Displaying similar documents to “A note on weakly-supplemented subgroups of finite groups”

A note on weakly-supplemented subgroups and the solvability of finite groups

Xin Liang, Baiyan Xu (2022)

Czechoslovak Mathematical Journal

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Suppose that G is a finite group and H is a subgroup of G . The subgroup H is said to be weakly-supplemented in G if there exists a proper subgroup K of G such that G = H K . In this note, by using the weakly-supplemented subgroups, we point out several mistakes in the proof of Theorem 1.2 of Q. Zhou (2019) and give a counterexample.

On weakly-supplemented subgroups of finite groups

Qingjun Kong (2019)

Czechoslovak Mathematical Journal

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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 = H K . In this paper, some interesting results with weakly-supplemented minimal subgroups to a smaller subgroup of G are obtained.

Finite p -nilpotent groups with some subgroups weakly -supplemented

Liushuan Dong (2020)

Czechoslovak Mathematical Journal

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Suppose that G is a finite group and H is a subgroup of G . Subgroup H is said to be weakly -supplemented in G if there exists a subgroup B of G such that (1) G = H B , and (2) if H 1 / H G is a maximal subgroup of H / H G , then H 1 B = B H 1 < G , where H G is the largest normal subgroup of G contained in H . We fix in every noncyclic Sylow subgroup P of G a subgroup D satisfying 1 < | D | < | P | and study the p -nilpotency of G under the assumption that every subgroup H of P with | H | = | D | is weakly -supplemented in G . Some recent results are generalized. ...

On weakly-supplemented subgroups and the solvability of finite groups

Qiang Zhou (2019)

Czechoslovak Mathematical Journal

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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 = H K . In this paper, some interesting results with weakly-supplemented minimal subgroups or Sylow subgroups of G are obtained.

Finite Groups with some s -Permutably Embedded and Weakly s -Permutable Subgroups

Fenfang Xie, Jinjin Wang, Jiayi Xia, Guo Zhong (2013)

Confluentes Mathematici

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Let G be a finite group, p the smallest prime dividing the order of G and P a Sylow p -subgroup of G with the smallest generator number d . There is a set d ( P ) = { P 1 , P 2 , , P d } of maximal subgroups of P such that i = 1 d P i = Φ ( P ) . In the present paper, we investigate the structure of a finite group under the assumption that every member of d ( P ) is either s -permutably embedded or weakly s -permutable in G to give criteria for a group to be p -supersolvable or p -nilpotent.

The p -nilpotency of finite groups with some weakly pronormal subgroups

Jianjun Liu, Jian Chang, Guiyun Chen (2020)

Czechoslovak Mathematical Journal

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For a finite group G and a fixed Sylow p -subgroup P of G , Ballester-Bolinches and Guo proved in 2000 that G is p -nilpotent if every element of P G ' with order p lies in the center of N G ( P ) and when p = 2 , either every element of P G ' with order 4 lies in the center of N G ( P ) or P is quaternion-free and N G ( P ) is 2 -nilpotent. Asaad introduced weakly pronormal subgroup of G in 2014 and proved that G is p -nilpotent if every element of P with order p is weakly pronormal in G and when p = 2 , every element of P with...

On R -conjugate-permutability of Sylow subgroups

Xianhe Zhao, Ruifang Chen (2016)

Czechoslovak Mathematical Journal

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A subgroup H of a finite group G is said to be conjugate-permutable if H H g = H g H for all g G . More generaly, if we limit the element g to a subgroup R of G , then we say that the subgroup H is R -conjugate-permutable. By means of the R -conjugate-permutable subgroups, we investigate the relationship between the nilpotence of G and the R -conjugate-permutability of the Sylow subgroups of A and B under the condition that G = A B , where A and B are subgroups of G . Some results known in the literature are improved...

Every 2 -group with all subgroups normal-by-finite is locally finite

Enrico Jabara (2018)

Czechoslovak Mathematical Journal

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A group G has all of its subgroups normal-by-finite if H / H G is finite for all subgroups H of G . The Tarski-groups provide examples of p -groups ( p a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a 2 -group with every subgroup normal-by-finite is locally finite. We also prove that if | H / H G | 2 for every subgroup H of G , then G contains an Abelian subgroup of index at most 8 .

Finite groups with some SS-supplemented subgroups

Mengling Jiang, Jianjun Liu (2021)

Czechoslovak Mathematical Journal

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A subgroup H of a finite group G is said to be SS-supplemented in G if there exists a subgroup K of G such that G = H K and H K is S-quasinormal in K . We analyze how certain properties of SS-supplemented subgroups influence the structure of finite groups. Our results improve and generalize several recent results.

On a generalization of a theorem of Burnside

Jiangtao Shi (2015)

Czechoslovak Mathematical Journal

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A theorem of Burnside asserts that a finite group G is p -nilpotent if for some prime p a Sylow p -subgroup of G lies in the center of its normalizer. In this paper, let G be a finite group and p the smallest prime divisor of | G | , the order of G . Let P Syl p ( G ) . As a generalization of Burnside’s theorem, it is shown that if every non-cyclic p -subgroup of G is self-normalizing or normal in G then G is solvable. In particular, if P a , b | a p n - 1 = 1 , b 2 = 1 , b - 1 a b = a 1 + p n - 2 , where n 3 for p > 2 and n 4 for p = 2 , then G is p -nilpotent or p -closed. ...

Some results on Sylow numbers of finite groups

Yang Liu, Jinjie Zhang (2024)

Czechoslovak Mathematical Journal

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We study the group structure in terms of the number of Sylow p -subgroups, which is denoted by n p ( G ) . The first part is on the group structure of finite group G such that n p ( G ) = n p ( G / N ) , where N is a normal subgroup of G . The second part is on the average Sylow number asn ( G ) and we prove that if G is a finite nonsolvable group with asn ( G ) < 39 / 4 and asn ( G ) 29 / 4 , then G / F ( G ) A 5 , where F ( G ) is the Fitting subgroup of G . In the third part, we study the nonsolvable group with small sum of Sylow numbers.