Let q be an odd prime, and let \(G_n(q)\) denote one of the finite classical groups—namely, the general linear, unitary, symplectic, or orthogonal group—of rank n over the field \(\textrm{GF}(q)\) with q elements. Given an integer \(M \ge 2\) , an element \(g \in G_n(q)\) is said to be a root of identity if \(g^M = 1\) . In this article, we derive generating functions for the probability that a randomly chosen element of \(G_n(q)\) is an M-th root of identity. Our results depend on the specific types of irreducible factors that appear in the factorization of the polynomial \( x^M-1\) over \(\textrm{GF}(q)\) .
