On the Exponent of Triple Tensor Product of p-Groups

Subject Areas : Statistics

H. Hadizadeh ¹ (Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran)
S. H. Jafari 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.

References:

[1] B. Ahmadi, H. Doostie, On the 2-generator p-groups with non-cyclic commutator subgroup, Math. 4: 73-78 (2014).

 

[2] R. Brown, J. L. Loday, Van Kampen theorems for diagrams of spaces, Topology. 26: 311-335 (1987).

 

 

[3] R. Brown, D. L. Johnson, E. F. Robertson, Some computations of nonabelian tensor products of groups, J. Algebra. 111: 177-202 (1987).

 

[4] J. Burns, G. Ellis, On the nilpotent multipliers of a group, Math. z. 226: 405-428 (1997).

 

[5] G. Ellis, On the tensor square of a prime power group, Arch. Math. 66: 467-469 (1996).

 

[6] G. Ellis, A. Mcdermott, Tensor products of prime-power groups, J. Pure. Appl. Algebra. 132: 119-128 (1998).

 

[7] G. Ellis, On the relation between upper central quotients and lower central series of a group, Trans. Amer. Math. Soc. 353: 4219-4234 (2001).

 

[8] S. H. Jafari, A bound on the order of non-abelian tensor square of a prime-power group, Comm. Algebra. 40: 528-530 (2012).

 

[9] S. H. Jafari, F. Saeedi, E. Khamseh, Characterization of finite groups by their non-abelian tensor square, Comm. Algebra. 41: 1954-1963 (2013).

 

[10] R. James, The groups of order  (  an odd prime), Math. Comp. 34: 613-637 (1980).

 

[11] E. Khamseh, M. R. R. Moghaddam, F. Saeedi, Characterization of finite -groups by their schur multiplier, J. Algebra. Appl. 12: 1250035_1-1250035_9 (2013).

 

[12] P. Moravec, The exponents of nonabelian tensor products of groups, J. Algebra. 212: 1840-1848 (2008).