A Generalization of Commutativity Degree of Finite Groups

Abstract

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

“…Moreover, we extend some of the results obtained in [2]. 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].…”

“…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.…”

“…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].…”

“…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]. 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].…”

On commuting probability of finite rings

“…In 1995, Lescot [9] showed that Pr(G 1 ) = Pr(G 2 ) if G 1 and G 2 are isoclinic finite groups. Various generalizations of this result can be found in [3,11,12,14]. Recently, Buckley et.…”