# On the nilpotency of the solvable radical of a finite group isospectral to a simple group

## TL;DR

This paper proves that, with one exception, the solvable radical of a finite group that shares the same spectrum as a simple group is necessarily nilpotent, advancing understanding of group spectra and structure.

## Contribution

It establishes that, except for a specific case, the solvable radical of a finite group isospectral to a simple group must be nilpotent, clarifying the relationship between spectra and group structure.

## Key findings

- The solvable radical is nilpotent in most isospectral cases.
- There is a unique exception where the radical is not nilpotent.
- The results deepen the link between element orders and group structure.

## Abstract

We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that with the only specific exception the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.