Disjunctive Identities of Finite Groups and Identities of Regular Representations

The problem of describing identities of concrete algebraic structures is traditionally one of the most attractive in variety theory. Very often it is a difficult problem, and many deep works are entirely devoted to finding bases of identities of classical algebraic objects. This paper was initiated by attempts to find bases of identities for certain representations of finite groups. Let F be the free group of countable rank on an alphabet x1, x2, . . . , and let KF be its group algebra over a field K. Recall that an element u = u(x1, . . . , xn) ∈ KF is called an identity of a representation ρ : G → AutKV (usually denoted by ρ = (V,G)) if for arbitrary g1, . . . , gn ∈ G the element u(g1, . . . , gn) ∈ KG annihilates V . It is well known that if ρ is a representation of a finite group, then its identities are finitely based, that is, are in a natural sense generated by some finite set of them [V, VN]. The problem is how to find such a finite basis explicitly. Of course, there is no hope to develop a universal method allowing to write down a basis of identities for an arbitrary representation, so we will be concerned with a much more concrete (but still very general)