Let
be a class of groups. A subgroup H of a group G is called
-s-supplemented in G, if there exists a subgroup K of G such that G = HK and K/K ∩ HG belongs to
where HG is the maximal normal subgroup of G which is contained in H. The main purpose of this paper is to study some subgroups of Fitting subgroup and generalized Fitting subgroup
-s-supplemented and some new criterions of p-nilpotency of finite groups are obtained.
*This research is supported by the grant of NSFC and TianYuan Fund of Mathematics of China (Grant #10626047). 相似文献
Let G be a finite group. A subgroup H of G is called an ?-subgroup in G if NG(H) ∩ Hx ≤ 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. 相似文献
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 ⩽ HsG, where HsG 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. 相似文献
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 HG, where HG 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). 相似文献
Let G be a finite group. A subgroup H of G is said to be weakly s-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are s-quasinormal in G. In this article, we investigate the structure of G under the assumption that some families of subgroups of G are weakly s-supplemented in G. Some recent results are generalized. 相似文献
We obtain several homotopy obstructions to the existence of non-closed connected Lie subgroupsH in a connected Lie groupG.First we show that the foliationF(G, H) onG determined byH is transversely complete [4]; moreover, forK the closure ofH inG, F(K, H) is an abelian Lie foliation [2].Then we prove that 1(K) and 1(H) have the same torsion subgroup, n(K)=n(H) for alln 2, and rank1(K) — rank1(H) > codimF(K, H). This implies, for instance, that a contractible (e.g. simply connected solvable) Lie subgroup of a compact Lie group must be abelian. Also, if rank1(G) 1 then any connected invariant Lie subgroup ofG is closed; this generalizes a well-known theorem of Mal'cev [3] for simply connected Lie groups.Finally, we show that the results of Van Est on (CA) Lie groups [6], [7] provide many interesting examples of such foliations. Actually, any Lie group with non-compact centre is the (dense) leaf of a foliation defined by a closed 1-form. Conversely, when the centre is compact, the latter is true only for (CA) Lie groups (e.g. nilpotent or semisimple). 相似文献
Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in Φ(G) have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified. 相似文献
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. 相似文献
A proper subgroup H of a group G is said to be strongly isolated if it contains the centralizer of any nonidentity element of H and 2-isolated if the conditions >CG(g) H 1 and 2(CG(g)) imply that CG(g)H. An involution i in a group G is said to be finite if |iig| < (for any g G). In the paper we study a group G with finite involution i and with a 2-isolated locally finite subgroup H containing an involution. It is proved that at least one of the following assertions holds:1) all 2-elements of the group G belong to H;2) (G,H) is a Frobenius pair, H coincides with the centralizer of the only involution in H, and all involutions in G are conjugate;3) G=FFCG(i) is a locally finite Frobenius group with Abelian kernel F;4) H=V D is a Frobenius group with locally cyclic noninvariant factor D and a strongly isolated kernel V, U=O2(V) is a Sylow 2-subgroup of the group G, and G is a Z-group of permutations of the set =Ug g G. 相似文献
We say that a subgroup H of a finite group G is nearly S-permutable in G if for every prime p such that (p, |H|) = 1 and for every subgroup K of G containing H the normalizer NK(H) contains some Sylow p-subgroup of K. We study the structure of G under the assumption that some subgroups of G are nearly S-permutable in G. 相似文献
A subgroup H of a group G is called µ-supplemented in G if there exists a subgroup K such that G = HK and H1K is a proper subgroup in G for every maximal subgroup H1 in H. For the initial values of p, we establish the p-supersolubility of a finite group with a μ-supplemented Sylow p-subgroup. 相似文献
Let G be a finite group and H a subgroup of G. We say that H is an ?-subgroup in G if NG(H) ∩ Hg ≤ H for all g ∈ G; H is called weakly ?-subgroup in G if G has a normal subgroup K such that G = HK and H ∩ K is an ?-subgroup in G. We say that H is weakly ? -embedded in G if G has a normal subgroup K such that HG = HK and H ∩ K is an ?-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that some subgroups of prime power order are weakly ?-embedded in G. Our results improve and generalize several recent results in the literature. 相似文献
A subgroup H is called Q-supplemented in a finite group G, if there exists a subgroup K of G such that G = HK and H ∩ K is contained in HQG, where HQG is the maximal quasinormal subgroup of G contained in H. In this article, we investigate the influence of Q-supplementation of some primary subgroups in finite groups. Some recent results are generalized. 相似文献
A subgroup H of a group G is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ HsG, where HsG is the maximal s-permutable subgroup of G contained in H. We improve a nice result of Skiba to get the following
Theorem. Let ? be a saturated formation containing the class of all supersoluble groups
and let G be a group with E a normal subgroup of G such that G/E ∈ ?. Suppose that each noncyclic Sylow p-subgroup P of F*(E) has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| are weakly s-permutable in G for all p ∈ π(F*(E)); moreover, we suppose that every cyclic subgroup of P of order 4 is weakly s-permutable in G if P is a nonabelian 2-group and |D| = 2. Then G ∈ ?.
