n-th relative nilpotency degree and relative n-isoclinism classes

Abstract

P. Hall introduced the notion of isoclinism between two groups more than 60 years ago. Successively, many authors have extended such a notion in different contexts. The present paper deals with the notion of relative n-isoclinism, given by N. S. Hekster in 1986, and with the notion of n-th relative nilpotency degree, recently introduced in literature.

Subgroup S-commutativity degrees of finite groups

“…The techniques of proof are straightforward applications of (2.3) and the details are omitted. However, it is good to note that Corollary 2.2 shows the stability with respect to forming direct products of spd(G): this fact was proved in [3,4,6,7,10,12,13,17] in different contexts. Another basic property is to relate spd(G) to quotients and subgroups of G.…”

“…The following notion has analogies with [6, Definitions 2.1,3.1,4.1] and [12, Equation 1.1] and will be treated as in [3,4,6,7,10,11,12,13,17].…”

“…There are several contributions on d(G) in [3,4,6,7,10,11,12,13]. The main strategy of investigation is to begin with the case of equality at 1 and then describe the situation, when we leave this extremal case.…”

“…In Section 2 we will describe a notion of probability on L(G), beginning from groups in which the subgroups in sn(G) permutes with those in M(G). The generality of the methods (we follow [3,4,6,7,10,11,12,13,17]) may be translated in terms of arbitrary sublattices, satisfying a prescribed restriction. Section 3 shows some consequences on the size of |L(G)|.…”

Considerations on the subgroup commutativity degree and related notions

“…In the present paper we deal only with finite group, even if there is a recent interest to the subject in the context of infinite groups [1,11,10,17,25]. The commutativity degree of a group G, given by (1.1)…”