Relative n-isoclinism classes and relative n-th nilpotency degree of finite groups

TL;DR

This paper explores the concept of relative n-isoclinism in finite groups, linking it to degrees of nilpotency and commutativity, and improves classical results in group theory classification.

Contribution

It introduces the notion of relative n-isoclinism and connects it to nilpotency degrees, enhancing understanding of finite group classifications.

Findings

Established connections between relative n-isoclinism and nilpotency degrees

Improved classical results in group classification literature

Provided new insights into prime power order groups

Abstract

The purpose of the present paper is to consider the notion of isoclinism between two finite groups and its generalization to n-isoclinism, introduced by J. C. Bioch in 1976. A weaker form of n-isoclinism, called relative n-isoclinism, will be discussed. This will allow us to improve some classical results in literature. We will point out the connections between a relative n-isoclinism and the notions of commutativity degree, n-th nilpotency degree and relative n-th nilpotency degree, which arouse interest in the classification of groups of prime power order in the last years.