On automorphisms fixing subnormal subgroups of soluble groups

Silvana Franciosi; Francesco de Giovanni

Displaying similar documents to “On automorphisms fixing subnormal subgroups of soluble groups”

On automorphisms fixing subnormal subgroups of soluble groups

Silvana Franciosi, Francesco de Giovanni (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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The group $Aut_{sn}G$ of all automorphisms leaving invariant every subnormal subgroup of the group $G$ is studied. In particular it is proved that $Aut_{sn}G$ is metabelian if $G$ is soluble, and that $Aut_{sn}G$ is either finite or abelian if $G$ is polycyclic.

Coleman automorphisms of finite groups with a self-centralizing normal subgroup

Jinke Hai (2020)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group with a normal subgroup $N$ such that $C_{G} (N) \leq N$ . It is shown that under some conditions, Coleman automorphisms of $G$ are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.

Polycyclic groups with automorphisms of order four

Tao Xu, Fang Zhou, Heguo Liu (2016)

Czechoslovak Mathematical Journal

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In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if $α$ is an automorphism of order four of a polycyclic group $G$ and the map $ϕ : G \to G$ defined by $g^{ϕ} = [g, α]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H^{''}$ is included in the centre of $H$ and $C_{H} (α^{2})$ is abelian, both $C_{G} (α^{2})$ and $G / [G, α^{2}]$ are abelian-by-finite. These results extend recent and classical results in...

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 $8$ .

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 $H \cap K$ 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.

Artinian automorphisms of infinite groups

Antonella Leone (2006)

Bollettino dell'Unione Matematica Italiana

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An automorphism a of a group G is called an artinian automorphism if for every strictly descending chain $H_{1}>H_{2}>\cdots>H_{n}>\cdots$ of subgroups of G there exists a positive integer $m$ such that $(H_{n})^{a}=H_{n}$ for every $n\geq m$ . In this paper we show that in many cases the group of all artinian automorphisms of $G$ coincides with the group of all power automorphisms of $G$ .

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 $R$ -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 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 $ℳ_{d} (P) = {P_{1}, P_{2}, \dots, P_{d}}$ 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 $ℳ_{d} (P)$ is either $s$ -permutably embedded or weakly $s$ -permutable in $G$ to give criteria for a group to be $p$ -supersolvable or $p$ -nilpotent.

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 property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent

Fares Gherbi, Nadir Trabelsi (2024)

Czechoslovak Mathematical Journal

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Let $𝔐$ be the class of groups satisfying the minimal condition on normal subgroups and let $Ω$ be the class of groups of finite lower central depth, that is groups $G$ such that $γ_{i} (G) = γ_{i + 1} (G)$ for some positive integer $i$ . The main result states that if $G$ is a finitely generated hyper-(Abelian-by-finite) group such that for every $x \in G$ , there exists a normal subgroup $H_{x}$ of finite index in $G$ satisfying $〈 x, x^{h} 〉 \in 𝔐 Ω$ for every $h \in H_{x}$ , then $G$ is finite-by-nilpotent. As a consequence of this result, we prove that a finitely generated...

On automorphisms of the Banach space $ℓ_{\infty} / c ₀$

Piotr Koszmider, Cristóbal Rodríguez-Porras (2016)

Fundamenta Mathematicae

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We investigate Banach space automorphisms $T : ℓ_{\infty} / c ₀ \to ℓ_{\infty} / c ₀$ focusing on the possibility of representing their fragments of the form $T_{B, A} : ℓ_{\infty} (A) / c ₀ (A) \to ℓ_{\infty} (B) / c ₀ (B)$ for A,B ⊆ ℕ infinite by means of linear operators from $ℓ_{\infty} (A)$ into $ℓ_{\infty} (B)$ , infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on $ℓ_{\infty} / c ₀$ . We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for...

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

Derrick Hart (2013)

Acta Arithmetica

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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 $σ$ -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 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).

Modular representations of finite groups with trivial restriction to Sylow subgroups

Paul Balmer (2013)

Journal of the European Mathematical Society

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Let $k$ be a field of characteristic $p$ . Let $G$ be a finite group of order divisible by $p$ and $P$ a $p$ -Sylow subgroup of $G$ . We describe the kernel of the restriction homomorphism $T (G) \to T (P)$ , for $T (-)$ the group of endotrivial representations. Our description involves functions $G \to k^{\times}$ that we call weak $P$ -homomorphisms. These are generalizations to possibly non-normal $P \leq G$ of the classical homomorphisms $G / P \to k^{\times}$ appearing in the normal case.

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.

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. ...