The number of elements whose square is the identity in the symmetric group S_n is recursive in n. This recursion may be proved combinatorially, and there is also a nice exponential generating function for this sequence. We study q-analogs of this phenomenon. We begin with sums involving q-binomial coefficients which come up naturally when counting elements in finite classical groups which square to the identity, and we obtain a recursive-like identity for the number of such elements in finite special orthogonal groups. We then study a q-analog for the number of elements in the symmetric group whose pth power is the identity, for some fixed prime p. We find an Eulerian generating function for these numbers, and we prove the q-analog of the recursion for these numbers by giving a combinatorial interpretation in terms of vector spaces over finite fields.