁢(χ)/N. Since |G⋆/N|=p4, any irreducible representation of Irr⁢(G⋆|χ) in this case will also be of dimension either 𝑝 or p2.

(ii) Assume that μ6=1, μ8≠1. Then, by (6.2),

g=x1a1⁢x2a2⁢y1b1⁢y2b2⁢n∈IG⋆⁢(χ)

if and only if a1=0 and a2=−t7⁢b1. Therefore, we have |IG⋆⁢(χ)/N|=p2 in this case. By Lemma 2.6 and the discussion before it, an irreducible representation of Irr⁢(IG⋆⁢(χ)|χ) will be of dimension one or 𝑝. Further it will be one-dimensional if and only if 𝜒 extends to IG⋆⁢(χ). By Lemma 2.4, this holds if and only if [IG⋆⁢(χ),IG⋆⁢(χ)]⊆Ker⁢(χ), that is

χ⁢([x2−t7⁢b1⁢y1b1⁢y2b2⁢z1c1⁢z2c2,x2−t7⁢b1′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′])=1.

By using (6.1), this is equivalent to

χ⁢(v2)(t3⁢t7−t4)⁢(b1⁢b2′−b1′⁢b2)=1,

where b1,b2,b1′,b2′ are arbitrary. So we must have t4=t3⁢t7.

This discussion altogether implies that, for t4=t3⁢t7, Irr⁢(G⋆|χ) consists of p2-dimensional irreducible representations, and for t4≠t3⁢t7, all representations of Irr⁢(G⋆|χ) are of dimension p3.

(iii) Similar to (ii), in this case, any representation of Irr⁢(G⋆|χ) is of dimension p2 if r1=−r3⁢r5 and is of dimension p3 if r1≠−r3⁢r5.

(iv) Assume that μ6≠1, μ8≠1. Then, by (6.2),

g=x1a1⁢x2a2⁢y1b1⁢y2b2⁢n∈IG⋆⁢(χ)

if and only if b1=−r8⁢a1 and b2=−r8⁢a2+(r7⁢r8−r5)⁢a1. Therefore, we have |IG⋆⁢(χ)/N|=p2 in this case. As earlier, we only need to answer whether 𝜒 extends to IG⋆⁢(χ) or not. Now 𝜒 extends to IG⋆⁢(χ) if and only if

χ⁢([x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2,x1a1′⁢x2a2′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′])=1.

By using (6.1) and substituting the values χ⁢(wi),χ⁢(vj) for 1≤i≤4, 1≤j≤2 in terms of χ⁢(u2), we get

χ⁢(u2)(a1′⁢a2−a1⁢a2′)⁢(r1−r2⁢r8−r3⁢(r7⁢r8−r5)+r4⁢r82−r82+r822)=1.

Since a1,a1′,a2,a2′ are arbitrary, we must have

(6.3)r1−r2⁢r8−r3⁢(r7⁢r8−r5)+r4⁢r82+(r82)=0.

As earlier, this implies that Irr⁢(G⋆|χ) consists of p2-dimensional irreducible representations if (6.3) holds and of dimension p3 otherwise. This completes our discussion regarding the irreducible representations of G⋆.

6.2 Non-degenerate 2-cocycles of H3⁢(Fp⁢[t]/(t2))

In this section, we describe the degenerate and non-degenerate cocycles of

H3⁢(Fp⁢[t](t2)).

Recall that every 2-cocycle 𝛼 of H3⁢(Fp⁢[t]/(t2)) is cohomologous to a cocycle of the following form:

(6.4)α⁢(x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2,x1a1′⁢x2a2′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′)=μ1a2⁢a1′⁢μ2b2⁢a1′⁢μ3b2⁢a2′⁢μ4b2⁢b1′⁢μ5b1⁢(a1′2)+a1′⁢c1μ6b2⁢(a1′2)+a1′⁢c2+a2′⁢c1⁢μ7a1′⁢(b12)−b1⁢c1′⁢μ8a2′⁢(b12)−b2⁢c1′−b1⁢c2′,

where μi∈C× such that μip=1 for all 𝑖. Following the notation of Section 6.1, if μ6≠1, we assume that μi=μ6ri for 1≤i≤8 with r6=1, and if μ8≠1, we assume that μi=μ8ti for 1≤i≤7 with t8=1. Let G⋆,N,χ be as given in Section 6.1.

