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 A u t s n G of all automorphisms leaving invariant every subnormal subgroup of the group G is studied. In particular it is proved that A u t s n G is metabelian if G is soluble, and that A u t s n G is either finite or abelian if G is polycyclic.

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 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...

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 > > H n > of subgroups of G there exists a positive integer m such that ( H n ) a = H n for every n 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 .

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...

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.

On automorphisms of the Banach space / c

Piotr Koszmider, Cristóbal Rodríguez-Porras (2016)

Fundamenta Mathematicae

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We investigate Banach space automorphisms T : / c / c focusing on the possibility of representing their fragments of the form T B , A : ( A ) / c ( A ) ( B ) / c ( B ) for A,B ⊆ ℕ infinite by means of linear operators from ( A ) into ( 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 / c . 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 * p

Derrick Hart (2013)

Acta Arithmetica

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Let A be a multiplicative subgroup of * p . Define the k-fold sumset of A to be k A = x 1 + . . . + x k : x i A , 1 i k . We show that 6 A * 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 .

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 ) T ( P ) , for T ( ) the group of endotrivial representations. Our description involves functions G k × that we call weak P -homomorphisms. These are generalizations to possibly non-normal P G of the classical homomorphisms G / P k × appearing in the normal case.

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 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...

A note on infinite a S -groups

Reza Nikandish, Babak Miraftab (2015)

Czechoslovak Mathematical Journal

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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...

Permutability of centre-by-finite groups

Brunetto Piochi (1989)

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

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Let G be a group and m be an integer greater than or equal to 2 . G is said to be m -permutable if every product of m elements can be reordered at least in one way. We prove that, if G has a centre of finite index z , then G is ( 1 + [ z / 2 ] ) -permutable. More bounds are given on the least m such that G is m -permutable.

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. ...

Automorphisms and derivations of a Fréchet algebra of locally integrable functions

F. Ghahramani, J. McClure (1992)

Studia Mathematica

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We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra L ¹ l o c of locally integrable functions on the half-line + . We show, among other things, that every automorphism θ of L ¹ l o c is of the form θ = φ a e λ X e D , where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and φ a is the dilation operator ( φ a f ) ( x ) = a f ( a x ) ( f L ¹ l o c , x + ). It is also shown that the automorphism group is a topological group with the topology of uniform convergence...

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 , , p 1 1 a 11 , 1 ) × × ( p r n r a r n r , , p r 1 a r 1 , 1 ) , where r > 1 , n i > 1 and p i j are distinct primes for i { 1 , 2 , , r } , j { 1 , 2 , , n i } .

Permutability of centre-by-finite groups

Brunetto Piochi (1989)

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

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Let G be a group and m be an integer greater than or equal to 2 . G is said to be m -permutable if every product of m elements can be reordered at least in one way. We prove that, if G has a centre of finite index z , then G is ( 1 + [ z / 2 ] ) -permutable. More bounds are given on the least m such that G is m -permutable.

Soluble Groups with Many Černikov Quotients

Silvana Franciosi, Francesco de Giovanni (1985)

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

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Si studiano i gruppi risolubili non di Černikov a quozienti propri di Černikov. Nel caso periodico tali gruppi sono tutti e soli i prodotti semidiretti H N con N p -gruppo abeliano elementare infinito e H gruppo irriducibile di automorfismi di N che sia infinito e di Černikov. Nel caso non periodico invece si riconduce tale studio a quello dei moduli a quozienti propri artiniani su un gruppo risolubile finito, e si fornisce una caratterizzazione di tali moduli.