On $p$-index extremal groups

TL;DR

This paper explores the relationship between the structure of finite groups and the properties of the p-indices of their elements, presenting new results and open problems in the study of p-index extremal groups.

Contribution

It introduces new findings and open questions regarding the properties of p-indices in finite groups, advancing understanding of their structural implications.

Findings

New results on p-index properties in finite groups

Identification of open problems in p-index extremal groups

Insights into the connection between group structure and element indices

Abstract

The question on connection between the structure of a finite group $G$ and the properties of the indices of elements of $G$ has been a popular research topic for many years. The $p$-index $|x^G|_p$ of an element $x$ of a group $G$ is the $p$-part of its index $|x^G|=|G:C_G(x)|$. The presented short note describes some new results and open problems in this direction, united by the concept of the $p$-index of a group element.