Theorem 6.1

The cocycle 𝛼 as given in (6.4) is non-degenerate if and only if one of the following holds.

μ6=1, μ8≠1 and t4≠t3⁢t7.
μ8=1, μ6≠1 and r1≠−r3⁢r5.
μ8≠1, μ6≠1 and r1−r2⁢r8−r3⁢(r7⁢r8−r5)+r4⁢r82+(r82)≠0.

In particular, the number of non-degenerate cocycles of the group H3⁢(Fp⁢[t]/(t2)) is p5⁢(p−1)2⁢(p+1).

Proof

Let

A=⟨ui,vi,wj,i=1,2, 1≤j≤4⟩≅(Z/pZ)8

be a subgroup of G⋆. Then it follows that 𝐴 is a central subgroup of G⋆ such that

1→A→G⋆→G→1

is a stem extension. For this case, note that tra⁢(χ|A)=[α]. This fact, Lemma 2.7 and Section 6.1 give the result. ∎

Proof of Corollary 1.6

By [2, Section 3], the groups H1⁢(Q,A^) and the quotient group

Ker⁢(resAG)/Im⁢(infGQ)

are isomorphic. Let

X=x1a1⁢x2a2⁢y1b1⁢y2b2⁢z1c1⁢z2c2andY=x1a1′⁢x2a2′⁢y1b1′⁢y2b2′⁢z1c1′⁢z2c2′

be two elements in 𝐺. By Theorem 1.3, it follows that

Im⁢(resAG)={[α]∣α⁢(X,Y)=μ3b2⁢a2′,μ3p=1}≅Z/p⁢Z.

Hence,

Ker⁢(resAG)≅(Z/p⁢Z)7.

By [5, Lemma 2.2 (ii)], it follows that every cocycle of H3⁢(Z/p⁢Z) is cohomologous to a cocycle of the form

σ⁢(x1a1⁢y1b1⁢z1c1,x1a1′⁢y1b1′⁢z1c1′)=λc1′⁢b1+a1′⁢b1⁢(b1−1)2⁢μa1′⁢c1+b1⁢a1′⁢(a1′−1)2,λp=μp=1.

Therefore, it is easy to see that

Im⁢(infQG)={[α]∣α⁢(X,Y)=μ5b1⁢(a1′2)+a1′⁢c1⁢μ7a1′⁢(b12)−b1⁢c1′,μ5p=μ7p=1}≅(Z/p⁢Z)2.

Thus, using Theorem 1.3, we have

(6.5)Ker⁢(resAG)/Im⁢(infGQ)={[α]∣α(X,Y)=μ1a2⁢a1′μ2b2⁢a1′μ4b2⁢b1′μ6b2⁢(a1′2)+a1′⁢c2+a2′⁢c1μ8a2′⁢(b12)−b2⁢c1′−b1⁢c2′}≅(Z/p⁢Z)5.

This proves (1).

By [2, Theorem A], the bijective classes in H1⁢(Q,A^) are in one to one correspondence with the non-degenerate cohomology classes in Ker⁢(resAG) modulo Im⁢(infGQ). By using the description of Ker⁢(resAG)/Im⁢(infGQ) given in (6.5) and Theorem 6.1, we obtain that the number of non-degenerate cocycles in Ker⁢(resAG) modulo Im⁢(infGQ) is exactly p2⁢(p−1)2⁢(p+1). This proves (2) ∎

Remark 6.2

In [2, Section 3.2], an explicit isomorphism between

Ker⁢(resAG)/Im⁢(infGQ)andH1⁢(Q,A^)

is given. Using the description of Ker⁢(resAG)/Im⁢(infGQ), as given in (6.5), all elements of H1⁢(Q,A^) can be explicitly described.

