Topic
Complement (group theory)
About: Complement (group theory) is a research topic. Over the lifetime, 677 publications have been published within this topic receiving 6229 citations.
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Papers
TL;DR: The main result of as mentioned in this paper is that all Sylow subgroups of odd order are cyclic, and moreover that a 2-Sylow subgroup is either a dihedral group, or a generalized quaternion group.
Abstract: The purpose of this paper is to determine the structure of some finite groups in which all Sylow subgroups of odd order are cyclic. This assumption on Sylow subgroups simplifies the structure of groups considerably, but the structure of 2-Sylow subgroups might be too complicated to make any definite statement on the structure of the groups. In this paper, therefore, we shall make another assumption on 2-Sylow subgroups, and our main result may be stated as follows. Let G be a non-solvable group of finite order. We assume that all Sylow subgroups of odd order are cyclic, and moreover that a 2-Sylow subgroup is either (a) a dihedral group, or (b) a generalized quaternion group. Then G contains a normal subgroup G1 such that [G: G1] ? 2 and G1 = Z X L, where Z is a solvable group whose Sylow subgroups are all cyclic, and L is isomor-phic with the linear fractional group LF (2, p) over the prime field of characteristic p in the case (a), and with the special linear group SL (2, p) in the case (b).
113 citations
TL;DR: A subgroup H of G is said to be ''pi$-quasinormal in G'' if it permute with every Sylow subgroup of G as discussed by the authors, and the main result in this paper is the following:
Abstract: A subgroup H of G is said to be $\pi$-quasinormal in G if it permute with every Sylow subgroup of G. In this paper, we extend the study on the structure of a finite group under the assumption that some subgroups of G are $\pi$-quasinormal in G. The main result we proved in this paper is the following:
109 citations
TL;DR: In this paper, a subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and HN∩N≤H G ǫ = core(H).
Abstract: A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H G = Core(H). We extend the study on the structure of a finite group under the assumption that all maximal or minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of G are c-normal in G. The main theorems we proved in this paper are: Theorem Let ℱ be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ℱ. If all maximal subgroups of any Sylow subgroup of F*(H) are c-normal in G, then G ∈ ℱ. Theorem Let ℱ be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ℱ. If all minimal subgroups and all cyclic subgroups of F*(H) are c-normal in G, then G ∈ ℱ.
97 citations
TL;DR: In this paper, a subgroup H of a group G is said to be SS-quasinormal if H possesses a supplement B such that H permutes with every Sylow subgroup of B.
97 citations
TL;DR: In this article, the influence of S-quasinormally embedded subgroups of groups of prime power order on the structure of finite groups has been investigated and the results improve and extend recent results of Ballester-Bolinches and Pedraza-Aguilera (J. Pure Appl. Algebra 127 (1998) 113).
93 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2022 | 1 |
| 2021 | 13 |
| 2020 | 12 |
| 2019 | 7 |
| 2018 | 10 |
| 2017 | 27 |