Finiteness of a Non Abelian Tensor Product of Groups

Some su cient 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 following works of A Lue and R K Dennis It was de ned for any groups A and B which act on themselves by conjugation y xyx and each of which acts on the other such that the following compatibility conditions hold b a a b a a a b b a b b for all a a A and b b B 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 that the tensor product of nite groups is nite The de nition of the non abelian tensor product was generalized in so as to deal with the case when the compatibility conditions do not hold The present paper is concerned solely with this generalized tensor product we obtain conditions which are su cient 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 de nition from Definition The non abelian tensor product A B is the group generated by the symbols a b a A b B subject to the relations aa b a a b a b a bb a b a b b a b a b a b a a b b a b a b a b a b b a a b a b The research described in this publication was made possible in part by Grant IFS MXH and by Grant INTAS The author would also like to thank the referee and editor for helpful comments Received by the editors January and in revised form August Published on August Mathematics Subject Classi cation G