Let V be a 3-dimensional vector space over a finite field. We show that any irreducible subgroup of GL(V ) that arises as the automorphism group of an abstract regular polytope preserves a nondegenerate symmetric bilinear form on V . In particular, the only classical groups on V that arise as automorphisms of such polytopes are the orthogonal groups.
In this paper, the author presents a new algorithm to recognise, constructively, when a given black-box group is a homomorphic image of the unitary group SU(d, q) for known d and q. The algorithm runs in polynomial time, assuming the existence of oracles for handling SL(2, q) subgroups, and for computing discrete logarithms in cyclic groups of order q ± 1.
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