On modular projective representations of finite nilpotent groups

Abstract

Abstract. Our aim is to determine necessary and sufficient conditions for a finite nilpotent group to have a faithful irreducible projective representation over a field of characteristic p ≥ 0.1. Introduction. Frucht [4] proved that a finite abelian group G admits a faithful irreducible projective representation over an algebraically closed field K of characteristic not dividing the order of the group G if and only if G is of symmetric type, i.e. it decomposes into a direct product of two isomorphic groups. Yamazaki [11] showed that sufficiency of Frucht's theorem holds for an arbitrary field L containing the primitive (exp G)th root of 1. Moreover, he established that the group G is of symmetric type if and only if for some factor system λ ∈ Z 2 (G, L * ) the twisted group algebra L λ G is a central simple algebra over the field L. Frucht's theorem is supplemented by Zhmud's result [14]: the minimal number of irreducible components of a faithful projective K-representation of the group G equals 1 if G is of symmetric type and equals 2 otherwise. A generalization of Frucht's and Zhmud's results to an arbitrary field with a restriction on the characteristic was given in [1]- [2]. A study of metabelian groups admitting a faithful irreducible projective representation over the field of complex numbers was performed by Ng [8]- [9]. Some general results on faithful projective representations of finite groups over a field with a restriction on the characteristic are obtained in [8]-[9] and [11]-[12]. Let us note that the above results are partially presented in Karpilovsky's monograph [6].In this paper we look for necessary and sufficient conditions for finite nilpotent groups to have faithful irreducible projective representations over a field of any characteristic. In Section 2 we prove a number of propositions about semisimple twisted group algebras of finite groups. Since for an algebraically closed field there exists a close connection between the existence of

On faithful irreducible projective representations of finite groups over a field of characteristic p

“…Barannyk [2,3] generalized Frucht's and Zhmud's results to an arbitrary field whose characteristic does not divide the order of A. Barannyk and Sobolewska [6] obtained some necessary and sufficient conditions for a nilpotent group to have a faithful irreducible projective representation over an arbitrary field. Some general results on faithful irreducible projective representations of arbitrary finite groups are obtained in [12,13] and [15,17].…”

“…In this paper we continue a study of faithful modular projective representations of finite groups as begun in [4] and [6]. We established that a finite group G with a normal Sylow p-subgroup G p admits a faithful irreducible projective representation over a field K of characteristic p if and only if K is not a perfect field, G = G p × B, G p is abelian and B has a faithful irreducible projective representation over K (Theorem 2.4).…”

Modular projective representations of direct products of finite groups

“…In paper [8] B. Fein generalized his results to the case of arbitrary finite dimensional L-algebras. Outer tensor products of irreducible modules over twisted group algebras were investigated as well in [2].…”

On indecomposable projective representations of finite groups over fields of characteristic p>0

“…Suppose 2 } is a basis of the left vector space V over the field K. It follows that any element of V is of the form 2 . Since for arbitrary x, y ∈ A there exists z ∈ A such that xy − yx = (u c − u e )z, we obtain x(y + (rad A) 2 ) = (y + (rad A) 2 )x for any x, y ∈ A.…”

“…48] the restriction of every cocycle λ ∈ Z 2 (G, F * ) to C p × C p is a coboundary. Therefore, statements (1) and (2) follow from the properties of natural homomorphisms of twisted group algebras ( [26, pp. 87-93]).…”