STRICT REGULARITY FOR $2$-COCYCLES OF FINITE GROUPS

Abstract

Let $\alpha $ be a complex-valued $2$-cocycle of a finite group $G.$ A new concept of strict $\alpha $-regularity is introduced and its basic properties are investigated. To illustrate the potential use of this concept, a new proof is offered to show that the number of orbits of G under its action on the set of complex-valued irreducible $\alpha _N$-characters of N equals the number of $\alpha $-regular conjugacy classes of G contained in $N,$ where N is a normal subgroup of $G.$

1 Introduction

Throughout this paper, G will denote a finite group and it will be implicitly assumed that all projective representations affording projective characters are defined over the field of complex numbers $\mathbb {C}.$

Definition 1.1. A $2$ -cocycle of G over $\mathbb {C}$ is a function $\alpha : G\times G\rightarrow \mathbb {C}^*$ such that $\alpha (1, 1) = 1$ and $\alpha (x, y)\alpha (xy, z) = \alpha (x, yz)\alpha (y, z)$ for all x, y, $z\in G.$

The set of all such $2$ -cocycles of G form a group $Z^2(G, \mathbb {C}^*)$ under multiplication. Let $\delta : G\rightarrow \mathbb {C}^*$ be any function with $\delta (1) = 1.$ Then $t(\delta )(x, y) = \delta (x)\delta (y)/\delta (xy)$ for all $x, y\in G$ is a $2$ -cocycle of G, which is called a coboundary. Two $2$ -cocycles $\alpha $ and $\beta $ are cohomologous if there exists a coboundary $t(\delta )$ such that $\beta = t(\delta )\alpha .$ This defines an equivalence relation on $Z^2(G, \mathbb {C}^*)$ and the cohomology classes $[\alpha ]$ form a finite abelian group, called the Schur multiplier $M(G).$

Definition 1.2. Let $\alpha $ be a $2$ -cocycle of G.

1. (a) Define $f_{\alpha }: G\times G\rightarrow \mathbb {C}^*$ by

   $$ \begin{align*}f_{\alpha}(g, x) = \frac{\alpha(g, x)\alpha(gx, g^{-1})}{\alpha(g, g^{-1})}.\end{align*} $$

2. (b) For each $x\in G$ , define $\alpha _x: C_G(x)\rightarrow \mathbb {C}^*$ by $\alpha _x(g) = \alpha (g, x)/\alpha (x, g).$

These two functions arise naturally in the twisted group algebra $(\mathbb {C}(G))_{\alpha }$ in which $\bar {x}\bar {y} = \alpha (x, y)\overline {xy}$ for all $x, y\in G$ (see [4, page 66]). Here, $\bar {g}\bar {x}\bar {g}^{-1} = f_{\alpha }(g, x)\overline {gxg^{-1}}$ for $g, x\in G$ and $\bar {g}\bar {x}\bar {g}^{-1} = \alpha _x(g)\bar {x}$ if $g\in C_G(x).$ Also, if $\beta = t(\delta )\alpha $ , then $f_{\beta }(g, x) = (\delta (x)/\delta (gxg^{-1}))f_{\alpha }(g, x)$ for all $g, x\in G$ and consequently $\alpha _x = \beta _x.$

Now $\alpha _x\in \operatorname {\mathrm {Lin}}(C_G(x))$ from [6, Lemma 4.2], where $\operatorname {\mathrm {Lin}}(C_G(x))$ is the group of linear characters of $C_G(x).$ The kernel of $\alpha _x$ is the absolute centraliser $C_{\alpha }(x)$ of x with respect to $\alpha $ and $C_G(x)/C_{\alpha }(x)\cong \langle \alpha _x\rangle .$

Definition 1.3. Let $\alpha $ be a $2$ -cocycle of G. Then $x\in G$ is $\alpha $ -regular if $\alpha _x$ is the trivial character of $C_G(x)$ (or equivalently $C_{\alpha }(x) = C_G(x)).$

First, every element of G is $\alpha $ -regular if $[\alpha ]$ is trivial. Second, setting $y = 1$ and $z = 1$ in Definition 1.1 yields $\alpha (x, 1) = 1$ and similarly $\alpha (1, x) = 1$ for all $x\in G,$ and hence $1$ is always $\alpha $ -regular. Third, if $x\in G$ is $\alpha $ -regular, then it is $\alpha ^k$ -regular for any integer $k.$ Finally, if $x\in G$ is $\alpha $ -regular, then so too is any conjugate of x (see [4, Lemma 2.6.1]), so that one may refer to the $\alpha $ -regular conjugacy classes of $G.$

Now let $\operatorname {\mathrm {Proj}}(G, \alpha )$ denote the set of all irreducible $\alpha $ -characters of G (see [4, page 184]). Then $x\in G$ is $\alpha $ -regular if and only if $\xi (x)\not = 0$ for some $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ (see [5, Proposition 1.6.3]) and $\vert \!\operatorname {\mathrm {Proj}}(G, \alpha )\vert $ is the number of $\alpha $ -regular conjugacy classes of G (see [5, Theorem 1.3.6]).

Let N be a normal subgroup of $G.$ Then G acts on $\operatorname {\mathrm {Proj}}(N, \alpha _N)$ by

$$ \begin{align*}\zeta^g(x) = f_{\alpha}(g, x)\zeta(gxg^{-1})\end{align*} $$

for $\zeta \in \operatorname {\mathrm {Proj}}(N, \alpha _N), g\in G$ and all $x\in N$ . Clifford’s theorem for projective characters applies to this action (see [5, Theorem 2.2.1]).

A new concept of strict $\alpha ^d$ -regularity, which refines the notion of $\alpha ^d$ -regularity, will be defined and investigated in Section 2 for d a divisor of the order of $[\alpha ].$ This concept will be used in Section 3 to give an alternative proof that the number of orbits of G under its action on $\operatorname {\mathrm {Proj}}(N, \alpha _N),$ for N a normal subgroup of $G,$ is equal to the number of $\alpha $ -regular conjugacy classes of G contained in N from [2, Lemma 3.1]. It is also easy to show that this result is independent of the choice of $2$ -cocycle from $[\alpha ].$ The result is well known when $\alpha $ is trivial (see [3, Corollary 6.33]); the method employed will be to apply this to the orbits of an $\alpha $ -covering group of G under its action on the irreducible characters and conjugacy classes of a normal subgroup, but to decompose these orbits into corresponding sets.

2 Strictly $\alpha ^d$ -regular elements

Let $o(\phantom {.})$ denote the order of an element in a group. Then for $[\beta ]\in M(G)$ , there exists $\alpha \in [\beta ]$ such that $o(\alpha ) = o([\beta ])$ and $\alpha $ is a class-function cocycle, that is, the elements of $\operatorname {\mathrm {Proj}}(G, \alpha )$ are class functions (see [5, Corollary 4.1.6]). To avoid repetition throughout the rest of this paper, it will be assumed that $\alpha $ has these two properties with $n = o(\alpha ).$ A consequence of the second property is that $x\in G$ is $\alpha $ -regular if and only if $f_{\alpha }(g, x) = 1$ for all $g\in G$ (see [5, page 33]). The first property allows us to make the following definition in terms of $\alpha ^d$ rather than for the more clumsy $\beta \in [\alpha ]^d.$

Definition 2.1. Define $x\in G$ to be strictly $\alpha ^d$ -regular if d is the smallest integer with $1\leq d\leq n$ such that x is $\alpha ^d$ -regular.

Next suppose $o(\alpha ^d)= o(\alpha ^k) = m.$ If $\omega $ is a primitive mth root of unity, then there exists a field automorphism $\tau $ of $\mathbb {Q}(\omega )$ over $\mathbb Q$ such that $\tau (\alpha ^d) =\alpha ^k.$ Consequently, $x\in G$ is $\alpha ^d$ -regular if and only if it is $\alpha ^k$ -regular. Thus, $d\mid n$ in Definition 2.1.

