### A Condition on the Probability that a Group Element Fixes a Set

#### Abstract

Let G be a finite non abelian group. The probability that a group element fixes a set is the probability that studies the number of orbits under group actions to the number of commuting elements of size two. The probability was computed and studied by several researchers. In this paper, we construct a new condition for the elements of the fixed set in which the set under the study is the set that consists of two commute elements a and b, where lcm(|a|, |b|)≥ 2^{i}, 1 < i ≤ 2n. This condition is applied for the dihedral groups, where the conjugate action is used to determine the probability.

#### Full Text:

PDF©2021 Science Asia