HALL SUBGROUPS AND $2$-COCYCLE REGULARITY

Abstract

Let H be a subgroup of a finite group G and let $\alpha $ be a complex-valued $2$-cocycle of $G.$ Conditions are found to ensure there exists a nontrivial element of H that is $\alpha $-regular in $G.$ However, a new result is established allowing a prime by prime analysis of the Sylow subgroups of $C_G(x)$ to determine the $\alpha $-regularity of a given $x\in G.$ In particular, this result implies that every $\alpha _H$-regular element of a normal Hall subgroup H is $\alpha $-regular in $G.$

1 Introduction

Throughout this paper, G will denote a finite group.

Definition 1.1. A $2$ -cocycle of G over $\mathbb {C}$ is a function $\alpha : G\times G\rightarrow \mathbb {C}^*$ such that $\alpha (x, 1) = 1$ 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 forms 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 $G.$ Then $x\in G$ is $\alpha $ -regular if $\alpha (x, g) = \alpha (g, x)$ for all $g\in C_G(x).$

Obviously, if $x\in G$ is $\alpha $ -regular, then it is $\alpha ^k$ -regular for any integer $k;$ also setting $y = 1$ and $z = x$ in Definition 1.1 yields $\alpha (1, x) = 1$ for all $x\in G$ and hence $1$ is $\alpha $ -regular. Let $\beta \in [\alpha ]$ . Then x is $\alpha $ -regular if and only if it is $\beta $ -regular and any conjugate of x is also $\alpha $ -regular (see [5, Lemma 2.6.1]), so that one may refer to the $\alpha $ -regular conjugacy classes of $G.$ Using this notation and $o(\phantom {.})$ for the order of a group element, we quote [3, Lemma 1.2(b)] for future reference.

Lemma 1.3. Suppose $o(x)$ and $o([\alpha ])$ are relatively prime. Then x is $\alpha $ -regular.

Let H be a subgroup of $G.$ Given a $2$ -cocycle $\alpha $ of G, one can define the $2$ -cocycle $\alpha _H$ of H by $\alpha _H(x, y) = \alpha (x, y)$ for all $x, y\in H.$ The mapping from $Z^2(G, \mathbb {C}^*)\rightarrow Z^2(H, \mathbb {C}^*)$ defined by $\alpha \mapsto \alpha _H$ maps coboundaries of G to those of H and consequently induces the restriction homomorphism $\operatorname {\mathrm {Res}}_{G, H}: M(G)\rightarrow M(H)$ defined by $[\alpha ]\mapsto [\alpha _H].$ Clearly, an element $h\in H$ that is $\alpha $ -regular in G is $\alpha _H$ -regular, but the converse is in general false. The twin aims of this paper are to find conditions under which first there exists a nontrivial element of H that is $\alpha $ -regular in G and second that every $\alpha _H$ -regular element of H is $\alpha $ -regular in $G.$

There are some circumstances in which it is possible to produce a nontrivial element $x\in G$ that is $\alpha $ -regular for all $[\alpha ]\in M(G).$ For example, this is true if $C_G(x) = \langle x\rangle ,$ since the Schur multiplier of a cyclic group is trivial (see [4, Proposition 2.1.1]). However, in general, $\alpha $ -regularity very much depends upon the choice of $[\alpha ]$ as the next example demonstrates, using the inflation homomorphism. Let N be a normal subgroup of G. Then the mapping from $Z^2(G/N, \mathbb {C}^*)\rightarrow Z^2(G, \mathbb {C}^*),\, \beta \mapsto \alpha ,$ where $\alpha (x, y) = \beta (xN, yN)$ for all $x, y\in G$ maps coboundaries of $G/N$ to those of G and hence induces $\operatorname {\mathrm {Inf}}: M(G/N)\rightarrow M(G), [\beta ]\mapsto [\alpha ].$ Using this notation, it is clear that every element of N is $\alpha $ -regular.

