On Capable groups of order p^2q

TL;DR

This paper characterizes capable groups of order p^2q for distinct primes p and q, extending previous classifications, and computes the number of element centralizers in groups with a central factor of order p^3.

Contribution

It extends the classification of capable groups of order p^2q and calculates centralizer counts for groups with central factor order p^3.

Findings

Characterization of capable groups of order p^2q.

Extension of previous theorems on group capability.

Calculation of element centralizers in groups with central factor order p^3.

Abstract

A group is said to be capable if it is the central factor of some group. In this paper, among other results we have characterized capable groups of order $p^2q$, for any distinct primes $p, q$, which extends Theorem 1.2 of S. Rashid, N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, {\em On the non abelian tensor square and capability of groups of order $p^2q$}, Arch. Math., \textbf{97} (2011), 299--306. We have also computed the number of distinct element centralizers of a group (finite or infinite) with central factor of order $p^3$, which extends Proposition 2.10 of S. M. Jafarian Amiri, H. Madadi and H. Rostami, {\em On $F$-groups with the central factor of order $p^4$}, Math. Slovaca, \textbf{67} (5) (2017), 1147--1154.