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 
$G.$


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 C. The set of all such 2-cocycles of G form a group Z 2 (G, C * ) under multiplication.Let δ : G → C * be any function with δ(1) = 1.Then t(δ)(x, y) = δ(x)δ(y)/δ(xy) for all x, y ∈ G is a 2-cocycle of G, which is called a coboundary.Two 2-cocycles α and β are cohomologous if there exists a coboundary t(δ) such that β = t(δ)α.This defines an equivalence relation on Z 2 (G, C * ) and the cohomology classes [α] form a finite abelian group, called the Schur multiplier M(G).DEFINITION 1.2.Let α be a 2-cocycle of G.
These two functions arise naturally in the twisted group algebra (C(G)) α in which xȳ = α(x, y)xy for all x, y ∈ G (see [4, page 66]).Here, ḡxḡ where Lin(C G (x)) is the group of linear characters of C G (x).The kernel of α x is the absolute centraliser C α (x) of x with respect to α and First, every element of G is α-regular if [α] is trivial.Second, setting y = 1 and z = 1 in Definition 1.1 yields α(x, 1) = 1 and similarly α(1, x) = 1 for all x ∈ G, and hence 1 is always α-regular.Third, if x ∈ G is α-regular, then it is α k -regular for any integer k.Finally, if x ∈ G is α-regular, then so too is any conjugate of x (see [4,Lemma 2.6.1]),so that one may refer to the α-regular conjugacy classes of G.
Let N be a normal subgroup of G. Then G acts on Proj(N, α N ) by for ζ ∈ Proj(N, α N ), g ∈ G and all x ∈ N. Clifford's theorem for projective characters applies to this action (see [5,Theorem 2

.2.1]).
A new concept of strict α d -regularity, which refines the notion of α d -regularity, will be defined and investigated in Section 2 for d a divisor of the order of [α].This concept will be used in Section 3 to give an alternative proof that the number of orbits of G under its action on Proj(N, α N ), for N a normal subgroup of G, is equal to the number of α-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 [α].The result is well known when α is trivial (see [3,Corollary 6.33]); the method employed will be to apply this to the orbits of an α-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.

Strictly α d -regular elements
Let o( ) denote the order of an element in a group.Then for ) and α is a class-function cocycle, that is, the elements of Proj(G, α) are class functions (see [5,Corollary 4.1.6]).To avoid repetition throughout the rest of this paper, it will be assumed that α has these two properties with n = o(α).A consequence of the second property is that x ∈ G is α-regular if and only if f α (g, x) = 1 for all g ∈ G (see [5, page 33]).The first property allows us to make the following definition in terms of α d rather than for the more clumsy β ∈ PROOF.For condition (a), if x is not α d/p -regular, then it is not α t -regular for all positive integers t with t | d/p.For condition (b), observe that x is α d -regular if and only if 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 θ(C H (r(x))) = C α (x) for x ∈ G and θ(C H (r(x)A)) = C G (x).Thus, working in H, we see that x is strictly α 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 α-covering group of G. Then x ∈ G is strictly α d -regular if and only if either:
(a) r(x) z m are the conjugates of r(x) in r(x)A, where z = A and dm = n; or (b) {r(x)z i : i = 1, . . ., m} is a maximal set of conjugacy class representatives of H in r(x)A.
is a homomorphism with kernel C H (r(x)), since λ(k r(x) ) = α 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 ).Now x is strictly α d -regular if and only if Im(k r(x) ) = z m , that is, if and only if the cosets of Im(k r(x) ) in A are z i z m for i = 1, . . ., m.