Let $\pi (d)$ denote the set of prime numbers that divide d and let $d_p$ denote the pth part of d for any prime number $p.$

Lemma 2.2. We have $x\in G$ is strictly $\alpha ^d$ -regular if and only if either:

1. (a) x is $\alpha ^d$ -regular but not $\alpha ^{d/p}$ -regular for each $p\in \pi (d);$ or

2. (b) $o(\alpha _x) = d$ in $\operatorname {\mathrm {Lin}}(C_G(x)).$

Proof. For condition (a), if x is not $\alpha ^{d/p}$ -regular, then it is not $\alpha ^t$ -regular for all positive integers t with $t\mid d/p.$ For condition (b), observe that x is $\alpha ^d$ -regular if and only if $\alpha _x^d$ is trivial, that is, $o(\alpha _x)\mid d.$ Now for $d> 1$ , x is strictly $\alpha ^d$ -regular if and only if $o(\alpha _x)\mid d$ , but $\alpha _x^{d/p}\not =1$ for each prime $p\in \pi (d)$ from condition (a). The latter is true if and only if $d_p\mid o(\alpha _x)$ for each prime $p\in \pi (d),$ that is, if and only if $d\mid o(\alpha _x).$

An equivalent way of stating Lemma 2.2(b) is that $x\in G$ is strictly $\alpha ^d$ -regular if and only if $\vert C_G(x)/C_{\alpha }(x)\vert = d.$

Now by definition for each $x\in G$ , there exists a unique $d\mid n$ such that x is strictly $\alpha ^d$ -regular. Thus, the conjugacy classes of G are partitioned into strictly $\alpha ^d$ -regular conjugacy classes. So for $d\mid n$ and N a normal subgroup of $G,$ let $t_d$ be the number of strictly $\alpha ^d$ -regular conjugacy classes of G contained in $N.$ Thus, the number of $\alpha ^d$ -regular conjugacy classes of G contained in N is $\sum _{s|d} t_s;$ in particular, $\sum _{d\mid n} t_d = t(N),$ where $t(N)$ is the number of conjugacy classes of G contained in $N.$

The choice of $2$ -cocycle $\alpha $ allows the construction of an $\alpha $ -covering group H of G with the following three properties (see [4, Section 4.1]):

1. (a) H has a cyclic subgroup $A\leq Z(H)\cap H'$ of order $n;$

2. (b) there exists a conjugacy-preserving transversal (see below) $\{r(g): g\in G\}$ of A in H such that $\theta : H\rightarrow G$ defined by $\theta (r(g)a) = g$ for all $g\in G$ and all $a\in A$ is a homomorphism with kernel $A;$

3. (c) there exists a faithful character $\lambda \in \operatorname {\mathrm {Lin}}(A)$ such that $\alpha (x, y) = \lambda (A(x, y))$ for all $x, y\in G,$ where $r(x)r(y) = A(x, y)r(xy).$

A conjugacy-preserving transversal means that $r(x)$ and $r(y)$ are conjugate in H if and only if x and y are conjugate in G (see [5, Lemma 4.1.1]).

It is easy to see that $\theta (C_H(r(x))) = C_{\alpha }(x)$ for $x\in G$ and $\theta (C_H(r(x)A)) = C_G(x).$ Thus, working in H, we see that x is strictly $\alpha ^d$ -regular if and only if the cyclic group $C_H(r(x)A)/C_H(r(x))$ has order $d.$

Proposition 2.3. Let H be an $\alpha $ -covering group of $G.$ Then $x\in G$ is strictly $\alpha ^d$ -regular if and only if either:

1. (a) $r(x)\langle z^m\rangle $ are the conjugates of $r(x)$ in $r(x)A,$ where $\langle z\rangle = A$ and $dm = n$ ; or

2. (b) $\{r(x)z^i: i = 1,\ldots , m\}$ is a maximal set of conjugacy class representatives of H in $r(x)A.$

Proof. Define $k_{r(x)} : C_H(r(x)A)\rightarrow A$ by $k_{r(x)}(h) = hr(x)h^{-1}(r(x))^{-1}.$ Then $k_{r(x)}$ is a homomorphism with kernel $C_H(r(x))$ , since $\lambda (k_{r(x)}) = \alpha _x.$ Now let z be a generator of A. Then $r(x)z^i$ and $r(x)z^j$ are conjugate if and only if $z^{j-i}\in \operatorname {\mathrm {Im}}(k_{r(x)}),$ that is, if and only if $z^i\operatorname {\mathrm {Im}}(k_{r(x)}) = z^j\operatorname {\mathrm {Im}}(k_{r(x)}).$

Now x is strictly $\alpha ^d$ -regular if and only if $\operatorname {\mathrm {Im}}(k_{r(x)}) = \langle z^m\rangle ,$ that is, if and only if the cosets of $\operatorname {\mathrm {Im}}(k_{r(x)})$ in A are $z^i\langle z^m\rangle $ for $i =1,\ldots , m.$

