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2 results

  • A NOTE ON PERMUTATION GROUPS AND THEIR REGULAR SUBGROUPS

  • Part of
  • MING-YAO XU
  • Journal:
    Journal of the Australian Mathematical Society / Volume 85 / Issue 2 / October 2008
    Published online by Cambridge University Press:
    01 October 2008, pp. 283-287
    Print publication:
    October 2008
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  • In this note we first prove that, for a positive integer n>1 with n≠p or p2 where p is a prime, there exists a transitive group of degree n without regular subgroups. Then we look at 2-closed transitive groups without regular subgroups, and pose two questions and a problem for further study.

  • Seminormal and subnormal subgroup lattices for transitive permutation groups

  • Part of
  • Cheryl E. Praeger
  • Journal:
    Journal of the Australian Mathematical Society / Volume 80 / Issue 1 / February 2006
    Published online by Cambridge University Press:
    09 April 2009, pp. 45-64
    Print publication:
    February 2006
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  • Various lattices of subgroups of a finite transitive permutation group G can be used to define a set of ‘basic’ permutation groups associated with G that are analogues of composition factors for abstract finite groups. In particular G can be embedded in an iterated wreath product of a chain of its associated basic permutation groups. The basic permutation groups corresponding to the lattice L of all subgroups of G containing a given point stabiliser are a set of primitive permutation groups. We introduce two new subgroup lattices contained in L, called the seminormal subgroup lattice and the subnormal subgroup lattice. For these lattices the basic permutation groups are quasiprimitive and innately transitive groups, respectively.