Showing papers by "Adolfo Ballester-Bolinches published in 1995"

The Jordan-Hölder theorem and prefrattini subgroups of finite groups

Abstract: by A. BALLESTER-BOLINCHES and L. M. EZQUERRO(Received 26 January, 1994)Introduction. All groups considered are finite. In recent years a number ofgeneralizations of the classic Jordan-Holder Theorem have been obtained (see [7],Theorem A.9.13): in a finite group G a one-to-one correspondence as in the Jordan-Holder Theorem can be defined preserving not only G-isomorphic chief factors but eventheir property of being Frattini or non-Frattini chief factors. In [2] and [13] a newdirection of generalization is presented: the above correspondence can be defined in sucha way that the corresponding non-Frattini chief factors have the same complement(supplement).In this paper we present a necessary and sufficient condition, named (JH), for a set Xof monolithic maximal subgroups of a group G to verify this wider version of theJordan-Holder Theorem: the Jordan-Holder bijection can be defined in such a way thatthe corresponding chief factors are G-isomorphic, have a supplement in X at the sametime and, in this case, it can be chosen to be a common supplement in X. It is remarkablethat (JH) is not only a sufficient condition but is indeed necessary.On the other hand, W. Gaschiitz introduced in [10] a conjugacy class of subgroups ofa finite soluble group called prefrattini subgroups. They form a characteristic conjugacyclass of subgroups which cover the Frattini chief factors and avoid the complementedones. These results were generalized by Hawkes [11] and Fbrster [8] in the soluble case.In [1], the authors introduced the concept of a system of maximal subgroups. Thesesystems can be used to select maximal subgroups in order to define prefrattini subgroupssimilar to those of Gaschiitz, Hawkes and Forster in the general non-soluble case. Theyenjoy most of the properties of the soluble case except the Cover and Avoidance Propertyand conjugacy. In fact, conjugacy characterizes solubility.Another generalization of Gaschiitz's work in the soluble universe is due to Kurzweil[12]. He introduced the //-prefrattini subgroups of a soluble group G, where H is asubgroup of G. The //-prefrattini subgroups are conjugate in G and they have the Coverand Avoidance Property; i 1f H the =y coincide with the classical prefrattini subgroups ofGaschiitz and if § is a saturated formation and H is an g-normalizer of G the//-prefrattini subgroups are those described by Hawkes.Tomkinson in [14] extended the results of Gaschiitz and Hawkes to a class It oflocally finite groups with a satisfactory Sylow structure. The intersection of II with theclass of all finite groups is just the class of all finite soluble groups.Our aim here is to present all the results of the finite universe in a more unifiedsetting. Our approach is based on Tomkinson's ideas. Using a property, denoted by (*),which is slightly stronger than (JH), we can define X-prefrattini subgroups of a finite

Correction to "On the Deskins Index Complex of a Maximal Subgroup of a Finite Group"

Abstract: This note is to correct a mistake in [1]. So, the notation, definitions, and references are those of that paper. If M is maximal subgroup of a finite group G and H/K is a chief factor of G supplemented by M we cannot say in general that H E I (M). For example, if G is the dihedral group of order 30, M a maximal subgroup of G isomorphic to Sym(3), and H = Soc(G), we have that H f I(M) since k(H) = H. (Compare this with the last paragraph of page 236 in [5].) This motivates that Proposition 1 in our paper does not hold. Changing that by Proposition 1* below, we prove below that the five theorems of the paper remain true.