Deaconescu's solubility conjecture for finite groups with many elements of the same order

Deaconescu’s conjecture asserts that a finite group is soluble if at least half its elements have one fixed order. The 2026 preprint claims this unrestricted statement is false and identifies both surviving cases and broad families of counterexamples.

Known results

• Wall proved the assertion for involutions ($k=2$).
• Liebeck and MacHale classified finite groups with at least half their elements of order $2$.
• Mann and Berkovich obtained further structural and classification results for $k=2$.

2026 arXiv preprint, version 2

The revised preprint claims the conjecture holds for $k=2$, $k=3$, $k=4$, and every prime power $k=p^a$ with $p>3$, but fails for every $k>4$ divisible by $2$ or $3$, as well as some other $k$. It further claims non-soluble examples with at least $2/15$ of elements of order $k$, plus stronger quantitative bounds. These claims have no independent verification or published referee assessment in the retrieved sources.

Current status (as of September 2026): The unrestricted conjecture is claimed false, and the revised preprint claims complete classifications in several cases, but those claims remain unverified in the retrieved record.