Finite groups with some SS-supplemented subgroups

Mengling Jiang; Jianjun Liu

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Displaying similar documents to “Finite groups with some SS-supplemented subgroups”

On TI-subgroups and QTI-subgroups of finite groups

Ruifang Chen, Xianhe Zhao (2020)

Czechoslovak Mathematical Journal

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Let $G$ be a group. A subgroup $H$ of $G$ is called a TI-subgroup if $H \cap H^{g} = 1$ or $H$ for every $g \in G$ and $H$ is called a QTI-subgroup if $C_{G} (x) \leq N_{G} (H)$ for any $1 \neq x \in H$ . In this paper, a finite group in which every nonabelian maximal is a TI-subgroup (QTI-subgroup) is characterized.

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 $| H / H_{G} | \leq 2$ for every subgroup $H$ of $G$ , then $G$ contains an Abelian subgroup of index at most $8$ .

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 $g \in G$ . 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...

Finite groups whose all proper subgroups are $𝒞$ -groups

Pengfei Guo, Jianjun Liu (2018)

Czechoslovak Mathematical Journal

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A group $G$ is said to be a $𝒞$ -group if for every divisor $d$ of the order of $G$ , there exists a subgroup $H$ of $G$ of order $d$ such that $H$ is normal or abnormal in $G$ . We give a complete classification of those groups which are not $𝒞$ -groups but all of whose proper subgroups are $𝒞$ -groups.

On solvability of finite groups with some $s s$ -supplemented subgroups

Jiakuan Lu, Yanyan Qiu (2015)

Czechoslovak Mathematical Journal

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A subgroup $H$ of a finite group $G$ is said to be $s s$ -supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G = H K$ and $H \cap K$ is $s$ -permutable in $K$ . In this paper, we first give an example to show that the conjecture in A. A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group $G$ is solvable if every subgroup of odd prime order of $G$ is $s s$ -supplemented in $G$ , and that $G$ is solvable if and only if every Sylow subgroup of odd order of $G$ is $s s$ -supplemented in $G$ . These results...

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 $asn (G) \neq 29 / 4$ , 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.

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.

Finite $p$ -nilpotent groups with some subgroups weakly $ℳ$ -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 $ℳ$ -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 $ℳ$ -supplemented in $G$ . Some recent results are generalized. ...

On $σ$ -permutably embedded subgroups of finite groups

Chenchen Cao, Li Zhang, Wenbin Guo (2019)

Czechoslovak Mathematical Journal

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Let $σ = {σ_{i} : i \in I}$ be some partition of the set of all primes $ℙ$ , $G$ be a finite group and $σ (G) = {σ_{i} : σ_{i} \cap π (G) \neq \emptyset}$ . A set $ℋ$ of subgroups of $G$ is said to be a complete Hall $σ$ -set of $G$ if every non-identity member of $ℋ$ is a Hall $σ_{i}$ -subgroup of $G$ and $ℋ$ contains exactly one Hall $σ_{i}$ -subgroup of $G$ for every $σ_{i} \in σ (G)$ . $G$ is said to be $σ$ -full if $G$ possesses a complete Hall $σ$ -set. A subgroup $H$ of $G$ is $σ$ -permutable in $G$ if $G$ possesses a complete Hall $σ$ -set $ℋ$ such that $H A^{x}$ = $A^{x} H$ for all $A \in ℋ$ and all $x \in G$ . A subgroup $H$ of $G$ is $σ$ -permutably embedded in...

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.

A note on weakly-supplemented subgroups of finite groups

Hong Pan (2018)

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 the paper, we extend one main result of Kong and Liu (2014).