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1.
In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G
N
of orderp is permutable inN
G
(P) and whenp = 2 either every cyclic subgroup ofP ∩G
N
of order 4 is permutable inN
G
(P) orP is quaternion-free. Some applications of this result are given.
The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR
China.
The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000). 相似文献
2.
Long Miao 《Bulletin of the Brazilian Mathematical Society》2009,40(4):495-509
A subgroup H of a group G is said to be M-supplemented in G if there exists a subgroup B of G such that G = HB and T B < G for every maximal sub-group T of H. Moreover, a subgroup H is called c-supplemented in G if there exists a subgroup K such that G = HK and H ∩ K ≤ H
G
where H
G
is the largest normal subgroup of G contained in H. In this paper we give some conditions of supersolv-ability of finite group under assumption that some primary subgroups
have some kinds of supplements, which are generalizations of some recent results. 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.
Long Miao 《Bulletin of the Brazilian Mathematical Society》2010,41(2):223-235
A subgroup H of a group G is said to be weakly s-permutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ H
sG
where H
sG
is the largest s-quasinormal subgroup of G contained in H. In this paper, we investigate the influence of weak s-permutability of some primary subgroups in finite groups. Some new results about p-supersolvability and p-nilpotency of finite groups are obtained. 相似文献
5.
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
6.
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
7.
A subgroup H of a finite group G is said to be c*-supplemented in G if there exists a subgroup K such that G = HK and H ⋂ K is permutable in G. It is proved that a finite group G that is S
4-free is p-nilpotent if N
G
(P) is p-nilpotent and, for all x ∈ G\N
G
(P), every minimal subgroup of
is c*-supplemented in P and (if p = 2) one of the following conditions is satisfied: (a) every cyclic subgroup of
of order 4 is c*-supplemented in P, (b)
, (c) P is quaternion-free, where P a Sylow p-subgroup of G and
is the p-nilpotent residual of G. This extends and improves some known results.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1011–1019, August, 2007. 相似文献
8.
Long Miao 《Bulletin of the Brazilian Mathematical Society》2007,38(4):585-594
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
9.
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. 相似文献
10.
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
11.
On complemented subgroups of finite groups 总被引:1,自引:0,他引:1
Long Miao 《Czechoslovak Mathematical Journal》2006,56(3):1019-1028
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. 相似文献
12.
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
13.
A subgroup H of G is said to be S-embedded in G if G has a normal subgroup N such that HN is s-permutable in G and H ∩ N ⩽ H
sG
, where H
sG
is the largest s-permutable subgroup of G contained in H. S-embedded subgroups are used to give novel characterizations for some classes of groups. New results are obtained and a number
of previously known ones are generalized. 相似文献
14.
A finite group G is called p
i
-central of height k if every element of order p
i
of G is contained in the k
th
-term ζ
k
(G) of the ascending central series of G. If p is odd, such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent p-groups, i.e., for such a finite p-group P the group P/Ω1(P) is also p-central of height p − 2 (Thm. B). In such a group P, the index of P
p
is less than or equal to the order of the subgroup Ω1(P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p − 1, p odd, and N
G
(P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Syl
p
(G) is p-central of height p − 2, then N
G
(P) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11]). 相似文献
15.
A subgroup H of a finite 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, it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of the p-focal subgroup of G is complemented in NG(P), where P is a Sylow p-subgroup of G. As some applications, some interesting results related with complemented minimal subgroups of focal subgroups are obtained. 相似文献
16.
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
17.
Martyn R. Dixon Martin J. Evans Antonio Tortora 《Central European Journal of Mathematics》2010,8(1):22-25
A subgroup H of a group G is inert if |H: H ∩ H
g
| is finite for all g ∈ G and a group G is totally inert if every subgroup H of G is inert. We investigate the structure of minimal normal subgroups of totally inert groups and show that infinite locally
graded simple groups cannot be totally inert. 相似文献
18.
Qin-hai ZHANG Cui-juan SUN Hai-peng QU & Ming-yao XU School of Mathematics Computer Sciences Shanxi Normal University Linfen China 《中国科学A辑(英文版)》2007,50(6):814-820
Following Blackburn, Deaconescu and Mann, a group G is called an equilibrated group if for any subgroups H,K of G with HK = KH, either H≤NG(K) or K≤NG(H). Continuing their work and based on the classification of metacyclic p-groups given by Newman and Xu, we give a complete classification of 2-generator equilibrated p-groups in this note. 相似文献
19.
Changwen Li 《印度理论与应用数学杂志》2011,42(5):291-306
A subgroup of H of a group G is called ss-quasinormally embedded in G if there exists a subgroup T of G such that G = HT and H ∩ T is squasinormally embedded in G. In this paper, we shall obtain some characterizations about p-nilpotency of G by assuming that some subgroups of prime power order of G are ss-quasinormally embedded in G. 相似文献
20.
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