Constructive Recognition of Normalizers of Small Extra-special Matrix Groups

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    Abstract

    The members of the class C-6 in Aschbacher's structure theorem for subgroups of the general linear group CL(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 r(3) 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.
    Original languageEnglish
    Pages (from-to)367-394
    JournalInternational Journal of Algebra and Computation
    Volume15
    Issue number2
    DOIs
    Publication statusPublished - 2005

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