Summary: 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 $p$th 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.

05A10, 05A15

q-analogs, q-binomial coefficients, vector spaces over finite fields, recursions, symmetric group, Eulerian generating functions