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

Fenfang Xie; Jinjin Wang; Jiayi Xia; Guo Zhong

Displaying similar documents to “Finite Groups with some $s$ -Permutably Embedded and Weakly $s$ -Permutable Subgroups”

On $R$ -conjugate-permutability of Sylow subgroups

Xianhe Zhao, Ruifang Chen (2016)

Czechoslovak Mathematical Journal

Similarity:

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...

A note on sumsets of subgroups in $ℤ *_{p}$

Derrick Hart (2013)

Acta Arithmetica

Similarity:

Let A be a multiplicative subgroup of $ℤ *_{p}$ . Define the k-fold sumset of A to be $k A = x_{1} + . . . + x_{k} : x_{i} \in A, 1 \leq i \leq k$ . We show that $6 A \supseteq ℤ *_{p}$ for $| A | > p^{11 / 23 + ϵ}$ . In addition, we extend a result of Shkredov to show that ${| 2 A | ≫ | A |}^{8 / 5 - ϵ}$ for $| A | ≪ p^{5 / 9}$ .

On $Π$ -property of some maximal subgroups of Sylow subgroups of finite groups

Zhengtian Qiu, Jianjun Liu, Guiyun Chen (2023)

Czechoslovak Mathematical Journal

Similarity:

Let $H$ be a subgroup of a finite group $G$ . We say that $H$ satisfies the $Π$ -property in $G$ if for any chief factor $L / K$ of $G$ , $| G / K : N_{G / K} (H K / K \cap L / K) |$ is a $π (H K / K \cap L / K)$ -number. We study the influence of some $p$ -subgroups of $G$ satisfying the $Π$ -property on the structure of $G$ , and generalize some known results.

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

Pengfei Guo, Jianjun Liu (2018)

Czechoslovak Mathematical Journal

Similarity:

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.

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

Jianjun Liu, Jian Chang, Guiyun Chen (2020)

Czechoslovak Mathematical Journal

Similarity:

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 $P \cap G^{'}$ with order $p$ lies in the center of $N_{G} (P)$ and when $p = 2$ , either every element of $P \cap G^{'}$ 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...

Some results on Sylow numbers of finite groups

Yang Liu, Jinjie Zhang (2024)

Czechoslovak Mathematical Journal

Similarity:

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 infinite $a S$ -groups

Reza Nikandish, Babak Miraftab (2015)

Czechoslovak Mathematical Journal

Similarity:

Let $G$ be a group. If every nontrivial subgroup of $G$ has a proper supplement, then $G$ is called an $a S$ -group. We study some properties of $a S$ -groups. For instance, it is shown that a nilpotent group $G$ is an $a S$ -group if and only if $G$ is a subdirect product of cyclic groups of prime orders. We prove that if $G$ is an $a S$ -group which satisfies the descending chain condition on subgroups, then $G$ is finite. Among other results, we characterize all abelian groups for which every nontrivial quotient group...

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

Enrico Jabara (2018)

Czechoslovak Mathematical Journal

Similarity:

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 $8$ .

Finite $p$ -nilpotent groups with some subgroups weakly $ℳ$ -supplemented

Liushuan Dong (2020)

Czechoslovak Mathematical Journal

Similarity:

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. ...

A note on weakly-supplemented subgroups of finite groups

Hong Pan (2018)

Czechoslovak Mathematical Journal

Similarity:

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).

On the number of isomorphism classes of derived subgroups

Leyli Jafari Taghvasani, Soran Marzang, Mohammad Zarrin (2019)

Czechoslovak Mathematical Journal

Similarity:

We show that a finite nonabelian characteristically simple group $G$ satisfies $n = | π (G) | + 2$ if and only if $G ≅ A_{5}$ , where $n$ is the number of isomorphism classes of derived subgroups of $G$ and $π (G)$ is the set of prime divisors of the group $G$ . Also, we give a negative answer to a question raised in M. Zarrin (2014).

Displaying similar documents to “Finite Groups with some s -Permutably Embedded and Weakly s -Permutable Subgroups”

Displaying similar documents to “Finite Groups with some $s$ -Permutably Embedded and Weakly $s$ -Permutable Subgroups”