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1.
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
2.
Joseph Kirtland 《Archiv der Mathematik》2011,97(5):399-406
Let G be a finite group. A subgroup H of G is a permutable subgroup of G if HK = KH for all subgroups K of G. It will be shown that if all subgroups not contained in the Frattini subgroup are permutable in a group G, then all subgroups are permutable in G. 相似文献
3.
David P. Sumner 《Discrete Mathematics》1978,22(1):49-55
A graph G is totally connected if both G and ? (its complement) are connected. The connected Ramsey number rc(F, H) is the smallest integer k ? 4 so that if G is a totally connected graph of order k then either F ? G or H ? ?. We show that if neither of F nor H contains a bridge, then rc = r(F, H), the usual generalized Ramsey number of F and H. We compute rc (PmPm), the connected Ramsey number for paths. 相似文献
4.
A group G is called a ${\mathcal {T}_{c}}$ -group if every cyclic subnormal subgroup of G is normal in G. Similarly, classes ${\mathcal {PT}_{c}}$ and ${\mathcal {PST}_{c}}$ are defined, by requiring cyclic subnormal subgroups to be permutable or S-permutable, respectively. A subgroup H of a group G is called normal (permutable or S-permutable) cyclic sensitive if whenever X is a normal (permutable or S-permutable) cyclic subgroup of H there is a normal (permutable or S-permutable) cyclic subgroup Y of G such that ${X=Y \cap H}$ . We analyze the behavior of a collection of cyclic normal, permutable and S-permutable subgroups under the intersection map into a fixed subgroup of a group. In particular, we tie the concept of normal, permutable and S-permutable cyclic sensitivity with that of ${\mathcal {T}_c}$ , ${\mathcal {PT}_c}$ and ${\mathcal {PST}_c}$ groups. In the process we provide another way of looking at Dedekind, Iwasawa and nilpotent groups. 相似文献
5.
A subgroup of a group G is said to be Sylow-quasinormal (S-quasinormal) in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be Supplement-Sylow-quasinormal (SS-quasinormal) in G if there is a supplement B of H to G such that H is permutable with every Sylow subgroup of B. In this article, we investigate the influence of SS-quasinormal of maximal or minimal subgroups of Sylow subgroups of the generalized Fitting subgroup of a finite group. 相似文献
6.
A subgroup H of a group G is said to be an SS-quasinormal (Supplement-Sylow-quasinormal) subgroup if there is a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B. A subgroup H of a group G is said to be S-quasinormally embedded inGif for every Sylow subgroup P of H, there is an S-quasinormal subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain SS-quasinormal or S-quasinormally embedded subgroups of prime power order are studied. 相似文献
7.
Adolfo Ballester-Bolinches James C. Beidleman Ramón Esteban-Romero Vicent Pérez-Calabuig 《Central European Journal of Mathematics》2013,11(6):1078-1082
A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions of Kaplan’s results, which enables a better understanding of the relationships between these classes. 相似文献
8.
A condition for the solvability of finite groups 总被引:1,自引:1,他引:0
A subgroup H is called ?-supplemented in a finite group G, if there exists a subgroup B of G such that G = HB and H 1 B is a proper subgroup of G for every maximal subgroup H 1 of H. We investigate the influence of ?-supplementation of Sylow subgroups and obtain a condition for solvability and p-supersolvability of finite groups. 相似文献
9.
We say that the subgroups G 1 and G 2 of a group G are mutually permutable if G 1 permutes with every subgroup of G 2 and G 2 permutes with every subgroup of G 1. Let G=G 1 G 2…G n be the product of its pairwise permutable subgroups G 1,G 2,…,G n such that the product G i G j is mutually permutable. We investigate the structure of the finite group G if special properties of the factors G 1,G 2,…,G n are known. Our results improve and extend some results of Asaad and Shaalan [1], Ezquerro and Soler-Escrivà [9] and Asaad and Monakhov [3]. 相似文献
10.
《代数通讯》2013,41(12):6135-6147
Abstract Two subgroups H and K of a group G are said to be totally permutable if every subgroup of H permutes with every subgroup of K. In this paper the behaviour of radicals and injectors associated to Fitting classes in a product of pairwise totally permutable finite groups is studied. 相似文献
11.
