Autoisoclinism classes and autocommutativity degrees of finite groups

Abstract

The notion of autoisoclinism was first introduced by Moghaddam et. al., in 2013. In this article we derive more properties of autoisoclinism and define autocommutativity degrees of finite groups. This work also generalizes some results of Lescot in 1995. Among the other results, we determine an upper bound for autocommutativity degree of finite groups.

Autocommuting probability of a finite group

“…Recently, in [12], Rismanchain and Sepehrizadeh have shown that Pr(G, Aut(G)) = Pr(H, Aut(H)) if G and H are autoisoclinic finite groups. We conclude this paper with the following generalization of [12, Lemma 2.5].…”

Probability that an autocommutator element of a finite group equals to a fixed element

“…We also develop and characterize a formula for probability of the pair (H, G), which generalizes the formula for Pr (H, G) given in [2]. Note that if H = G then Paut(H, G) = P aut(G), which coincides with autocommutativty degree Paut(G) of G, if we take = 1, the identity element of G. We note that this case is treated in [7]. It may be recalled that…”

“…In this article G denotes a finite group, H a subgroup of G, and an element of G. In [6], Moghaddam et al have considered the probability Paut(H, G) for an element randomly chosen of H, which is fixed by an automorphism randomly chosen of Aut(G). On the other hand, in [2], Das where [x, α] = x −1 x α , is called autocommutator element of G (see also [7]). Moreover, we extend some of the results obtained in [2].…”

Generalized autocommuting probability of a finite group relative to its subgroups

“…where [x, α] is the autocommutator of x and α defined as x −1 α(x). The ratio Pr(G, Aut(G)) is called autocommuting probability of G. The case when G is non-abelian is considered in [1,3,12]. Few generalizations of Pr(G, Aut(G)) can also be found in [3,4,9,12].…”