6.3 Projective representations of H3⁢(Z/p2⁢Z)

In this section, we first give a construction of projective representations of

G=H3⁢(Z/p2⁢Z).

As mentioned earlier, it is enough to give a construction of all ordinary irreducible representations of a representation group of 𝐺. From [5, Theorem 1.2],

G⋆=⟨x,y∣[y,x]=z,[z,x]=z1,[z,y]=z2,xp2=yp2=zp2=1⟩

is a representation group of 𝐺. Note that xa⁢yb⁢zc,xa′⁢yb′⁢zc′∈G⋆ satisfy

[xa⁢yb⁢zc,xa′⁢yb′⁢zc′]=za′⁢b−a⁢b′⁢z1b⁢(a′2)−b′⁢(a2)+(a′⁢c−a⁢c′)⁢z2a′⁢(b2)−a⁢(b′2)+(b′⁢c−b⁢c′).

Consider N=⟨z,z1,z2⟩≅(Z/p2⁢Z)3, a normal subgroup of G⋆. Any character χ:N→C× is given by χ⁢(zi)=λi for i=1,2 and χ⁢(z)=λ, where λip2=1 for i=1,2 and λp2=1. The inertia group Sχ=IG⋆⁢(χ) of 𝜒 consists of the elements xa⁢yb⁢n, n∈N, such that χ⁢([xa⁢yb,z])=1, that is χ⁢(z1)a⁢χ⁢(z2)b=1. We consider the various cases of λi’s.

(i) Assume λ1=λ2=1. In this case, Sχ=G⋆. We note that

|Sχ/N|=|G⋆|/|N|=p4,

Sχ/N is abelian. Hence, by Theorem 2.3 and Lemma 2.6, depending on χ⁢(z), Irr⁢(G⋆|χ) consists of representations of dimension either 1, 𝑝 or p2.

(ii) Assume λ1=1, λ2≠1 such that λ2p=1. Then Sχ=⟨N,x,yp⟩. Therefore, |Sχ/N|=p3 and Sχ/N is abelian. Hence Irr⁢(G⋆|χ) consists of 𝑝-dimensional representations if 𝜒 extends to Sχ, and if 𝜒 does not extend to Sχ, then Irr⁢(G⋆|χ) consists of p2-dimensional representations. We use Lemma 2.4 and

χ⁢(z2)p⁢t=χ⁢(z2)p⁢t1=1

to observe that 𝜒 extends to Sχ if and only if

χ⁢([xa⁢yp⁢t⁢zc,xa′⁢yp⁢t1⁢zc′])=χ⁢(z)p⁢(t⁢a′−t1⁢a)=1.

Therefore, 𝜒 extends if and only if λp=1.

(iii) Assume λ1=1 and order of λ2 is p2. In this case, we obtain Sχ=⟨N,x⟩. Since Sχ/N is cyclic of order p2, the character 𝜒 extends to Sχ by Lemma 2.4. Therefore, by Theorem 2.3, Irr⁢(G⋆∣χ) consists of p2-dimensional irreducible representations.

(iv) For λ2=1 and λ1≠1, we proceed as above.

(v) Suppose λ1≠1, λ2≠1 and λ1p=λ2p=1. Assume that χ⁢(z1)=χ⁢(z2)r for some 1≤r<p−1. For the inertia group computations, 𝑎 and 𝑏 are such that χ⁢(z1)a⁢χ⁢(z2)b=1, that is χ⁢(z2)b+r⁢a=1. This implies

b∈{−r⁢a+p⁢t∣0≤t≤p−1}.

