We show that for all integers m 2, and all integers k 2, the orthogonal groups O ± (2m, F 2 k ) act on abstract regular polytopes of rank 2m, and the symplectic groups Sp(2m, F 2 k ) act on abstract regular polytopes of rank 2m + 1.

Abstract. We construct generic extensions and polynomials for certain linear algebraic groups over finite fields. In particular, we show the existence of generic polynomials for the cyclic group C 2 m in characteristic p for all odd primes p.

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