On the orders of the non-Frattini elements of a finite group

TL;DR

This paper investigates the relationship between elements of specific orders in finite groups and their containment in the Frattini subgroup, establishing conditions for the existence of non-Frattini elements with composite orders.

Contribution

It proves that if a finite group has an element of order equal to the product of distinct primes, then a non-Frattini element with order divisible by that product also exists.

Findings

Existence of non-Frattini elements with composite orders under certain conditions

Connection between element orders and Frattini subgroup structure

Extension of known results on element orders in finite groups

Abstract

Let $G$ be a finite group and let $p_1,\dots,p_n$ be distinct primes. If $G$ contains an element of order $p_1\cdots p_n,$ then there is an element in $G$ which is not contained in the Frattini subgroup of $G$ and whose order is divisible by $p_1\cdots p_n.$