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

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 $2^{365}\cdot 3^{105}\cdot 7^{104}\approx 7.3\cdot 10^{247}$ of the same order type, where only one is solvable. In this note we produce a much smaller example of order $2^{13}\cdot 3^4\cdot 7^3=227598336$.