Therefore, we get Sχ=⟨N,xa⁢y−r⁢a+p⁢t⟩. So |Sχ/N|=p3 and Sχ/N is abelian. As in (ii), Irr⁢(G⋆|χ) consists of 𝑝-dimensional representations if 𝜒 extends to Sχ, and if 𝜒 does not extend to Sχ, then Irr⁢(G⋆|χ) consists of p2-dimensional representations. So it remains to determine the conditions for which 𝜒 extend to Sχ. By Lemma 2.4, 𝜒 extends to Sχ if and only if

χ⁢([xa⁢y−r⁢a+p⁢t⁢zc,xa′⁢y−r⁢a′+p⁢t′⁢zc′])=1.

This is equivalent to

χ⁢(z)p⁢(a′⁢t−a⁢t′)⁢χ⁢(z2)r2⁢a⁢a′⁢(a−a′)=1,

where a,a′,t,t′ are arbitrary. Therefore, we must have λ2r2=1 and χ⁢(z)p=1. This implies that 𝑟 is a multiple of 𝑝. This in turn gives that λ1=λ2r=1, a contradiction. Hence 𝜒 does not extend to Sχ in this case. Therefore, Irr⁢(G⋆|χ) consists of p2-dimensional representations.

(vi) Assume that λ1≠1 and λ2 is of order p2. Assume that λ1=(λ2)r. It is easy to see that Sχ=⟨N,xa⁢y−r⁢a⟩, and therefore Sχ/N is an abelian group of order p2. As in (v), we obtain that 𝜒 extends to Sχ if and only if λ1p=1. Hence, in this case, Irr⁢(G⋆|χ) consists of p2-dimensional representations. If both λ1 and λ2 are of order p2, then Irr⁢(G⋆|χ) consists of p3-dimensional representations.

(vii) Assume that the order of λ1 is p2 and that of λ2 is 𝑝. This case can be done parallel to (v). Here we obtain that Irr⁢(G⋆|χ) consists of p2-dimensional representations.

6.4 Non-degenerate 2-cocycles of H3⁢(Z/p2⁢Z)

In this section, we describe the non-degenerate cocycles of the group H3⁢(Z/p2⁢Z).

Theorem 6.3

The cocycle 𝛼 as given in (6.6) is non-degenerate if and only if λ1 and λ2 are of order p2. In particular, the number of non-degenerate cocycles is (p−1)2⁢p2.

Proof

Recall that every 2-cocycle of G=H3⁢(Z/p2⁢Z) is cohomologous to a cocycle of the following form:

(6.6)α⁢(xa1⁢yb1⁢zc1,xa1′⁢yb1′⁢zc1′)=λ1c1′⁢a1+b1⁢(a1′2)+a1⁢b1⁢a1′⁢λ2c1′⁢b1+a1′⁢(b12),λ1p2=λ2p2=1.

Let 𝑍 be a subgroup of G⋆ generated by z1 and z2. Then Z⊆N and

1→Z→G⋆→G→1

is a stem extension. For any character 𝜒 of 𝑁 as given above, let χ|Z denote its restriction to the group 𝑍. We note that tra⁢(χ|Z)=[α]. The result now follows from the above description of ordinary irreducible representations of G⋆, Theorem 2.2 and Lemma 2.7. ∎

Funding source: Indian Institute of Science

Award Identifier / Grant number: R(IA)/CVR-PDF/2020/2700

Funding source: Science and Engineering Research Board

Award Identifier / Grant number: MTR/2018/000501

Award Identifier / Grant number: MTR/2018/000094

Funding source: Ministry of Education, India

Award Identifier / Grant number: SPARC/2018-2019/P88/SL

Funding statement: S. Hatui acknowledges the financial support received from IISc in the form of Raman postdoctoral fellowship (R(IA)/CVR-PDF/2020/2700). E. K. Narayanan thanks SERB, India for the financial support through MATRICS grant MTR/2018/000501. P. Singla thanks SERB, India and MHRD, India for the financial support through MATRICS grant MTR/2018/000094 and MHRD-SPARC grant (SPARC/2018-2019/P88/SL).