Counting orbits of projective characters
Let N be a subgroup of G. Let H be an α-covering group of G and, using the notation of Section 2, let M be the subgroup of H containing A such that θ(M) = N.Finally, for any integer k, let Irr(M|λ k ) = {χ ∈ Irr(M) : χ A = χ(1)λ k }, where Irr(M) is the set of irreducible characters of M. Then the mapping from Proj(N, [4, pages 134-135] or [5,Corollary 4.1.3]).Now suppose N is normal in G, then it is easy to check that ζ g = χ r(g) for all g ∈ G and hence the orbit length of ζ under the action of G equals that of χ under the action of H.By definition, for each x ∈ G, there exists a unique d | n such that x is strictly α d -regular.Thus, the conjugacy classes of H are partitioned according to |C H (r(x)A)/C H (r(x)| for r(x)a, where x ∈ G and a ∈ A. However, if x is a strictly α 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 α d -regular conjugacy classes of G contained in N is s|d (n/s)t s ; in particular, 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(α d ) = o(α k ).Let σ be a field automorphism of C that extends τ, as described in Section 2, so that for all x ∈ N.
Lemma 3.1 sets up a one-to-one correspondence between the orbits of G under its action on Proj(N, α d N ) and those under its action on Proj(N, α k N ) in which orbit lengths are preserved.We next just restate Lemma 3.1 for an α-covering group H of G.The total number of orbits of H under its action on Irr(M) is t(M), so the total number of orbits of H under its actions on Irr(M|λ c ), for the φ(n) values of c with 1 ≤ c ≤ n that are relatively prime to n, is Hence, the number of orbits of H under its action on Irr(M|λ) (and the number of orbits of G under its action on Proj(N, α N )) is t 1 , as required.

x 1 for
each prime p ∈ π(d) from condition (a).The latter is true if and only if d p | o(α x ) for each prime p ∈ π(d), that is, if and only if d | o(α x ).An equivalent way of stating Lemma 2.2(b) is that x ∈ G is strictly α d -regular if and only if |C G (x)/C α (x)| = d.Now by definition for each x ∈ G, there exists a unique d | n such that x is strictly α d -regular.Thus, the conjugacy classes of G are partitioned into strictly α d -regular conjugacy classes.So for d | n and N a normal subgroup of G, let t d be the number of strictly α d -regular conjugacy classes of G contained in N. Thus, the number of α d -regular conjugacy classes of G contained in N is s|d t s ; in particular, d|n t d = t(N), where t(N) is the number of conjugacy classes of G contained in N. The choice of 2-cocycle α allows the construction of an α-covering group H of G with the following three properties (see [4, Section 4.1]): (a) H has a cyclic subgroup A ≤ Z(H) ∩ H of order n; (b) there exists a conjugacy-preserving transversal (see below) {r(g) : g ∈ G} of A in H such that θ : H → G defined by θ(r(g)a) = g for all g ∈ G and all a ∈ A is a homomorphism with kernel A; (c) there exists a faithful character λ ∈ Lin(A) such that α(x, y) = λ(A(x, y)) for all x, y ∈ G, where r(x)r(y) = A(x, y)r(xy).

COROLLARY 3 . 2 .
Suppose that o(λ d ) = o(λ k ) in λ = Lin(A).Let σ be as in Lemma 3.1, so that σ(λ d ) = λ k .Then χ h = χ if and only if σ(χ) h = σ(χ ) for h ∈ H and χ ∈ Irr(M|λ d ).Let φ denote Euler's totient function.We use the well-known result from number theory that d|n φ(d) = d|n φ(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 Proj(N, α N ) is equal to the number of α-regular conjugacy classes of G contained in N. PROOF.Proceeding by induction, we count the number of α 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 Irr(N).So assume by induction that the number of orbits of G under its action on Proj(N, α d N ) is equal to the number of α d -regular conjugacy classes of G contained in N for each d | n with d 1.Let H be an α-covering group of G and let M denote the subgroup of H containing A such that θ(M) = N.Now for d | n and d 1, G has s|d t s orbits under its action on Proj(N, α d N ).Thus, H has the same number of orbits under its action on Irr(M|λ d ).Now o(λ k ) = o(λ d ) for φ(n/d) values of k with 1 ≤ k ≤ n.Thus, using Corollary 3.2, the total number of orbits of H under its actions on Irr(M|λ c ), for the n − φ(n) values of c with 1 ≤ c ≤ n that are not relatively prime to n, is

Then from [ 1 ,
Lemma 1.4], we see that Proj(N,β N ) = {δ N ζ : ζ ∈ Proj(N, α N )} and, for g ∈ G, ζ g = ζ if and only if (δ N ζ) g = δ N ζ for ζ ∈ Proj(N, α N ).In particular, this establishes a one-to-one correspondence between and only if it is α k -regular.Thus, d | n in Definition 2.1.Let π(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 ∈ G is strictly α d -regular if and only if either: