Proportions of elements with given 2-part order in finite classical groups of odd characteristic

TL;DR

This paper establishes lower bounds on the proportion of elements with specific 2-part orders in finite classical groups of odd characteristic, with implications for group recognition algorithms.

Contribution

It provides new lower bounds on element proportions with certain 2-part orders in finite classical groups, including rank-independent bounds and applications to recognition algorithms.

Findings

Proportion of odd order elements in symplectic and orthogonal groups is at least C/ell^{3/4}.

Positive constant lower bounds for elements with specific 2-part orders, independent of rank.

Results inform analysis of Yalçınkaya's Black Box recognition algorithm.

Abstract

For an element $g$ in a group $X$, we say that $g$ has 2-part order $2^{a}$ if $2^{a}$ is the largest power of 2 dividing the order of $g$. We prove lower bounds on the proportion of elements in finite classical groups in odd characteristic that have certain 2-part orders. In particular, we show that the proportion of odd order elements in the symplectic and orthogonal groups is at least $C/\ell^{3/4}$, where $\ell$ is the Lie rank, and $C$ is an explicit constant. We also prove positive constant lower bounds for the proportion of elements of certain 2-part orders independent of the Lie rank. Furthermore, we describe how these results can be used to analyze part of Yal\c{c}inkaya's Black Box recognition algorithm for finite classical groups in odd characteristic.