Acknowledgements

The authors thank the referee for a very careful reading and valuable suggestions which improved the paper greatly.

Communicated by: Benjamin Klopsch

References

[1] N. B. Backhouse and C. J. Bradley, Projective representations of abelian groups, Proc. Amer. Math. Soc. 36 (1972), 260–266. 10.1090/S0002-9939-1972-0308329-5Search in Google Scholar

[2] N. Ben David and Y. Ginosar, On groups of central type, non-degenerate and bijective cohomology classes, Israel J. Math. 172 (2009), 317–335. 10.1007/s11856-009-0075-0Search in Google Scholar

[3] N. Blackburn and L. Evens, Schur multipliers of 𝑝-groups, J. Reine Angew. Math. 309 (1979), 100–113. 10.1515/crll.1979.309.100Search in Google Scholar

[4] P. Etingof and S. Gelaki, The classification of triangular semisimple and cosemisimple Hopf algebras over an algebraically closed field, Int. Math. Res. Not. IMRN 2000 (2000), no. 5, 223–234. Search in Google Scholar

[5] S. Hatui and P. Singla, On Schur multiplier and projective representations of Heisenberg groups, J. Pure Appl. Algebra 225 (2021), no. 11, Paper No. 106742. 10.1016/j.jpaa.2021.106742Search in Google Scholar

[6] R. J. Higgs, Projective representations of abelian groups, J. Algebra 242 (2001), no. 2, 769–781. 10.1006/jabr.2000.8751Search in Google Scholar

[7] G. Hochschild and J.-P. Serre, Cohomology of group extensions, Trans. Amer. Math. Soc. 74 (1953), 110–134. 10.1090/S0002-9947-1953-0052438-8Search in Google Scholar

[8] I. M. Isaacs, Character Theory of Finite Groups, AMS Chelsea, Providence, 2006. 10.1090/chel/359Search in Google Scholar

[9] N. Iwahori and H. Matsumoto, Several remarks on projective representations of finite groups, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 129–146. Search in Google Scholar

[10] U. Jezernik, Schur multipliers of unitriangular groups, J. Algebra 399 (2014), 26–38. 10.1016/j.jalgebra.2013.10.004Search in Google Scholar

[11] G. Karpilovsky, Projective Representations of Finite Groups, Monogr. Textb. Pure Appl. Math. 94, Marcel Dekker, New York, 1985. Search in Google Scholar

[12] G. Karpilovsky, The Schur Multiplier, London Math. Soc. Lecture Note Ser. 2, Oxford University, New York, 1987. Search in Google Scholar

[13] G. W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. 10.1007/BF02392428Search in Google Scholar

[14] A. O. Morris, Projective representations of Abelian groups, J. Lond. Math. Soc. (2) 7 (1973), 235–238. 10.1112/jlms/s2-7.2.235Search in Google Scholar

[15] A. O. Morris, M. Saeed-Ul-Islam and E. Thomas, Some projective representations of finite abelian groups, Glasg. Math. J. 29 (1987), no. 2, 197–203. 10.1017/S0017089500006844Search in Google Scholar

[16] J. Schur, Über die Darstellung der endlichen Gruppen durch gebrochen lineare Substitutionen, J. Reine Angew. Math. 127 (1904), 20–50. 10.1515/crll.1904.127.20Search in Google Scholar

[17] J. Schur, Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 132 (1907), 85–137. 10.1515/crll.1907.132.85Search in Google Scholar

[18] J. Schur, Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155–250. 10.1515/crll.1911.139.155Search in Google Scholar

Received: 2022-02-16

Revised: 2023-03-18

Published Online: 2023-06-27

Published in Print: 2024-03-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2022-0033