A note about solvable and non-solvable finite groups of the same order type

TL;DR

This paper presents a smaller example of two finite groups with the same order type where only one is solvable, addressing a question about the relationship between order type and solvability.

Contribution

It provides a significantly smaller counterexample to Thompson's question, demonstrating that solvability does not necessarily follow from having the same order type.

Findings

Constructed a smaller group pair with same order type and different solvability

Confirmed that order type alone does not determine solvability

Reduced the size of known counterexamples significantly

Abstract

Two finite groups are said to have the same order type if for each positive integer $n$ both groups have the same number of elements of order $n$. In 1987 John G. Thompson asked if in this case the solvability of one group implies the solvability of the other group. In 2024 Pawel Piwek gave a negative example. He constructed two groups of order $2^{365}\cdot3^{105}\cdot7^{104}\approx7.3\cdot10^{247}$ of the same order type, where only one is solvable. In this note we produce a much smaller example of order $2^{13}\cdot3^4\cdot7^3=227598336$.