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We prove a structure theorem for the isometry group of an Her- mitian map b: V V ! W , where V and W are vector spaces over a nite eld of odd order. We also present a Las Vegas polynomial-time algorithm to nd generators for this isometry group, and to determine its structure. The algorithm can be adapted to construct the intersection of the members in a set of classical subgroups of GL(V ), yielding the rst polynomial-time solu- tion of this old problem. Our approach develops new computational tools for algebras with involution, which in turn have applications to other algorith- mic problems of interest. An implementation of our algorithm in the Magma system demonstrates its practicability.

Computing isometry groups of Hermitian maps