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1.
A subgroup H of a group G is weakly normal in G if H
g
≤ N
G
(H) implies that g ∈ N
G
(H). In this paper, we shall obtain some characterizations about the supersolvability and nilpotency of G by assuming that some subgroups of prime power order of G are weakly normal in G. 相似文献
2.
Given a class ℑ of finite groups, a subgroup H of a group G is called ℑ
n
-normal in G if there exists a normal subgroup T of G such that HT is a normal subgroup of G and (H ∩ T)H
G
/H
G
is contained in the ℑ-hypercenter Z
∞ℑ (G/H
G
) of G/H
G
. We obtain some results about the ℑ
n
-normal subgroups and use them to study the structure of some groups. 相似文献
3.
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 ⩽ H
G
= Core
G
(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
4.
A subgroup H of a finite group G is called a TI-subgroup if H ∩ H
x
= 1 or H for any x ∈ G. In this short note, the finite groups all of whose nonabelian subgroups are TI-subgroups are classified. 相似文献
5.
Chuu-Lian Terng 《Journal of Geometric Analysis》1995,5(1):129-150
An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h
1,h
2) ·x = h
1
xh
2
−1
· LetP(G, H) denote the group ofH
1-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H
0([0, 1], g) isometrically byg * u = gug
−1 −g′g
−1. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine
Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples
ofP(G, H)-actions for suitable choice ofH andG.
Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut für Mathematik in Bonn. 相似文献
6.
Alessio Russo 《代数通讯》2013,41(10):3950-3954
A subgroup H of a group G is said to be weakly normal if H g = H whenever g is an element of G such that H g ≤ N G (H). There is a strictly relation between weak normality and groups in which normality is a transitive relation ( T-groups). In [Ballester-Bolinches, A., Esteban-Romero, R. (2003). On finite T-groups. J. Aust. Math. Soc. 75:181–191] it is proved that a finite group G is a soluble T-group if and only if every subgroup of G is weakly normal. In this article, we extend the above result to infinite groups having no infinite simple sections. Moreover, it will be shown that every locally graded non-periodic group, all of whose subgroups are weakly normal, is abelian. 相似文献
7.
Let G be a finite group and H a subgroup of G. We say that: (1) H is τ-quasinormal in G if H permutes with all Sylow subgroups Q of G such that (|Q|, |H|) = 1 and (|H|, |Q
G
|) ≠ 1; (2) H is weakly τ-quasinormal in G if G has a subnormal subgroup T such that HT = G and T ∩ H ≦ H
τG
, where H
τG
is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. Our main result here is the following. Let ℱ be a saturated formation containing all supersoluble groups and let X ≦ E be normal subgroups of a group G such that G/E ∈ ℱ. Suppose that every non-cyclic Sylow subgroup P of X has a subgroup D such that 1 < |D| < |P| and every subgroup H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is non-Abelian) not having a supersoluble supplement in G is weakly τ-quasinormal in G. If X is either E or F* (E), then G ∈ ℱ. 相似文献
8.
Let H be a subgroup of a group G. We say that H satisfies the power condition with respect to G, or H is a power subgroup of G, if there exists a non-negative integer m such that H = G m = 〈 g m |g ? G 〉. In this note, the following theorem is proved: Let G be a group and k the number of nonpower subgroups of G. Then (1) k = 0 if and only if G is a cyclic group (theorem of F. Szász); (2) 0 < k < ∞ if and only if G is a finite noncyclic group; (3) k = ∞ if and only if G is a infinte noncyclic group. Thus we get a new criterion for the finite noncyclic groups. 相似文献
9.
Daniel T. Wise 《Inventiones Mathematicae》2002,149(3):579-617
A subgroup M⊂G is almost malnormal provided that for each g∈G−M, the intersection M
g
∩M is finite. It is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal
subgroup, is residually finite. A consequence of a generalization of this result is that an acute-angled n-gon of finite groups is residually finite if n≥4. Another consequence is that if G acts properly discontinuously and cocompactly on a 2-dimensional hyperbolic building whose chambers have acute angles and
at least 4 sides, then G is residually finite.
Oblatum 17-VII-2000 & 13-II-2002?Published online: 29 April 2002 相似文献
10.
Shi Rong Li 《数学学报(英文版)》2008,24(4):647-654
Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown: (1) G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; (2) G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F(H), the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G. 相似文献
11.
In a finite group G every element can be factorized in such a way that there is one factor for each prime divisor p of | G |, and the order of this factor is pα for some integer α ≧ 0. We define g ∈G to be uniquely factorizable if it has just one such factorization (whose factors must be pairwise commuting). We consider the existence of uniquely factorizable
elements and its relation to the solvability of the group. We prove that G is solvable if and only if the set of all uniquely factorizable elements of G is the Fitting subgroup of G. We also prove various sufficient conditions for the non-existence of uniquely factorizable elements in non-solvable groups.
Received: 9 June 2005 相似文献
12.
A subgroup H of a finite group G is said to be permutable in G if it permutes with every subgroup of G. In this paper, we determine the finite groups which have a permutable subgroup of prime order and whose maximal subgroups
are totally (generalized) smooth groups. 相似文献
13.
Gordan Savin 《Israel Journal of Mathematics》1992,80(1-2):195-205
LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L
2(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP
f(g)=ΣH∩ΓΓ
f(γ{}itg) is an automorphic form on Γ\G. We show thatP
f is cuspidal in some cases, whenH ∩Γ\H is compact.
Partially supported by NSF Grant # DMS 9103608. 相似文献
14.
Andrea Lucchini 《Israel Journal of Mathematics》2011,181(1):53-64
It has been conjectured by Mann that the infinite sum Σ
H
μ(H,G)/|G:H|
s
, where H ranges over all open subgroups of a finitely generated profinite group G, converges absolutely in some half right plane if G is positively finitely generated. We prove that the conjecture is true if the nonabelian crowns of G have bounded rank. In particular Mann’s conjecture holds if G has polynomial subgroup growth or is an adelic profinite group. 相似文献
15.
XIAJIANGUO 《高校应用数学学报(英文版)》1998,13(1):109-116
Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal sub-group of G. For principal fibre bundle (E,p, E,/G;G) tmd (E/H,p‘,E/G;G/H), the relation between auta(E) (resp. autce (E)) and autG/H(E/H) (resp. autGe/H(E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group JG(E) (resp. SG(E)) while the group J G/u(E/H) is known. 相似文献
16.
Shi Rong LI Xiu Yun GUO 《数学学报(英文版)》2007,23(4):731-734
A subgroup H of a finite group G is called a TI-subgroup if H ∩ H^x = 1 or H for all x ∈ G. In this paper, a complete classification for finite p-groups, in which all abelian subgroups are TI-subgroups, is given. 相似文献
17.
We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ⩽ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S
4-free and every minimal subgroup of P ∩ G
N
is c-supplemented in N
G
(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized. 相似文献
18.
A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H
x
in 〈H, H
x
〉. A finite group G is called PRN-group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all PRN-groups and classify minimal non-PRN-groups (non-PRN-group all of whose proper subgroups are PRN-groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are PRN-groups. 相似文献
19.
Zhixin Zhao 《Mathematica Slovaca》2012,62(3):461-472
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. 相似文献
20.
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