The Probability of an Automorphism Fixing a Subgroup Element of a Finite Group

Abstract

In 1975, Sherman introduced the probability of an automorphism of a finite group, which fixes an arbitrary element of the group. In this paper we introduce a new probability concept, namely the probability of an automorphism of a given finite group such that it fixes a subgroup element of the group. Among other results, we construct some upper and lower bounds for both probabilities.

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

“…The following theorem is a generalization of the work of Moghaddam et al which in turn is similar to Theorem 2.1 of [6].…”

“…In this section we first give the following definition which generalizes definitions of Das et al and Moghaddam et al, see [2,6]. Definition 2.1.…”

“…As an immediate consequence, we have the following generalization of the well-known formula Paut(G) = k |G| , where k is the number of the disjoint orbits of G under the group automorphism Aut(G) (see also [6,8]). Proof.…”

“…The concept of commutativity degree started by Gustafson in 1975, which is probability that two elements of a finite group G commute. 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]).…”

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. Let H and K be two subgroups of a finite group G such that H ⊆ K. Motivated by the works in [2,6], we define Prg(H, Aut(K)) = |{(x, α) ∈ H × Aut(K) : [x, α] = g}| |H|| Aut(K)| (1.1)…”