MAXIMAL SUBSETS OF PAIRWISE NONCOMMUTING ELEMENTS OF SOMEp-GROUPS OF MAXIMAL CLASS

Abstract

Let G be a group. A subset X of G is a set of pairwise noncommuting elements if x y = yx for any two distinct elements x and y in X . If |X | ≥ |Y | for any other set of pairwise noncommuting elements Y in G, then X is said to be a maximal subset of pairwise noncommuting elements. In this paper we determine the cardinality of a maximal subset of pairwise noncommuting elements for some p-groups of maximal class. Specifically, we determine this cardinality for all 2-groups and 3-groups of maximal class.2010 Mathematics subject classification: primary 20D15; secondary 20D60.

Groups with a few nonabelian centralizers

“…Let G be a finite group and let X be a subset of pairwise noncommuting elements of G such that |X| ≥ |Y | for any other set of pairwise noncommuting elements Y in G. Then the subset X is said to have the maximum size and this size is denoted by ω(G). Various attempts have been made to find ω(G) for some groups G (see for example [3][4][5][6] and [12]).…”

On Maximal Subsets of Pairwise Noncommuting Elements in Finite -Groups

“…Pyber [12] proved that there is some constant c such that the index of the centre Z(G) in G satisfies |G : Z(G)| ≤ c ω (G) . The value of ω(G) has been computed for various groups G (see for example [1,3,[5][6][7]).…”

“…Fouladi and Orfi [7] proved that ω(G) = |G |(p + 1)/p, where G is a finite nonabelian metacyclic p-group with p > 2. Further, Fouladi and Orfi [6] determined ω(G) for some p-groups G of maximal class.…”

Maximal subset of pairwise non-commuting elements of finite minimal non-Abelian groups

“…Pyber [13] has shown that there is some constant c such that |G : Z(G)| ≤ c ω(G) . Moreover various attempts have been made to find ω(G) for some groups G, see for example [1], [2], [3], [7], [8] and [9].…”