3 Counting orbits of projective characters

Let N be a subgroup of $G.$ Let H be an $\alpha $ -covering group of G and, using the notation of Section 2, let M be the subgroup of H containing A such that $\theta (M) = N.$ Finally, for any integer k, let $\operatorname {\mathrm {Irr}}(M\vert \lambda ^k) = \{\chi \in \operatorname {\mathrm {Irr}}(M): \chi _A = \chi (1)\lambda ^k\},$ where $\operatorname {\mathrm {Irr}}(M)$ is the set of irreducible characters of $M.$ Then the mapping from $\operatorname {\mathrm {Proj}}(N, \alpha _N^k)$ to $\operatorname {\mathrm {Irr}}(M\vert \lambda ^k), \zeta \mapsto \chi $ is a bijection, where $\zeta (x) = \chi (r(x))$ for all $x\in N$ (see [4, pages 134–135] or [5, Corollary 4.1.3]). Now suppose N is normal in G, then it is easy to check that $\zeta ^g = \chi ^{r(g)}$ for all $g\in G$ and hence the orbit length of $\zeta $ under the action of G equals that of $\chi $ under the action of $H.$ By definition, for each $x\in G$ , there exists a unique $d\mid n$ such that x is strictly $\alpha ^d$ -regular. Thus, the conjugacy classes of H are partitioned according to $\vert C_H(r(x)A)/C_H(r(x)\vert $ for $r(x)a,$ where $x\in G$ and $a\in A.$ However, if x is a strictly $\alpha ^d$ -regular conjugacy class representative of $G,$ then $n/d$ corresponding conjugacy class representatives of H are obtained as detailed in Proposition 2.3. So the number of conjugacy classes of H in M corresponding to the number of $\alpha ^d$ -regular conjugacy classes of G contained in N is $\sum _{s|d} (n/s)t_s;$ in particular, $\sum _{d|n} (n/d)t_d = t(M),$ where $t(M)$ is the number of conjugacy classes of H contained in $M.$

Lemma 3.1. Let N be a normal subgroup of G and suppose that $o(\alpha ^d) = o(\alpha ^k).$ Let $\sigma $ be a field automorphism of $\mathbb {C}$ that extends $\tau $ , as described in Section 2, so that $\sigma (\alpha ^d) = \alpha ^k.$ Then $\zeta ^g = \zeta '$ if and only if $\sigma (\zeta )^{g} = \sigma (\zeta ')$ for $g\in G$ and $\zeta \in \operatorname {\mathrm {Proj}}(N, \alpha _N^d).$

Proof. If $\zeta \in \operatorname {\mathrm {Proj}}(N, \alpha _N^d),$ then $\sigma (\zeta )\in \operatorname {\mathrm {Proj}}(N, \sigma (\alpha _N^d)).$ Now

$$ \begin{align*}\sigma(\zeta)^g(x) = f_{\sigma(\alpha)}(g, x)\sigma(\zeta(gxg^{-1})) = \sigma(f_{\alpha}(g, x)\zeta(gxg^{-1}))\end{align*} $$

for all $x\in N.$

Lemma 3.1 sets up a one-to-one correspondence between the orbits of G under its action on $\operatorname {\mathrm {Proj}}(N, \alpha _N^d)$ and those under its action on $\operatorname {\mathrm {Proj}}(N, \alpha _N^k)$ in which orbit lengths are preserved. We next just restate Lemma 3.1 for an $\alpha $ -covering group H of $G.$

Corollary 3.2. Suppose that $o(\lambda ^d) = o(\lambda ^k)$ in $\langle \lambda \rangle = \operatorname {\mathrm {Lin}}(A).$ Let $\sigma $ be as in Lemma 3.1, so that $\sigma (\lambda ^d)=\lambda ^k.$ Then $\chi ^h = \chi '$ if and only if $\sigma (\chi )^h = \sigma (\chi ')$ for $h\in H$ and $\chi \in \operatorname {\mathrm {Irr}}(M\vert \lambda ^d).$

Let $\phi $ denote Euler’s totient function. We use the well-known result from number theory that $\sum _{d\mid n}\phi (d) = \sum _{d\mid n}\phi (n/d) = n.$

