Finiteness of a Non-abelian Tensor Product of Groups Nick Inassaridze

Some suucient conditions for niteness of a generalized non-abelian tensor product of groups are established extending Ellis' result for compatible actions. The non-abelian tensor product of groups was introduced by Brown and Loday 2,3] following works of A.Lue 4] and R.K.Dennis 7]. It was deened for any groups A and B which act on themselves by conjugation (x y = xyx ?1) and each of which acts on the other such that the following compatibility conditions hold: (a b) a 0 = a (b (a ?1 a 0)); (b a) b 0 = b (a (b ?1 b 0)) (1) for all a; a 0 2 A and b; b 0 2B. These compatibility conditions are very important in the subsequent theory of the tensor product. In particular they play a crucial role in Ellis's proof 5] that the tensor product of nite groups is nite. The deenition of the non-abelian tensor product was generalized in 6] so as to deal with the case when the compatibility conditions (1) do not hold. The present paper is concerned solely with this generalized tensor product; we obtain conditions which are suucient for its niteness. Henceforth, let A and B be groups with a chosen action of A on B and a chosen action of B on A. We assume that A and B act on themselves by conjugation. These actions yield, in an obvious way, actions of the free product A B on A and on B. We recall the following deenition from 6]. 1. Definition. The non-abelian tensor product A B is the group generated by the symbols a b; (a 2 A; b 2 B) subject to the relations aa 0 b = (a a 0 a b)(a b) a bb 0 = (a b)(b a b b 0) (a b)(a 0 b 0) = (a;b] a 0 a;b] b 0)(a b) (a 0 b 0)(a b) = (a b)(b;a] a 0 b;a] b 0)