On the commuting probability and supersolvability of finite groups

Abstract

For a finite group G, let d(G) denote the probability that a randomly chosen pair of elements of G commute. We prove that if d(G) > 1/s for some integer s > 1 and G splits over an abelian normal nontrivial subgroup N , then G has a nontrivial conjugacy class inside N of size at most s − 1. We also extend two results of Barry, MacHale, and Ní Shé on the commuting probability in connection

Commuting Probability of Compact Groups

“…The set A(G) := {(x, y) ∈ G × G | xy = yx} is a measurable set (actually a closed set) of G × G and its measure in G × G will be denoted by cp(G). In the case G is finite, cp(G) has been extensively studied for example see [1] and [3] and references therein. It is famous (see e.g.…”

Commutativity Degree of Crossed Modules

“…In this section, we introduce two dimensional version of commutativity degree and give fundamental idea can be found in [17]. Another investigation of that is given in [18]. Before this, we recall commutativity degree of finite groups and give some bounds for this commutativity degree.…”

“…Before this, we recall commutativity degree of finite groups and give some bounds for this commutativity degree. For details, see [10,14,[16][17][18].…”

“…gap> all48 := AllXMods( [4,8]);; gap> Length(all48); 336 gap> iso_all48 := AllXModsUpToIsomorphism(all48);; gap> Length(iso_all48); 59 gap> f_1 := IsoclinicXModFamily(iso_all48 [1],iso_all48); [ 1,3,4,6,9,10,12,27,29,31,33,35,38,40,43,55,57,59 ] gap> f_2 := IsoclinicXModFamily(iso_all48 [2],iso_all48); [ 2,5,7,8,11,13,14,28,30,32,34,36,37,39,41,42,56,58 ] gap> f_3 := IsoclinicXModFamily(iso_all48 [15],iso_all48); [ 15,18,22,25,44,47,51,53 ] gap> f_4 := IsoclinicXModFamily(iso_all48 [16],iso_all48); [ 16,17,19,…”

Finite Groups with Five Relative Commutativity Degrees

“…The next remarkable and significant result is due to Rusin [17] in 1979 who classified all finite groups with commutativity degrees greater than 11/32. Since then the commutativity degree of finite groups is studied actively and we may refer the interested reader to [2,4,8,10,12,13,14,15] for some major contributions to the field. While the commutativity degree can be applied to distinguish between groups but it is not so strong to reveal internal structure of groups in general.…”