Maximal Subsets of Pairwise Non-Commuting Elements in Finite $A_2$-Groups

Authors
Reza Orfi
Shirin Fouladi
Kharazmi University
Abstract
Let $G$ be a finite group. A subset $X subseteq G$ is called a pairwise non-commuting set if $xy neq yx$ for all distinct elements $x, y in X$. If $|X| geq |Y|$ for every other pairwise non-commuting set $Y$ in $G$, then $X$ is called a maximal pairwise non-commuting set. A $p$-group $G$ is defined as an $A_2$-group if it contains a nonabelian subgroup of index $p$ such that all subgroups of index $p^2$ are abelian. In this paper, we determine the exact maximum size of such sets in finite $A_2$-groups.
Keywords
20D15
20D60
A. Azad,S.FauladiandR.Orfi,Maximalsubsetsofpairwisenon-commutingelementsofsomefinite p-groups,Bull.Iran.Math.Soc.,39,1,187-192(2013)ꞏ

Y.Berkovich,Groupsofprimepowerorder.Vol.1,WalterdeGruyter,Berlin,2008

Y.BerkovichandZ.Janko, Groupsofprimepowerorder,Vol.2,WalterdeGruyter.Berlin,2008.

A. Y. M. Chin, On non-commuting sets in an extraspecial p-group, J. Group Theory 8(2) (2005), 189-194.

S.FouladiandR.Orfi,Maximalsubsetsofpairwisenon-commutingelementsofsomefinitep-groups ofmaximalclass,Bull.Aust.Math.Soc84(2011),447-451.

S. Fouladi and R. Orfi, Maximal subsets of pairwise non-commuting elements of some finite metacyclicp-groupsBull.Aust.Math.Soc,87(2013),18-23.

R.Orfi,MaximalSubsetsofpairwisenon-commutingelementsofp-groupsoforderlessthanp6,Int. J.GroupTheory3(2014),65-72.

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