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

Abstract

For an element g in a group X, we say that g has 2-part order 2b if 2b 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/n3/4, where n 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 Yalçinkayaʼs Black Box recognition algorithm for finite classical groups in odd characteristic.