A Note on c-Supplemented Subgroups of Finite Groups

A subgroup H of a finite group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is contained in H _G , where H _G is the largest normal subgroup of G contained in H. In this article, we prove that G is solvable if every subgroup of prime odd order of G is c-supplemented in G. Moreover, we prove that G is solvable if and only if every Sylow subgroup of odd order of G is c-supplemented in G. These results improve and extend the classical results of Hall's articles of (1937) and the recent results of Ballester-Bolinches and Guo's article of (1999), Ballester-Bolinches et al.'s article of (2000), and Asaad and Ramadan's article of (2008).     

On a Class of Hungarian Semigroups and the Factorization Theorem of Khinchin

Let G be a connected reductive Lie group and K be a maximal compact subgroup of G. We prove that the semigroup of all K-biinvariant probability measures on G is a strongly stable Hungarian semigroup. Combining with the result [see Rusza and Szekely⁽⁹⁾], we get that the factorization theorem of Khinchin holds for the aforementioned semigroup. We also prove that certain subsemigroups of K-biinvariant measures on G are Hungarian semigroups when G is a connected Lie group such that Ad G is almost algebraic and K is a maximal compact subgroup of G. We also prove a p-adic analogue of these results.     

Equivariant holomorphic Hermitian principal bundles over a generalized flag manifold

Let G be a simple and simply connected complex linear algebraic group. Fix a maximal compact subgroup K(G) ì G{K(G) \subset G}, and let P be a parabolic subgroup of G. Let H be any connected reductive complex linear algebraic group. We classify the K(G)-equivariant holomorphic Hermitian principal H-bundles over G/P.     

Homological properties of Fourier algebras on homogeneous spaces

Let K be a compact subgroup of a locally compact group G. Completely complemented ideals in A(G/K) are characterised. Biprojectivity and biflatness for the Fourier algebra A(G/K) are studied. A(G/K) is operator biprojective precisely when K is open and if this happens, then G does not contain the free group on two generators as a closed subgroup.     

Nearly m-embedded subgroups and solvability of finite groups

A subgroup A of a finite group G is called {1≤G}-embedded in G if for each two subgroups K≤H of G, where K is a maximal subgroup of H, A either covers the pair (K,H) or avoids it. Moreover, a subgroup H of G is called nearly m-embedded in G if G has a subgroup T and a {1≤G}-embedded subgroup C such that G?=?HT and H∩T≤C≤H. In this paper, we mainly prove that G is solvable if and only if its Sylow 3-subgroups, Sylow 5-subgroups and Sylow 7-subgroups are nearly m-embedded in G.     

On <Emphasis Type="Italic">c</Emphasis>*-normal subgroups in finite groups

A subgroup H of a finite group G is called a c*-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an S-quasinormal embedded subgroup of G. In this paper, the structure of a finite group G with some c*-normal maximal subgroups of Sylow subgroups is characterized and some known related results are generalized.     

Orbits and Invariants Associated with a Pair of Spherical Varieties: Some Examples

Let H and K be spherical subgroups of a reductive complex group G. In many cases, detailed knowledge of the double coset space H\G/K is of fundamental importance in group theory and representation theory. If H or K is parabolic, then H\G/K is finite, and we recall the classification of the double cosets in several important cases. If H=K is a symmetric subgroup of G, then the double coset space K\G/K (and the corresponding invariant theoretic quotient) are no longer finite, but several nice properties hold, including an analogue of the Chevalley restriction theorem. These properties were generalized by Helminck and Schwarz (Duke Math. J. 106(2) (2001), pp. 237–279) to the case where H and K are fixed point groups of commuting involutions. We recall Helminck and Schwarz's main results. We also give examples to show the difficulty in extending these results if we allow H=K to be a reductive spherical (nonsymmetric) subgroup or if we have H symmetric and K spherical reductive.     

On Π-Supplemented Subgroups of a Finite Group

A subgroup H of a finite group G is said to satisfy Π-property in G if for every chief factor L/K of G, |G/K: N_G/K(HK/K ∩ L/K)| is a π(HK/K ∩ L/K)-number. A subgroup H of G is called Π-supplemented in G if there exists a subgroup T of G such that G = HT and H ∩ T ≤ I ≤ H, where I satisfies Π-property in G. In this article, we investigate the structure of a finite group G under the assumption that some primary subgroups of G are Π-supplemented in G. The main result we proved improves a large number of earlier results.     

A Note on Solitary Subgroups of Finite Groups

We say that a subgroup H of a finite group G is solitary (respectively, normal solitary) when it is a subgroup (respectively, normal subgroup) of G such that no other subgroup (respectively, normal subgroup) of G is isomorphic to H. A normal subgroup N of a group G is said to be quotient solitary when no other normal subgroup K of G gives a quotient isomorphic to G/N. We show some new results about lattice properties of these subgroups and their relation with classes of groups and present examples showing a negative answer to some questions about these subgroups.     

