CONSTRUCTIVE RECOGNITION OF NORMALIZERS OF SMALL EXTRA-SPECIAL MATRIX GROUPS.

The members of the class C₆ in Aschbacher's structure theorem for subgroups of the general linear group GL(d,q) are the normalizers of certain absolutely irreducible, symplectic-type r-groups, where r is a prime, d a power of r and q ≡ 1 (mod r). For a prime r > 2 and d = r, we present a constructive one-sided Monte Carlo algorithm to recognize whether or not a given subgroup G of GL(r,q) contains a normal extra-special r-group of order r³ and exponent r. In the former case the algorithm returns a homomorphism from G into SL(2,r) with kernel being the extra-special r-group times the center. [ABSTRACT FROM AUTHOR]