Example 1.4. Let $C_n^{(m)}$ denote the direct product of m copies of the cyclic group of order $n.$ Let $G\cong C_{n_1}\times \cdots \times C_{n_k},$ where $n_{i+1}\mid n_i$ for $i = 1,\ldots , k-1$ and $k\geq 2.$ Then $M(G) \cong C_{n_2}\times C_{n_3}^{(2)}\times \cdots \times C_{n_k}^{(k-1)}$ (see [4, Corollary 2.2.12]). Also, the group of elements that are $\alpha $ -regular for all $[\alpha ]\in M(G)$ is isomorphic to $C_{n_1/n_2}$ (see [5, Theorem 11.8.19]). Let $R\cong C_2^{(2)},$ then $M(R)\cong C_2$ and so only the trivial element of $C_2^{(2)}$ is $\alpha $ -regular for $[\alpha ]$ nontrivial. However, if $H \neq R$ is a subgroup of R, then every element of H is $\alpha _H$ -regular. Now let $S \cong C_2^{(3)},$ so that $M(S)\cong C_2^{(3)}.$ Let x be a nontrivial element of $S.$ Then $\operatorname {\mathrm {Inf}}: M(S/\langle x\rangle )\rightarrow M(S)$ is an injective map (see [4, Theorem 2.3.10]) that produces a subgroup $\langle [\alpha ]\rangle $ of order $2$ of $M(S)$ in which $1$ and x are the only $\alpha $ -regular elements. Thus, for any two different nontrivial elements $[\alpha ], [\beta ]\in M(S)$ , the intersection of the set of $\alpha $ -regular elements and $\beta $ -regular elements of S contains only the identity element.

2 Subgroups and regularity

Definition 2.1. Let $\alpha $ be a $2$ -cocycle of $G.$ Then an $\alpha $ -representation of G of dimension n is a function $P:G\rightarrow GL(n, \mathbb {C})$ such that $P(x)P(y) = \alpha (x, y)P(xy)$ for all x, $y\in G.$

An $\alpha $ -representation P is also called a projective representation of G with $2$ -cocycle $\alpha ,$ its trace function $\xi $ is its $\alpha $ -character and $\xi (1),$ which is the dimension of $P,$ is called the degree of $\xi .$

To avoid repetition, all $\alpha $ -representations of G in this section are defined over $\mathbb {C}.$ Let $\operatorname {\mathrm {Proj}}(G, \alpha )$ denote the set of all irreducible $\alpha $ -characters of $G,$ the relationship between $\operatorname {\mathrm {Proj}}(G, \alpha )$ and $\alpha $ -representations is much the same as that between $\operatorname {\mathrm {Irr}}(G)$ and (ordinary) representations of G (see [5, page 184] for details) so, for example, $\sum _{\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )}\xi (1)^2 = \vert G\vert $ (see [6, Lemma 1.4.4]). Next, $x\in G$ is $\alpha $ -regular if and only if $\xi (x)\not = 0$ for some $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ (see [6, Proposition 1.6.3]) and $\vert \!\operatorname {\mathrm {Proj}}(G, \alpha )\vert $ is the number of $\alpha $ -regular conjugacy classes of G (see [6, Theorem 1.3.6]).

For $[\beta ]{\kern-1pt}\in{\kern-1pt} M(G)$ , there exists $\alpha {\kern-1pt}\in{\kern-1pt} [\beta ]$ such that $o(\alpha ) {\kern-1pt}={\kern-1pt} o([\beta ])$ and $\alpha $ is class-preserving, that is, the elements of $\operatorname {\mathrm {Proj}}(G, \alpha )$ are class functions (see [6, Corollary 4.1.6]). Henceforward, it will be assumed, without loss of generality, that the initial choice of $2$ -cocycle $\alpha $ has these two properties. Under these assumptions, the ‘standard’ inner product $\langle \phantom {\xi }, \phantom {.}\rangle $ may be defined on $\alpha _H$ -characters of subgroups H of G and the ‘normal’ orthogonality relations hold (see [6, Section 1.11.D]).

The main result in this section is the following simple observation.

Lemma 2.2. Let $\alpha $ be a $2$ -cocycle of G and let H be a subgroup of $G.$ Let $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ and $\gamma \in \operatorname {\mathrm {Proj}}(H, \alpha _H).$ Suppose that either $\langle \xi _H, \gamma \rangle = 0$ or $\vert H\vert \nmid \xi (1)\gamma (1).$ Then there exists a nontrivial $h\in H$ such that $\xi (h)\overline {\gamma (h)}\not = 0$ and, in particular, all such elements are $\alpha $ -regular in $G.$

Proof. The inner product of $\xi _H$ and $\gamma ,$ which is a nonnegative integer, is defined by

$$ \begin{align*}\langle \xi_H, \gamma \rangle = \frac{1}{\vert H\vert}\bigg(\xi(1)\gamma(1) + \sum_{h\in H -\{1\}}\xi(h)\overline{\gamma(h)}\bigg).\end{align*} $$

Thus, under the two specified conditions, the summation on the right-hand side must be nonzero.

Using Frobenius reciprocity, similar results can be obtained to those in Lemma 2.2 using induction instead of restriction and replacing $\vert H\vert $ by $\vert G\vert .$

Corollary 2.3. Let $\alpha $ be a $2$ -cocycle of G and let P be a Sylow p-subgroup of $G.$

1. (a) Suppose that G contains a nontrivial $\alpha $ -regular element. Then G contains a nontrivial $\alpha $ -regular element of prime power order.

