A note on groups with self-normalizing cyclic subgroups

TL;DR

This paper characterizes groups where all non-trivial cyclic subgroups are self-normalizing and a specific order implication holds, showing such groups are either p-groups or simple, contributing to group theory classification.

Contribution

It proves a new classification result for groups with self-normalizing cyclic subgroups under a particular order implication condition.

Findings

Groups are either p-groups or simple under the given conditions.

The order implication constrains the group's structure.

Self-normalizing cyclic subgroups influence group classification.

Abstract

In this article, we prove that if all non-trivial cyclic subgroups of a group $G$ are self normalizing and $G$ satisfies the implication $$ \ o(x)\neq o(y)\Rightarrow o(xy)\neq o(x), o(y), $$ for all non-trivial elements $x$ and $y$, then $G$ is a $p$-group or simple.