Finding involutions with small support

TL;DR

This paper investigates the frequency of permutations and classical group elements with specific involution properties, showing that such elements with small support are proportionally common in large groups.

Contribution

It establishes lower bounds on the proportion of permutations and classical group elements with involutions of small support, extending previous results to broader group classes.

Findings

Proportion of permutations with small support involutions is at least proportional to epsilon.

Similar bounds are obtained for elements in classical groups with certain eigenvalue properties.

Results apply to permutations in symmetric and alternating groups, and to classical groups of odd characteristic.

Abstract

We show that the proportion of permutations $g$ in $S_n$ or $A_n$ such that $g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality at most $\lceil n^\varepsilon \rceil$ is at least a constant multiple of $\varepsilon$. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension $n$ in odd characteristic that have even order and power up to an involution with $(-1)$-eigenspace of dimension at most $\lceil n^\varepsilon \rceil$ for a linear or unitary group, or $2\lceil \lfloor n/2 \rfloor^\varepsilon \rceil$ for a symplectic or orthogonal group.