Let G be a finite group, and let A be a proper subgroup of G. Then any chief factor H/AG of G is called a G-boundary factor of A. For any Gboundary factor H/AG of A, the subgroup (A ∩ H)/AG of G/AG is called a G-trace of A. In this paper, we prove that G is p-soluble if and only if every maximal chain of G of length 2 contains a proper subgroup M of G such that either some G-trace of M is subnormal or every G-boundary factor of M is a p′-group. This result give a positive answer to a recent open problem of Guo and Skiba. We also give some new characterizations of p-hypercyclically embedded subgroups. 相似文献
A proper subgroup H of a group G is said to be strongly embedded if 2 (H) and 2(HHg) (for all
). An involution i of G is said to be finite if
(for all gG). As is known, the structure of a (locally) finite group possessing a strongly embedded subgroup is determined by the theorems of Burnside and Brauer--Suzuki, provided that the Sylow 2-subgroup contains a unique involution. In this paper, sufficient conditions for the equality m2(G)= 1 are established, and two analogs of the Burnside and Brauer—Suzuki theorems for infinite groups G possessing a strongly embedded subgroup and a finite involution are given. 相似文献
Suppose that H is a subgroup of a finite group G. H is called π-quasinormal in G if it permutes with every Sylow subgroup of G; H is called π-quasinormally embedded in G provided every Sylow subgroup of H is a Sylow subgroup of some π-quasinormal subgroup of G; H is called c-supplemented in G if there exists a subgroup N of G such that G = HN and H ∩ N ⩽ HG = CoreG(H). In this paper, finite groups G satisfying the condition that some kinds of subgroups of G are either π-quasinormally embedded or c-supplemented in G, are investigated, and theorems which unify some recent results are given.
相似文献
We look at the structure of a soluble group G depending on the value of a function m(G)= max mpG), where mp(G)=max{logp|G:M| | M< G, |G:M|=pa}, p (G). Theorem 1 states that for a soluble group G, (1) r(G/ (G))= m(G); (2) d(G/ (G)) 1+ (m(G)) 3+m(G); (3) lp(G) 1+t, where 2t-1<mp(G) 2t. Here, (G) is the Frattini subgroup of G, and r(G), d(G), and lp(G) are, respectively, the principal rank, the derived length, and the p-length of G. The maximum of derived lengths of completely reducible soluble subgroups of a general linear group GL(n,F) of degree n, where F is a field, is denoted by (n). The function m(G) allows us to establish the existence of a new class of conjugate subgroups in soluble groups. Namely, Theorem 2 maintains that for any natural k, every soluble group G contains a subgroup K possessing the following properties: (1) m(K); k; (2) if T and H are subgroups of G such that KT <max <maxHG then |H:T|=pt for some prime p and for t>k. Moreover, every two subgroups of G enjoying (1) and (2) are mutually conjugate. 相似文献
Using the classification of finite simple groups, we prove that if H is an insoluble normal subgroup of a finite group G, then H contains a maximal soluble subgroup S such that G=HNG(S). Thereby Problem 14.62 in the Kourovka Notebook is given a positive solution. As a consequence, it is proved that in every finite group, there exists a subgroup that is simultaneously a
-projector and a
-injector in the class,
, of all soluble groups. 相似文献