Groups whose elements are not conjugate to their powers

TL;DR

This paper studies finite groups where no element is conjugate to a different power of itself, proving such groups are solvable and characterizing specific classes with this property for certain prime elements.

Contribution

It introduces the concept of irrational groups, proves their solvability, and characterizes classes where the property applies only to specific prime elements.

Findings

Irrational groups are solvable.

Characterization of classes with prime-specific properties.

Extension of the property to certain p-elements.

Abstract

We call a finite group irrational if none of its elements is conjugate to a distinct power of itself. We prove that those groups are solvable and describe certain classes of these groups, where the above property is only required for $p$-elements, for $p$ from a prescribed set of primes.