Search results for "Subgroup"

On Join Properties of Hall π-Subgroups of Finite π-Soluble Groups

1998

All groups considered in the sequel are finite. K. Doerk and T. Hawkes, in Section I.4 of their recent comprehensive w x volume on finite soluble groups 1 , include background material and a proof of the following result: Let S be a Hall system of a soluble group G and let U and V be subgroups into which S reduces. Then S reduces into U l V, and if , in addition, U permutes with V, then S reduces into UV. It is clear that the second part of the above result holds equally well with a single Hall subgroup in place of a Hall system; in other words, if a Hall p-subgroup of G contains Hall p-subgroups of U and V and U permutes with V, then it also contains a Hall p-subgroup of UV.

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.

Nilpotent-like fitting formations of finite soluble groups

2000

[EN] In this paper the subnormal subgroup closed saturated formations of finite soluble groups containing nilpotent groups are fully characterised by means of extensions of well-known properties enjoyed by the formation of all nilpotent groups.

On the lattice of J-subnormal subgroups

1992

Extension of a Schur theorem to groups with a central factor with a bounded section rank

2013

Abstract A well-known result reported by Schur states that the derived subgroup of a group is finite provided its central factor is finite. Here we show that if the p-section rank of the central factor of a locally generalized radical group is bounded, then so is the p-section rank of its derived subgroup. We also give an explicit expression for this bound.

A Question of R. Maier Concerning Formations

1996

The formation f is said to be saturated if the group G belongs to f Ž . whenever the Frattini factor group GrF G is in f. Let P be the set of all prime numbers. A formation function is a Ž . function f defined on P such that f p is a, possibly empty, formation. A formation f is said to be a local formation if there exists a formation Ž function f such that f s G g G : if HrK is a chief factor of G and p < < Ž . Ž .. divides HrK , then GrC HrK g f p ; G is the class of all finite G groups. If f is a local formation defined by a formation function f , then Ž . we denote f s LF f and f is a local definition of f. Among all possible local definitions of a local formation f there exists exactly …

On prefrattini residuals

1998

All groups considered in the sequel are finite. Let (ℭ and denote the formations of groups which consist of collections of groups that respectively either split over each normal subgroup (nC-groups) or for which the groups do not possess nontrivial Frattini chief factors [8]. The purpose of this article is to develop and expand a concept that arises naturally with the residuals for these formations, namely each G-chief factor is non-complemented (Frattini). With respect to a solid set X of maximal subgroups, these properties are generalized respectively to so-called X-parafrattini (X-profrattini) normal subgroups for which each type is closed relative to products. The relationships among th…

Blocks and Normal Subgroups

1998

Squaring a conjugacy class and cosets of normal subgroups

2015

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.