We obtain upper bounds on the composition length of a finite permutation group in terms of the degree and the number of orbits, and analogous bounds for primitive, quasiprimitive and semiprimitive groups. Similarly, we obtain upper bounds on the composition length of a finite completely reducible linear group in terms of some of its parameters. In almost all cases we show that the bounds are sharp, and describe the extremal examples.Problem 1.7. Which other infinite families of permutation groups have composition lengths bounded above by a logarithmic function of the degree?Our final main result gives examples of two such families. A permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive. It is semiprimitive if each of its normal subgroups is either semiregular or transitive. (A permutation group is semiregular if the only element fixing a point is the identity.) Theorem 1.8. Let G be a permutation group of degree n.(a) If G is quasiprimitive but not primitive, thenwhere c na = 10 3 log 2 5 = 1.43 · · · as in Theorem 1.6. (b) If G is semiprimitive but not quasiprimitive, then c(G) 8 3 log 2 n − 3.We give infinitely many examples to show that the bound in Theorem 1.8(b) is best possible (see Example 6.2). For a semiprimitive group G, a normal subgroup of G which is minimal † The statement in [24, Theorem 2.10] refers to a paper 'in preparation' (reference [Py5] in [24]).
In this paper, we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. These groups have recently been investigated in terms of their abstract structure, in a similar way to the O'Nan–Scott Theorem for primitive groups. Our goal here is to explore aspects of such groups which may be useful in place of precise structural information. We give bounds on the order, base size, minimal degree, fixed point ratio, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we study the structure of a finite semiprimitive group that induces the alternating or symmetric group on the set of orbits of an intransitive minimal normal subgroup.
In this paper we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. We give bounds on the order, base size, minimal degree, fixity, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we classify finite semiprimitive groups that induce the alternating or symmetric group on the set of orbits of an intransitive normal subgroup.
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