Search results for "combinatoric"

A non-linear Bishop–Phelps–BollobÁs type theorem

2018

Self-normalizing Sylow subgroups

2003

Using the classification of finite simple groups we prove the following statement: Let p > 3 p>3 be a prime, Q Q a group of automorphisms of p p -power order of a finite group G G , and P P a Q Q -invariant Sylow p p -subgroup of G G . If C N G ( P ) / P ( Q ) \mathbf {C}_{\mathbf {N}_G(P)/P}(Q) is trivial, then G G is solvable. An equivalent formulation is that if G G has a self-normalizing Sylow p p -subgroup with p > 3 p >3 a prime, then G G is solvable. We also investigate the possibilities when p = 3 p=3 .

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.

A Note on the δ-length of Maximal Subgroups in Finite Soluble Groups

1994

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 …