Zeros of characters and orders of elements in finite groups

TL;DR

This paper studies Wilde's conjecture on character values and element orders in finite groups, reducing it to prime-specific cases and verifying it for many nearly simple groups.

Contribution

It reduces Wilde's conjecture to prime-by-prime checks on nearly simple groups and verifies the conjecture in many key cases for primes greater than 5.

Findings

Strong form of Wilde's conjecture holds for many nearly simple groups.

Verification achieved for most classes of nearly simple groups for primes p>5.

Remaining cases require further information on character extensions.

Abstract

We investigate a beautiful conjecture of T. Wilde on character values and element orders of finite groups. We reduce it to a statement on nearly simple groups that can be checked ``prime by prime". For these groups, we show that a strong form of Wilde's conjecture holds in many important cases, and for primes $p>5$ we are able to show the required statement for most classes of nearly simple groups. The few remaining cases, however, seem to require information on extensions of irreducible characters that are not available at the present time.