On two problems about order sequences of finite groups

TL;DR

This paper addresses open problems about the order sequences of finite groups, demonstrating limitations in their ability to characterize group properties like solvability and supersolvability.

Contribution

It proves that order sequences do not always compare in a way that reflects group solvability or supersolvability, resolving two open problems in the field.

Findings

Order sequences do not always dominate those of solvable groups.

Supersolvability cannot be inferred solely from order sequences.

Counterexamples show limitations of order sequences in group classification.

Abstract

The order sequence of a finite group $G$ is a non-decreasing finite sequence formed of the element orders of $G$. Several properties of order sequences were studied by P. J. Cameron and H. K. Dey in a recent paper that concludes with a list of open problems. In this paper we solve two of these problems by showing the following facts: 1) if there is a non-supersolvable/non-solvable group of order $n$, it is not always true that its order sequence is properly dominated by the order sequence of any supersolvable/solvable group of order $n$; 2) the supersolvability of a finite group cannot be described by its order sequence.