The Superimprimitive Subgroups Of The Alternating Group Of Degree 8

Abstract

A transitive permutation group G is called superimprimitive if it is imprimitive with non-trivial block systems of imprimitivity of lengths all the non-trivial divisors of the degree of G; The superimprimitivity concepts was studied first by Omar (2), and later by the authors (3). In the present paper we shall give some results concerning this concept m part 1, and determine in part 2, all superimprimitive subgroups of the alternating group of degree 8. We proved the following: Lemma (I): Let G be a transitive group acting on a set X and m is the number of non-trivial divisors of |X). If G contains m intransitive normal proper subgroups each having different orbit lengths then G is superimprimitive. The orbits of each subgroup form a block system of imprimitivity. Lemma (2): (a) Let G be a superimprimitive group. For every non-trivial divisor d of the degree of G and for x£X, there exists a group Z which lies property between G, and G such that the set {x^ has length d. (b) I^G,CZ,cG holds, where Zi, i=l,...,m are proper subgroups of G and the sets {x '} have different lengths, then G is superimprimitive. Then we show that, among the 48337 subgroups of Ag, which split into 137 classes there are 4425 superimprimitive subgroups which split into 18 classes, their generators are given.