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

Derrick Hart

Displaying similar documents to “A note on sumsets of subgroups in $ℤ *_{p}$ ”

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...

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...

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.

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.

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 the number of isomorphism classes of derived subgroups

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

Czechoslovak Mathematical Journal

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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).

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

Zhengtian Qiu, Jianjun Liu, Guiyun Chen (2023)

Czechoslovak Mathematical Journal

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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.

On the conjugate type vector and the structure of a normal subgroup

Ruifang Chen, Lujun Guo (2022)

Czechoslovak Mathematical Journal

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Let $N$ be a normal subgroup of a group $G$ . The structure of $N$ is given when the $G$ -conjugacy class sizes of $N$ is a set of a special kind. In fact, we give the structure of a normal subgroup $N$ under the assumption that the set of $G$ -conjugacy class sizes of $N$ is $(p_{1 n_{1}}^{a_{1 n_{1}}}, \dots, p_{11}^{a_{11}}, 1) \times \dots \times (p_{r n_{r}}^{a_{r n_{r}}}, \dots, p_{r 1}^{a_{r 1}}, 1)$ , where $r > 1$ , $n_{i} > 1$ and $p_{i j}$ are distinct primes for $i \in {1, 2, \dots, r}$ , $j \in {1, 2, \dots, n_{i}}$ .

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...

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. ...

Displaying similar documents to “A note on sumsets of subgroups in ℤ * p ”

Displaying similar documents to “A note on sumsets of subgroups in $ℤ *_{p}$ ”