Conjugacy class properties of the extension of GL(n,q) generated by the inverse transpose involution

Letting τ denote the inverse transpose automorphism of GL(n,q), a formula is obtained for the number of g in GL(n,q) so that ggτ is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, ggτ behaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL(n,q) and of symmetric function theory, including a new identity for Hall–Littlewood polynomials. We obtain information about random elements of finite symplectic groups in even characteristic, and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of GL(n,q) generated by the inverse transpose automorphism. We give a second approach to these results using the theory of bilinear forms over a field. The results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups, and we give a few examples in this direction.