Na Tang Wenbin Guo V. V. Kabanov 《Proceedings of the Steklov Institute of Mathematics》2007,257(1):S189-S194
A subgroup H of a group G is called s-semipermutable in G if H is permutable with every Sylow p-subgroup of G with (p, |H|) = 1. In this paper, we use s-semipermutable subgroups to determine the structure of finite groups. Some of the previous results are generalized. 相似文献
12.
Long Miao 《Mathematical Notes》2009,86(5-6):655-664
A subgroup H of a group G is said to be ?-supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H. In this paper, we obtain the following statement: Let ? be a saturated formation containing all supersolvable groups and H be a normal subgroup of G such that G/H ε ?. Suppose that every maximal subgroup of a noncyclic Sylow subgroup of F*(H), having no supersolvable supplement in G, is ?-supplemented in G. Then G ε ?. 相似文献
13.
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
14.
M. Asaad 《Acta Mathematica Hungarica》2014,144(2):499-514
Let G be a finite group and H a subgroup of G. We say that H is s-permutable in G if HP = PH for all Sylow subgroups P of G; H is s-semipermutable in G if HP = PH for all Sylow subgroups P of G with (|P|, |H|) = 1. Let H s G be the subgroup of H generated by all those subgroups of G which are s-permutable in G and H sG the intersection of all such s-permutable subgroups of G contain H. We say that H is nearly s-embedded in G if G has an s-permutable subgroup T such that H sG = HT and \({H \cap T \leqq H_{ssG}}\) , where H ssG is an s-semipermutable subgroup of G contained in H. In this paper, we study the structure of a finite group G under the assumption that some subgroups of prime power order are nearly s-embedded in G. A series of known results are improved and extended. 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.
A subgroup H of a finite group G is weakly-supplemented in G if there exists a proper subgroup K of G such that G = HK. In the 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 P∩G′ is weakly-supplemented in N G (P), where P is a Sylow p-subgroup of G. As applications, some interesting results with weakly-supplemented minimal subgroups of P∩G′ are obtained. 相似文献
17.
Suppose that G is a finite group and H is a subgroup of G. H is said to be s-quasinormally embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal subgroup of G; H is called c*-quasinormally embedded in G if there is a subgroup T of G such that G = HT and H??T is s-quasinormally embedded in G. We investigate the influence of c*-quasinormally embedded subgroups on the structure of finite groups. Some recent results are generalized. 相似文献
18.
The paper is devoted to the study of properties of a class of subgroups H in Lie groups G that was recently introduced by the author. A closed subgroup H in a Lie group G is said to be plesio-uniform if there is a closed subgroup P of G that contains H and for which P is uniform in G and H is quasi-uniform in P. In the paper we give answers to several natural questions concerning plesio-uniform subgroups. It is proved that one obtains the same notion of plesio-uniformity when transposing the conditions of uniformity and quasi-uniformity in the definition of plesio-uniformity of a subgroup. If a closed subgroup H of G contains a plesio-uniform subgroup, then H is also plesio-uniform. Other properties of plesio-uniform subgroups are also considered. 相似文献
19.
Let G be a finite group. A subgroup H of G is said to be weakly S-embedded in G if there exists a normal subgroup K of G such that HK is S-quasinormal in G and H ∩ K ≤ H seG , where H seG is the subgroup generated by all those subgroups of H which are S-quasinormally embedded in G. We say that a subgroup H of G is weakly τ-embedded in G if there exists a normal subgroup K of G such that HK is S-quasinormal in G and H ∩ K ≤ H seG , where H seG is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. In this paper, we study the properties of weakly S-embedded and weakly τ-embedded subgroups, and use them to determine the structure of finite groups. 相似文献
20.
Ping Kang 《Periodica Mathematica Hungarica》2018,76(2):198-206
For a subgroup of a finite group we introduce a new property called weakly c-normal. Suppose that G is a finite group and H is a subgroup of G. H is said to be weakly c-normal in G if there exists a subnormal subgroup K of G such that \(G=HK\) and \(H\cap K\) is s-quasinormally embedded in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying \(1<|D|<|P|\) and study the structure of G under the assumption that every subgroup H of P with \(|H|=|D|\) is weakly c-normal in G. Some recent results are generalized and unified. 相似文献