2. (b) Suppose that P contains a nontrivial $\alpha _P$ -regular element. Then P contains a nontrivial $\alpha $ -regular element of $G.$

Proof. Let $c_{\alpha }(G)$ denote the greatest common divisor of the degrees of the elements of $\operatorname {\mathrm {Proj}}(G, \alpha ).$ Then $(c_{\alpha }(G))_p = \min \{\gamma (1): \gamma \in \operatorname {\mathrm {Proj}}(P, \alpha _P)\}$ (see [6, Lemma 1.4.11]), where $n_p$ denotes the pth part of $n.$

For item (a), $\vert \!\operatorname {\mathrm {Proj}}(G, \alpha )\vert> 1$ and so there exists a prime number q such that $(c_{\alpha }(G))_q^2 < \vert Q\vert ,$ where Q is a Sylow q-subgroup of $G.$ Let $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ and let $\gamma \in \operatorname {\mathrm {Proj}}(Q, \alpha _Q)$ with $(\xi (1))_q = \gamma (1) = (c_{\alpha }(G))_q.$ Then Q contains a nontrivial $\alpha $ -regular element of G from Lemma 2.2.

For item (b), $\vert \!\operatorname {\mathrm {Proj}}(P, \alpha _P)\vert> 1$ and the proof is the same as for item (a).

These results give little control over the nontrivial $\alpha $ -regular element of G produced, so in the next section, we will seek conditions under which a given element of G is $\alpha $ -regular.

3 Hall subgroups and regularity

Let H be a subgroup of G and let $\alpha $ be a $2$ -cocycle of $H.$ Then for $g\in G$ , one can define the $2$ -cocycle $\alpha ^g$ of $Z^2(gHg^{-1}, \mathbb {C}^*)$ by $\alpha ^g(x, y)= \alpha (g^{-1}xg, g^{-1}yg)$ for all $x, y\in gHg^{-1}.$ The mapping from $Z^2(H, \mathbb {C}^*)\rightarrow Z^2(gHg^{-1}, \mathbb {C}^*)$ defined by $\alpha \mapsto \alpha ^g$ maps coboundaries of H to those of $gHg^{-1}$ and therefore induces a homomorphism called conjugation by $g, \operatorname {\mathrm {Con}}_H^g: M(H)\rightarrow M(gHg^{-1})$ defined by $[\alpha ]\mapsto [\alpha ^g].$ So, in particular, $h\in H$ is $\alpha $ -regular if and only if $ghg^{-1}$ is $\alpha ^g$ -regular in $gHg^{-1}.$ Next, $[\alpha ]$ is G-stable if for all $g\in G,$

$$ \begin{align*}\operatorname{\mathrm{Res}}_{H, H(g)}([\alpha]) = \operatorname{\mathrm{Res}}_{gHg^{-1}, H(g)}(\operatorname{\mathrm{Con}}_H^g([\alpha])),\end{align*} $$

where $H(g) = H\cap gHg^{-1}.$ The G-stable elements of $M(H)$ form a subgroup $M(H)^G$ of $M(H).$ In the next result, another homomorphism is mentioned, this is corestriction from $M(H)$ into $M(G),$ but as it will not subsequently be used, the reader is referred to [4, page 10] for details.

Next, some notation and definitions. Let $\pi $ denote a set of prime numbers and let n be a positive integer. Then $n_{\pi}$ denotes the $\pi $ th part of n and n is a $\pi $ -number if $n_{\pi } = n.$ An element $x\in G$ and a (sub)group H of G are a $\pi $ -element and $\pi $ -(sub)group if $o(x)$ and $\vert H\vert $ are respectively $\pi $ -numbers. Also let $x_{\pi }$ and $x_{\pi '}$ be the unique elements in $\langle x\rangle $ such that $x = x_{\pi }x_{\pi '}$ with $o(x_{\pi })$ a $\pi $ -number and $o(x_{\pi '})$ a $\pi '$ -number, where $\pi '$ is the complement to $\pi $ in the set of all prime numbers. A Sylow $\pi $ -subgroup S of G is a maximal $\pi $ -subgroup of $G; S$ is a Hall $\pi $ -subgroup of G if, in addition, $\vert G : S\vert $ is relatively prime to $\vert S\vert .$ The first result generalises to Hall subgroups a theorem on the connection between the Schur multiplier of G and those of its Sylow subgroups (see [4, Theorem 2.1.2]).

