on the nse of some particular groups

Subject Areas : Statistics

Bahareh Asadian ¹ , Neda Ahanjideh ²

1 - Dep. Pure Math., Shahrekord Univ., Shahrekord, Iran
2 - Dep. Pure math, sharekord univ. Shahrekord, Iran

Received: 2019-04-10 Accepted : 2021-03-06 Published : 2021-11-22

Keywords: گروه های پوچ توان, مجموعه $nse$, گروه های ۲-فروبنیوس, گروه های فروبنیوس,

Abstract :

For a finite group $ G $, let $ { \rm n s e } ( G ) $ be the set of the number of the elements of the same order in$ G $. In this paper, we first study the set $ n s e $ of a Frobenius group , the set $ { \rm n s e } $ of a $ 2 $- Frobenius group and the set $ { \rm n s e } $ of a nilpotent group. Then, we show that for the finite non-solvable Frobenius group $ G $ with the certain structure and an arbitrary group $ L $ , if $ {\rm n s e } ( G ) = { \rm n s e } ( L ) $, then $ G ≅ L $. Also, a new criterion is presented to recognize nilpotent groups by their $ {\rm n s e }$.

References:

[1] N. Ahanjideh, B. Asadian, NSE characterization of some Alternating groups, Journal of Algebra and Its Applications 14(2): 1550012 (2015).

 

[2] Y. Bugeaud, Z. Cao, M. Mignotte, On simple K₄ groups, Journal of Algebra 241: 658-668 (2001).

 

[3] G. Y. Chen, On the structure of Frobenius groups and 2-Frobenius groups, Journal of Southwest China Normal University 20(5): 485-487 (1995).

 

[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, (1985).

 

[5] M. Herzog, On finite simple groups of order divisible by three primes only, Journal of Algebra 10(3): 383-388 (1968).

 

[6] B. Huppert, Character theory of finite groups, Walter de Gruyter, Berlin, (1998).

 

[7] A. Khalili Asboei, S. S. Salehi Amiri, A. Iranmanesh, A new of PSL₂(q) for some q, Ukrainian Mathematical Journal 67(9): 1297-1305 (2016).

 

[8] E. I. Khukhro, V. D. Mazurov, Unsolved Problems in Group Theory: The Kourovka Notebook, Sobolev Institute of Mathematics, Novosibirsk, 17th edition, (2010).

 

[9] A. S. Kondratev, Prime graph components of finite simple groups, Mathematics of the USSR-Sbornik 67(1): 235-247 (1990).

 

[10] R. Shen, On same order type groups, SAṺ. Fen Bilimleri Dergisi 15(2): 156-158 (2011).

 

[11] R. Shen, C. Shao, Q. Jiang,W. Shi, V. Mazurov, A new characterization of A₅, Monatshefte für Mathematik 160(3): 337-341 (2010).

 

[12] A. V. Vasilev, On connection between the structure of a finite group and the properties of its prime graph 46(36): 396-404 (2005).

 

[13] J. S. Williams, Prime graph components of finite groups, Journal of Algebra 69(2): 487-513 (1981).