Search results for "Maximal subgroup"
showing 10 items of 20 documents
Maximal subgroups and PST-groups
2013
A subgroup H of a group G is said r to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maxmial subgroups, Arch. Math. (Basel), 2011, 96(1), 19-25)] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions o…
On the abnormal structure of finite groups
2014
We study finite groups in which every maximal subgroup is supersoluble or normal. Our results answer some questions arising from papers of Asaad and Rose.
On groups with abelian Sylow 2-subgroups
1970
Finite groups with abelian Sylow 2-subgroups have been classified by Walter [8]. In this note I want to describe an alternate proof of some partial result of Walter's work, namely the theorem stated below. It represents the first major reduction step in that classification. The approach used here is to some extent derived from [1]. ! Besides the groups L 2 (q)= PSL(2, q) another class of simple groups enters our discussion: We say that a simple group G with abelian Sz-subgroups is of type JR (Janko-Ree) if, for any involution t in G, CG (t) is a maximal subgroup of G isomorphic to ( t ) | E where PSL(2, q)~ E ~_ PFL(2, q) with odd q > 5. In fact, E = L 2 (q), as proved by Walter 1-7] ; and …
Nilpotent length and system permutability
2022
Abstract If C is a class of groups, a C -injector of a finite group G is a subgroup V of G with the property that V ∩ K is a C -maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschutz and B. Hartley states the existence and conjugacy of F -injectors in finite soluble groups for Fitting classes F . We shall show that for groups of nilpotent length at most 4, F -injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.
On the Deskins index complex of a maximal subgroup of a finite group
1999
AbstractLet M be a maximal subgroup of a finite group G. A subgroup C of G is said to be a completion of M in G if C is not contained in M while every proper subgroup of C which is normal in G is contained in M. The set, I(M), of all completions of M is called the index complex of M in G. Set P(M) = {C ϵ I(M) ¦ C} is maximal in I(M) and G = CM. The purpose of this note is to prove: A finite group G is solvable if and only if, for each maximal subgroup M of G, P(M) contains element C with CK(C) nilpotent.
On the normal index of maximal subgroups in finite groups
1990
AbstractFor a maximal subgroup M of a finite group G, the normal index of M is the order of a chief factor H/K where H is minimal in the set of normal supplements of M in G. We use the primitive permutation representations of a finite group G and the normal index of its maximal subgroups to obtain results about the influence of the set of maximal subgroups in the structure of G.
OnF-Subnormal Subgroups andF-Residuals of Finite Soluble Groups
1996
All groups that we consider are finite and soluble. Recall that a formation is a class of groups which is closed under homomorphic images and subdirect products. Hence, if F is a formation and G is a group which is a direct product of the subgroups A and B, then G is in F if and only if A and B lie in F. More generally, Doerk and w x Hawkes 4, IV, 1.18 proved that if G is a group such that G s A = B, then G s A = B , where G is the F-residual of G, that is, the smallest normal subgroup of G with quotient in F. The main purpose of this paper is the development of this result by means of the concept of F-subnormal subgroup. Suppose that F is a saturated formation. A maximal subgroup M of a Ž …
Sufficient conditions for supersolubility of finite groups
1998
Abstract In this paper sufficient conditions for the supersolubility of finite groups are given under the assumption that the maximal subgroups of Sylow subgroups of the group and the maximal subgroups of Sylow subgroups of the Fitting subgroup are well-situated in the group. That will improve earlier results of Srinivasan [7], Asaad et al. [1] and Ballester-Bolinches [2].
ℏ-Normalizers and local definitions of saturated formations of finite groups
1989
We define, in each finite groupG, h-normalizers associated with a Schunck class ℏ of the formEΦ f with f a formation. We use these normalizers in order to give some sufficient conditions for a saturated formation of finite groups to have a maximal local definition.
C-Supplemented subgroups of finite groups
2000
A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup K of G such that HKa G and H\ K is contained in CoreGOHU .W e follow Hall's ideas to characterize the structure of the finite groups in which every subgroup is c-supplemented. Properties of c-supplemented subgroups are also applied to determine the structure of some finite groups.