Roots of identity in finite groups of Lie type

Abstract

Given an integer $M\ge 2$, we deploy the generating function techniques to compute the number of $M$-th roots of identity in some of the well-known finite groups of Lie type, more precisely for finite general linear groups, symplectic groups, orthogonal groups of all types and unitary groups over finite fields of odd characteristics.

Revised version; Suggestions are welcome;