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.

يقال لزمرة التبديلات الأنتقالية أنها متعددة غير الأولية اذا كانت غير الأولية ولها نظام من البلوكات الفصلية لكل قاسم فعلي من قواسم درجة الزمرة . ولقد قدم هذا البحث نظريتين لشروط مكافئة للتعريف . للتعرف عل الزمر متعددة غير الأولية . ثم وضحنا أنه بين كل الزمر الجزئية لزمرة التبديلات الزوجية من درجة ثمانية ، م هـ ، وعددهم 48337 زمرة جزئية مقسمين الى 137 فصل تكافؤ . يوجد25 4 4 زمرة جزثية متعددة غيرالأولية مقسمين الى 18 فصل تكافؤ.