Theorem 3.3. Let N be a normal subgroup of $G.$ Then the number of orbits of G under its action on $\operatorname {\mathrm {Proj}}(N, \alpha _N)$ is equal to the number of $\alpha $ -regular conjugacy classes of G contained in $N.$

Proof. Proceeding by induction, we count the number of $\alpha ^d$ -regular conjugacy classes of G contained in N. First, if $d = n,$ then, as previously stated, the number of conjugacy classes of G contained in N is equal to the number of orbits of G under its action on $\operatorname {\mathrm {Irr}}(N).$ So assume by induction that the number of orbits of G under its action on $\operatorname {\mathrm {Proj}}(N, \alpha _N^d)$ is equal to the number of $\alpha ^d$ -regular conjugacy classes of G contained in N for each $d\mid n$ with $d\not = 1.$ Let H be an $\alpha $ -covering group of G and let M denote the subgroup of H containing A such that $\theta (M) = N.$

Now for $d\mid n$ and $d\not = 1$ , G has $\sum _{s\mid d} t_s$ orbits under its action on $\operatorname {\mathrm {Proj}}(N, \alpha _N^d).$ Thus, H has the same number of orbits under its action on $\operatorname {\mathrm {Irr}}(M\vert \lambda ^d).$ Now $o(\lambda ^k) = o(\lambda ^d)$ for $\phi (n/d)$ values of k with $1\leq k\leq n.$ Thus, using Corollary 3.2, the total number of orbits of H under its actions on $\operatorname {\mathrm {Irr}}(M\vert \lambda ^c),$ for the $n - \phi (n)$ values of c with $1\leq c\leq n$ that are not relatively prime to $n,$ is

$$ \begin{align*} \sum_{\substack{d\mid n\\ d\not=1}}\phi\bigg(\frac{n}{d}\bigg)\bigg(\sum_{s\mid d} t_s\bigg) &= \sum_{s\mid d} t_s\bigg(\sum_{\substack{d\mid n\\ d\not=1}}\phi\bigg(\frac{n}{d}\bigg)\bigg) \\ &= \sum_{s\mid n} t_s\bigg(\sum_{\substack{r\mid (n/s)\\ (r, s)\not=(1,1)}}\phi\bigg(\frac{n/s}{r}\bigg)\bigg) \\ &= t_1(n-\phi(n))+ \sum_{\substack{s\mid n\\ s\not=1}}t_s\frac{n}{s}. \end{align*} $$

The total number of orbits of H under its action on $\operatorname {\mathrm {Irr}}(M)$ is $t(M),$ so the total number of orbits of H under its actions on $\operatorname {\mathrm {Irr}}(M\vert \lambda ^c),$ for the $\phi (n)$ values of c with $1\leq c\leq n$ that are relatively prime to $n,$ is

$$ \begin{align*}t(M) - t_1(n-\phi(n)) - \sum_{\substack{s\mid n\\ s\not=1}} t_s\frac{n}{s} = t_1\phi(n).\end{align*} $$

Hence, the number of orbits of H under its action on $\operatorname {\mathrm {Irr}}(M\vert \lambda )$ (and the number of orbits of G under its action on $\operatorname {\mathrm {Proj}}(N, \alpha _N)$ ) is $t_1,$ as required.

Suppose that $\beta = t(\delta )\alpha .$ Then from [1, Lemma 1.4], we see that $\operatorname {\mathrm {Proj}}(N, \beta _N) = \{\delta _N\zeta : \zeta \in \operatorname {\mathrm {Proj}}(N, \alpha _N)\}$ and, for $g\in G$ , $\zeta ^g = \zeta '$ if and only if $(\delta _N\zeta )^g = \delta _N\zeta '$ for $\zeta \in \operatorname {\mathrm {Proj}}(N, \alpha _N).$ In particular, this establishes a one-to-one correspondence between the orbits of G under its action on $\operatorname {\mathrm {Proj}}(N, \beta _N)$ and those under its action on $\operatorname {\mathrm {Proj}}(N, \alpha _N)$ in which orbit lengths are preserved. So from this and Lemma 3.1, the result of Theorem 3.3 is independent of the choice of $2$ -cocycle from $[\alpha ]^c$ for c relatively prime to $n.$