On the non abelian tensor product of a group and its central automorphisms

Subject Areas : Statistics

Monireh Seifi ¹ , S. Hadi Jafari ²

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

Received: 2020-02-02 Accepted : 2020-04-19 Published : 2021-11-22

Keywords: " گروه‌های پوچتوان", "گروه خودریختی‌ها", "حاصلضرب تانسوری نا‌آبلی", "خودریختی‌های مرکزی",

Abstract :

The non-abelian tensor product of groups has it's origin in K-algebraic theory and topology and was first introduced by R. Brown and J.L. Loday in 1987 .One of the first topics which was studied on G⊗G is that whether the properties of G⊗G inherited from G or not? For instance, Bacon in 1994 determined an upper bound for the number of minimal generators of G⊗G in terms of the number of minimal generators of G.Let G be a group and Autz (G) be the group of It's central automorphisms, which is a normal subgroup of Aut(G). Our goal is to obtain an estimate for the number of minimal generators of G⊗Autz(G). For this, we first identify it's minimal generators. Then, when both G and Autz(G) are nilpotent groups of class two, we give an upper bound for d(G⊗Autz(G)) in terms of d(G) and d(Autz(G)), where d(X) is the minimal number of generators of X.

References:

[1] R. Brown and J.-L. Loady, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311-335.

 

[2] H. Hadizadeh and S. H. Jafari, On the Exponent of Triple Tensor Product of p-Groups, Journal of New Researches in Mathematics. 5(22) (2020), 77-84.

 

[3] H. Golmakani, A. Jafarzadeh and P. Niroomand, The Tensor Degree of a Pair of Finite Groups, Journal of New Researches in Mathematics. 4(13) (2018), 41-46.

 

[4] M. Ghaffarzadeh, On minimal degrees of faithful quasi-permutation representations of nilpotent groups, Journal of New Researches in Mathematics. 3(12) (2018), 87-98.

 

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

 

[6] M. R. Bacon, On the nonabelian tensor square of a nilpotent group of class two, Glasgow Math. J. 36 (1994), 291-296.

 

[7] M. R. R. Moghaddam and M. J. Sadeghifard, Nonabelian tensor analogues of 2-auto Engel groups, Bull. Korean Math. Soc. 52(4) (2015), 1097-1105.

 

[8] The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.7.6, 2014. Available at: http://www. gap-system.org/.

 

[9] M. L. Curran and D. J. McCaughan, Central automorphisms that are almost inner, Comm. Algebra 29(5) (2001), 2081-2087.