On the Exponent of Triple Tensor Product of p-Groups

The non-abelian tensor product of groups which has its origins in algebraic K-theory as well as inhomotopy theory, was introduced by Brown and Loday in 1987. Group theoretical aspects of non-abelian tensor products have been studied extensively. In particular, some studies focused on the relationship between the exponent of a group and exponent of its tensor square. On the other hand, computation of c-fold tensor products is generally a difficult problem. Several authors have given upper bounds for the order of G, when G is a finite p-group. In this paper we determine a bound for the exponent triple tensor product which sharpens a bound of G. Ellis.Let G be a nilpotent group of nilpotency class k≥3 and prime power exponent P^e (where p is a prime and is not equal 3). In this paper, we show that the exponent of triple tensor product of G, that is, (G⨂G)⨂G, divides P^([k/2]-1)e where [k/2] denotes the smallest integer n such that n≥k/2. In this way, the exponent provided by Ellis is improved.