Forum Problem Lists Definitions FAQ

21.43 (2026)

Open

Conjecture: Suppose that for a fixed positive integer $k$ at least half of the elements of a finite group $G$ have order $k$. Then $G$ is solvable.

Contributor: M. Deaconescu

Progress

*This is not always true. In a direct product of $N$ copies of $A_5$ there are $60^N$ elements, while the number of elements of order exactly 30 is $60^N - 45^N - 40^N - 36^N + 25^N + 21^N + 16^N - 1$. As $N \to \infty$, the ratio of elements of order 30 converges to 1. (L. Tae Young, Letter of 9 January 2026 ; see also (R. McCulloch, L. Tae Young, Preprint, 2026, https://arxiv.org/abs/2602.19340)).

Comments

All comments are the responsibility of the user. Comments appearing on this page are not verified for correctness. Please keep posts mathematical and on topic. If you want to submit a proof (or a partial proof), please use the dedicated proof submission form rather than posting it in the comments.
Order by newest first or oldest first.

No comments yet. Be the first to comment.

Proof claims

Proof claims are the responsibility of the submitter. Appearance here does not mean the claim has been checked for mathematical correctness. Moderators only screen for spam, abuse, and obviously low-effort submissions.

No proof claims yet.