On Zappa's question in the case of alternating groups

TL;DR

This paper investigates Zappa's question regarding cosets of Sylow p-subgroups in finite groups, proving that the smallest such groups cannot be alternating simple groups for any prime p, thus narrowing the search for minimal examples.

Contribution

It establishes that the minimal groups satisfying Zappa's condition are not alternating simple groups, extending previous results and refining the understanding of the problem.

Findings

Smallest groups satisfying Zappa's condition are not alternating simple groups.

The result applies to all primes p, ruling out alternating groups as minimal examples.

Provides new constraints on the structure of groups related to Zappa's question.

Abstract

In 1962, Guido Zappa asked whether a non-trivial coset of a Sylow $p$-subgroup of a finite group could contain only elements whose orders are powers of $p$. Marston Conder gives a positive answer to this question in the case of $p=5$. It is known that the smallest group satisfying the conditions of this problem must be a non-abelian simple group. In this paper, we prove that the smallest group of the Zappa problem could not be an alternating simple group for any prime $p$.