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We investigate the structure of certain types of subalgebras of Galilei algebras and the relationship between the conjugacies of these subalgebras under different groups of automorphisms.

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

Communicated by Efim Zelmanov Keywords: Faithful projective representations Faithful representations Modular projective representations Modular representations Projective representations Twisted group algebrasLet K be a field of finite characteristic p and G a finite group with a normal Sylow p-subgroup. We give necessary and sufficient conditions for G to have a faithful irreducible projective representation over K . In the case when G is an abelian p-group and K is not a perfect field we find also all dimensions of faithful irreducible projective representations of G over K and show that the group G has infinitely many isomorphism classes of faithful irreducible projective representations of each possible dimension d = 1.