Proposition 3.1. Suppose H is a Hall $\pi $ -subgroup of $G.$ Then:

1. (a) corestriction from $M(H)$ into $M(G)$ maps $M(H)^G$ isomorphically onto the Hall $\pi $ -subgroup of $M(G)$ ;

2. (b) restriction from $M(G)$ into $M(H)$ induces an injective homomorphism, $\operatorname {\mathrm {res}}$ , from the Hall $\pi $ -subgroup of $M(G)$ into $M(H)$ ;

3. (c) $M(H)^G$ is a direct factor of $M(H)$ and $M(H)^G$ is the image of $\operatorname {\mathrm {res}}$ .

The proof is the same as for the aforementioned theorem with a few very minor modifications, but it relies on the fact that $\vert H\vert $ and $\vert G : H\vert $ are relatively prime. Consequently, Proposition 3.1 does not hold in general for a Sylow $\pi $ -subgroup of G. However, the next result is an immediate consequence of Proposition 3.1(a).

Corollary 3.2. Suppose $H_1$ and $H_2$ are Hall $\pi $ -subgroups of $G.$ Then $M(H_1)^G$ and $M(H_2)^G$ are isomorphic.

Despite this corollary, it is possible for two Hall $\pi $ -subgroups to possess nonisomorphic Schur multipliers as the following example illustrates.

Example 3.3. Using the nomenclature and results from [2], the Mathieu group $M_{23}$ has trivial Schur multiplier and has two conjugacy classes of Hall $\pi $ -subgroups for $\pi = \{2, 3, 5, 7\}.$ Also, these Hall $\pi $ -subgroups are either isomorphic to $L_3(4):2_2$ or $2^4:A_7$ and the first of these groups has a cyclic Schur multiplier of order $4,$ whereas for the second, it is cyclic of order $6$ using Magma [1].

Given the close relationship between the Schur multiplier of a Hall $\pi $ -subgroup H of G and the Hall $\pi $ -subgroup of $M(G)$ , one might expect a corresponding relationship between the $\alpha _H$ -regular elements of H and the $\alpha $ -regular $\pi $ -elements of $G.$

Theorem 3.4. Let $\alpha $ be a $2$ -cocycle of $G.$ Let $x\in G$ and let $\pi $ be the set of prime numbers that divide $o(x).$ For each $p_i\in \pi ,$ let $P_i$ be a Sylow $p_i$ -subgroup of $C = C_G(x)$ and suppose that $\alpha (g, x) = \alpha (x, g)$ for all $g\in P_i.$ Then x is $\alpha $ -regular in $G.$

Proof. Using the assumption that $o(\alpha ) = o([\alpha ]), x$ is $\alpha $ -regular if and only if it is $\alpha _{\pi }$ -regular and $\alpha _{\pi '}$ -regular. Now, x is $\alpha _{\pi '}$ -regular from Lemma 1.3, so we may assume $\alpha = \alpha _{\pi }.$ Now, $\alpha ': C\times \langle x\rangle \rightarrow \mathbb {C^*},$ defined by $\alpha '(g, x^i) = \alpha (g, x^i)/\alpha (x^i, g)$ for all $g\in C$ and all integers $i,$ is a pairing (see [4, Lemma 2.3.8]). The kernel K of the linear character $\alpha '(g, x)$ for all $g\in C$ has order divisible by $\vert P\vert $ for all Sylow p-subgroups P of $C,$ by supposition for $p\in \pi $ and by Lemma 1.3 otherwise. (Alternatively, $\vert K\vert $ is divisible by $\vert P_i\vert $ for all $p_i\in \pi $ by supposition and the group generated by the pairing $\alpha '$ is isomorphic to a subgroup of $C/K\otimes \langle x\rangle .$ This tensor product is trivial since the first group is a $\pi '$ -group whereas the second is a $\pi $ -group.)

Two applications of Theorem 3.4 are recorded in the following corollaries.

Corollary 3.5. Let $\alpha $ be a $2$ -cocycle of G and let $x\in S$ be $\alpha _S$ -regular for S, a Sylow $\pi $ -subgroup of $G.$ For each prime number $p_i\in \pi ,$ let $P_i$ be a Sylow $p_i$ -subgroup of $C_S(x)$ and suppose that $P_i$ is a Sylow $p_i$ -subgroup of $C_G(x).$ Then x is $\alpha $ -regular in $G.$

Proof. The set of prime numbers that divide $o(x)$ is a subset of $\pi $ and so x is $\alpha $ -regular in G from Theorem 3.4.

Corollary 3.6. Let $\alpha $ be a $2$ -cocycle of G and let S be a Sylow $\pi $ -subgroup of $G.$ If S is normal in $G,$ then every $\alpha _S$ -regular element of S is $\alpha $ -regular in $G.$

Proof. Let $x\in S$ be $\alpha _S$ -regular. Then $C_S(x) = C_G(x)\cap S$ is a normal Sylow $\pi $ -subgroup of $C_G(x)$ and Corollary 3.5 applies.