FINDING INVOLUTIONS WITH SMALL SUPPORT

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 $\lceil n^{{\it\varepsilon}}\rceil$ is at least a constant multiple of ${\it\varepsilon}$. 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 $\lceil n^{{\it\varepsilon}}\rceil$ for a linear or unitary group, or $2\lceil \lfloor n/2\rfloor ^{{\it\varepsilon}}\rceil$ for a symplectic or orthogonal group.