  Letting tau denote the inverse transpose automorphism of GL(n,q), a formula
is obtained for the number of g in GL(n,q) so that gg^{tau} 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, gg^{tau} 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.
