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Publication - Dr Tim Burness

    Generation and random generation: From simple groups to maximal subgroups

    Citation

    Burness, TC, Liebeck, MW & Shalev, A, 2013, ‘Generation and random generation: From simple groups to maximal subgroups’. Advances in Mathematics, vol 248., pp. 59-95

    Abstract

    Let G be a finite group and let d(G) be the minimal number of generators
    for G. It is well known that d(G) = 2 for all (non-abelian) finite simple groups. We
    prove that d(H) 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.
    We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.
    We then apply our results to the study of permutation groups. In particular we
    show that if G is a finite primitive permutation group with point stabilizer H, then
    d(H) is at most d(G) + 4.

    Full details in the University publications repository