On the Exponent of Triple Tensor Product of p-Groups

Subject Areas : Statistics

H. Hadizadeh ^{1 *} , S. H. Jafari ²

1 - Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran
2 - Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran

Received: 2018-01-28 Accepted : 2019-03-02 Published : 2020-01-21

Keywords: رده پوچ‌توانی, حاصل‌ضرب تانسوری ناآبلی, نمای گروه, گروه پوچ‌توان,

Abstract :

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.

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