Pos Groups Revisited

TL;DR

This paper investigates properties of POS-groups, constructs new nonabelian examples, and proves that the alternating groups $A_n$ for $n ge 3$ are not POS-groups, advancing understanding of these algebraic structures.

Contribution

It introduces new nonabelian POS-groups and establishes that alternating groups $A_n$ are not POS-groups for $n ge 3$, expanding the classification of POS-groups.

Findings

Constructed new families of nonabelian POS-groups

Proved $A_n$ is not a POS-group for $n ge 3

Analyzed properties of arbitrary POS-groups

Abstract

A finite group $G$ is said to be a POS-group if for each $ x $ in $G$ the cardinality of the set $\{y \in G | o(y) =o(x)\}$ is a divisor of the order of $G$. In this paper we study some of the properties of arbitrary POS-groups, and construct a couple of new families of nonabelian POS-groups. We also prove that the alternating group $A_n$, $n \ge 3$, is not a POS-group.