Probability of having $n^{th}$-roots and n-centrality of two classes of groups

Subject Areas : History and biography

M. Hashemi ¹ , M. Polkouei ²

1 - Faculty of Mathematical Sciences, University of Guilan, P.O.Box 41335-19141, Rasht, Iran
2 - Faculty of Mathematical Sciences, University of Guilan, P.O.Box 41335-19141, Rasht, Iran

Received: 2015-12-08 Accepted : 2016-04-15 Published : 2016-06-01

Keywords: Nilpotent groups, n-central groups, $n^{th}$-roots,

Abstract :

In this paper, we consider the finitely 2-generated groups $K(s,l)$ and $G_m$ as follows:$$K(s,l)=\langle a,b|ab^s=b^la, ba^s=a^lb\rangle,\\G_m=\langle a,b|a^m=b^m=1, {[a,b]}^a=[a,b], {[a,b]}^b=[a,b]\rangle$$and find the explicit formulas for the probability of having nth-roots for them. Also, weinvestigate integers n for which, these groups are n-central.

References:

[1] C. M. Campbell, P. P. Campel, H. Doostie and E. F. Robertson, Fibonacci length for metacyclian groups. Algebra Colloq. 11 (2004), 215-222.

[2] C. M. Campbell, E. F. Robertson, On a group presentation due to Fox. Canada. Math. Bull. 19 (1967), 247-248.

[3] H. Doostie, M. Hashemi, Fibonacci lengths involving the Wall number K(n). J. Appl. Math. Computing. 20 (2006), 171-180.

[4] A. Sadeghieh, H. Doostie And M. Azadi, Certain numerical results on the Fibonacci length and nth-roots of Hamiltonian groups. International Mathematical Forum. 39 (2009), 1923-1938.

[5] A. Sadeghieh, H. Doostie, The n-th roots of elements in finite groups. Mathematical Sciences. 4 (2008), 347-356.

[6] C. Delizia, A. Tortora and A. Abdollahi, Some special classes of n-abelian groups. International journal of Group Theory. 1 (2012), 19-24.