Finite groups with many elements of the same order

Abstract

We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by presenting some counterexamples. By restricting to a fixed $k$, the conjecture may or may not hold depending on $k$. We prove that if $k$ is a power of a prime other than $2$ or $3$, or if $k=2,3$ or $4$, then the conjecture holds, while it fails for many other choices of $k$ including all multiples of $2$ and $3$ which are larger than $5$. For $k=4$ we also find the sharp upper bound of the ratio of elements of order $4$ in non-soluble groups. We also prove that for all $k>1$, it is always possible to find a finite non-soluble group where at least $2/15$ of the elements have order $k$.