On g-s-supplemented subgroups of finite groups

A subgroup H of a group G is said to be g-s-supplemented in G if there exists a subgroup K of G such that HK⊴G and H ∩ K ⩽ H _sG , where H_sG is the largest s-permutable subgroup of G contained in H. By using this new concept, we establish some new criteria for a group G to be soluble.     

Groups with normal restriction property

Let G be a finite group. A subgroup M of G is said to be an NR-subgroup if, whenever K\trianglelefteq M{K\trianglelefteq M}, then K ^G ∩ M = K where K ^G is the normal closure of K in G. Using the Classification of Finite Simple Groups, we prove that if every maximal subgroup of G is an NR-subgroup then G is solvable. This gives a positive answer to a conjecture posed in Berkovich (Houston J. Math. 24 (1998), 631–638).     

Homogenous projective factors for actions of semi-simple Lie groups

We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup Q⊂G, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫ _K kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q. Oblatum 14-X-1997 & 18-XI-1998 / Published online: 20 August 1999     

The Kernel of the Adjoint Representation of a p-Adic Lie Group Need Not Have an Abelian Open Normal Subgroup

Let G be a p-adic Lie group with Lie algebra 𝔤 and Ad: G → Aut(𝔤) be the adjoint representation. It was claimed in the literature that the kernel K?ker(Ad) always has an abelian open normal subgroup. We show by means of a counterexample that this assertion is false. It can even happen that K = G, but G has no abelian subnormal subgroup except for the trivial group. The arguments are based on auxiliary results on subgroups of free products with central amalgamation.     

On c-Normal Subgroups of Finite Groups

A subgroup H is said to be c-normal in a group G if there exists a normal subgroup K of G such that G = HK and H K is contained in H_G, where H_G is the maximal normal subgroup of G. We determine the structures of some groups in which some primary subgroups is c-normal.AMS Mathematics Subject Classification (2000) 20D10 20D20     

The automorphism group of a nonsplit metacyclic p-group

This note considers a finite group G = HK, which is a product of a subgroup H and a normal subgroup K, and determines subgroups of Aut G. The special case when G is a nonsplit metacyclic p-group, where p is odd, is then considered and the structure of its automorphism group Aut G is given. Received: 13 September 2007, Revised: 22 November 2007     

Quasi-Purifiable Subgroups and Minimal Direct Summands

Let G be an arbitrary Abelian group. A subgroup A of G is said to be quasi-purifiable in G if there exists a pure subgroup H of G containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup H is called a “quasi-pure hull” of A in G. We prove that if G is an Abelian group whose maximal torsion subgroup is torsion-complete, then all subgroups A are quasi-purifiable in G and all maximal quasi-pure hulls of A are isomorphic. Every subgroup A of a torsion-complete p-primary group G is contained in a minimal direct summand of G that is a minimal pure torsion-complete subgroup containing A. An Abelian group G is said to be an “ADE decomposable group” if there exist an ADE subgroup K of G and a subgroup T′ of T(G) such that G = K⊕ T′. An Abelian group whose maximal torsion subgroup is torsion-complete is ADE decomposable. Hence direct products of cyclic groups are ADE decomposable groups.     

Normal Restriction in Finite Groups

A subgroup H of a finite group G is called an NR ? subgroup (Normal Restriction) if whenever K ? H, then K ^G ∩ H = K, where K ^G is the normal closure of K in G. In this article, we will prove some sufficient conditions for the solvability of finite groups which possess many NR-subgroups. We also prove a criterion for the existence of a normal p-complement in finite groups.     

On Weakly ℋ-Subgroups of Finite Groups

Let G be a finite group. A subgroup H of G is called an ?-subgroup in G if N _G (H) ∩ H ^x  ≤ H for all x ∈ G. A subgroup H of G is called weakly ?-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an ?-subgroup in G. In this article, we investigate the structure of the finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup of G are weakly ?-subgroups in G. Some recent results are extended and generalized.     

On complemented subgroups of finite groups

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that G = HK and H ⋂ K = 1. In this paper we determine the structure of finite groups with some complemented primary subgroups, and obtain some new results about p-nilpotent groups.     

Finite groups with weakly <Emphasis Type="Italic">S</Emphasis>-quasinormal subgroups

We introduce a new subgroup embedding property in a finite group called weakly S-quasinormality. We say a subgroup H of a finite group G is weakly S-quasinormal in G if there exists a normal subgroup K such that HK ⊴ G and H ∩ K is S-quasinormally embedded in G. We use the new concept to investigate the properties of some finite groups. Some previously known results are generalized.