Recognizing $A_7$ by its set of element orders

TL;DR

This paper proves that the alternating group A7 can be uniquely identified among all groups solely by the set of element orders it contains, highlighting a spectral characterization.

Contribution

It establishes that A7 is uniquely determined by its spectrum within the class of all groups, a novel spectral recognition result.

Findings

A7 is uniquely characterized by its element order set

Spectral recognition can distinguish A7 from other groups

The result advances understanding of group spectra and their identifying power

Abstract

Let $G$ be a periodic group, the spectrum $\omega(G) \subseteq \mathbb{N}$ of $G$ is the set of orders of elements in $G$. In this paper we prove that the alternating group $A_{7}$ is uniquely defined by its spectrum in the class of all groups.