Finite 2-groups all of whose non-abelian subgroups are self-centralizing

2020 ◽  
pp. 2150219
Author(s):  
Lijian An ◽  
Haipeng Qu ◽  
Junqiang Zhang
Keyword(s):  
Abelian Subgroup ◽  
Abelian Subgroups

In this paper, we consider finite 2-groups having the property that every non-abelian subgroup contains its centralizer. Such groups are classified completely up to isomorphism.

scholarly journals On infinite groups in which all abelian subgroups are locally cyclic

2021 ◽  
Author(s):  
Costantino Delizia ◽  
Chiara Nicotera
Keyword(s):  
Finite Groups ◽  
Abelian Subgroup ◽  
Periodic Case ◽  
Infinite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Periodic Groups ◽  
Abelian Subgroups

AbstractThe structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

Groups with few conjugacy classes of non-normal subgroups

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Maria Falco ◽  
Francesco Giovanni ◽  
Carmela Musella
Keyword(s):  
Abelian Subgroup ◽  
Soluble Group ◽  
Commutator Subgroup ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
Abelian Subgroups ◽  
Locally Soluble Group

AbstractThe structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.

Maximal Abelian Subgroups of the Symmetric Groups

1971 ◽  
Vol 23 (3) ◽  
pp. 426-438 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Symmetric Group ◽  
General Problem ◽  
Abelian Subgroup ◽  
Symmetric Groups ◽  
Specific Class ◽  
Maximal Abelian Subgroup ◽  
Number Of Generators ◽  
Abelian Subgroups ◽  
Almost All

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

scholarly journals Abelian p-subgroups of the general linear group

1970 ◽  
Vol 11 (3) ◽  
pp. 257-259 ◽  
Author(s):  
J. T. Goozeff
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
Finite Fields ◽  
Abelian Subgroup ◽  
General Linear Group ◽  
General Linear ◽  
Normal Abelian Subgroup ◽  
Abelian Subgroups

A. J. Weir [1] has found the maximal normal abelian subgroups of the Sylow p-subgroups of the general linear group over a finite field of characteristic p, and a theorem of J. L. Alperin [2] shows that the Sylow p-subgroups of the general linear group over finite fields of characteristic different from p have a unique largest normal abelian subgroup and that no other abelian subgroup has order as great.

scholarly journals On connected transversals to Abelian subgroups

1994 ◽  
Vol 49 (1) ◽  
pp. 121-128 ◽  
Author(s):  
Markku Niemenmaa ◽  
Tomas Kepka
Keyword(s):  
Cyclic Group ◽  
Abelian Subgroup ◽  
Loop Theory ◽  
Special Cases ◽  
Inner Mapping Group ◽  
Abelian Subgroups ◽  
Mapping Group ◽  
Finite Loop

In this paper we investigate the situation where a group G has an abelian subgroup H with connected transversals. We show that if H is finite then G is solvable. We also investigate some special cases where the structure of H is very close to the structure of a cyclic group. Finally we apply our results to loop theory and we show that if the inner mapping group of a finite loop Q is abelian then Q is centrally nilpotent.

Finite Groups in Which the Automizers of Some Abelian Subgroups are Trivial

2018 ◽  
Vol 25 (04) ◽  
pp. 701-712
Author(s):  
Pengfei Bai ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Group Of Automorphisms ◽  
Normal Abelian Subgroup ◽  
Abelian Subgroups ◽  
A Finite Group

If H is a subgroup of a finite group G, then the automizer AutG(H) of H in G is defined as the group of automorphisms of H induced by conjugation by elements of NG(H). A finite group G is called an NNC-group if for any non-normal abelian subgroup A, either [Formula: see text] or [Formula: see text]. In this paper, classifications of nilpotent NNC-groups and non-solvable NNC-groups are given. We also investigate the solvable NNC-groups and describe the structure of solvable NNC-groups.

scholarly journals CSA-groups and separated free constructions

1995 ◽  
Vol 52 (1) ◽  
pp. 63-84 ◽  
Author(s):  
D. Gildenhuys ◽  
O. Kharlampovich ◽  
A. Myasnikov
Keyword(s):  
Abelian Subgroup ◽  
Dihedral Group ◽  
Hyperbolic Groups ◽  
Maximal Abelian Subgroup ◽  
Hnn Extensions ◽  
Abelian Subgroups ◽  
Amalgamated Products ◽  
Free Constructions ◽  
Torsion Free

A group G is called a CSA-group if all its maximal Abelian subgroups are malnormal; that is, Mx ∩ M = 1 for every maximal Abelian subgroup M and x ∈ G − M. The class of CSA-groups contains all torsion-free hyperbolic groups and all groups freely acting on λ-trees. We describe conditions under which HNN-extensions and amalgamated products of CSA-groups are again CSA. One-relator CSA-groups are characterised as follows: a torsion-free one-relator group is CSA if and only if it does not contain F2 × Z or one of the nonabelian metabelian Baumslag-Solitar groups B1, n = 〈x, y | yxy−1 = xn〉, n ∈ Z ∂ {0, 1}; a one-relator group with torsion is CSA if and only if it does not contain the infinite dihedral group.

scholarly journals CommutativeC⁎-Algebras of Toeplitz Operators via the Moment Map on the Polydisk

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Mauricio Hernández-Marroquin ◽  
Armando Sánchez-Nungaray ◽  
Luis Alfredo Dupont-García
Keyword(s):  
Abelian Subgroup ◽  
Toeplitz Operators ◽  
The Other ◽  
Moment Map ◽  
Maximal Abelian Subgroup ◽  
Other Hand ◽  
Algebras Of Toeplitz Operators ◽  
Abelian Subgroups ◽  
The Moment

We found that in the polydiskDnthere exist(n+1)(n+2)/2different classes of commutativeC⁎-algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that theC⁎-algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutativeC⁎-algebra of Toeplitz operators.

Finite groups in which every non-abelian subgroup is a TI-subgroup or a subnormal subgroup

2019 ◽  
Vol 18 (08) ◽  
pp. 1950159
Author(s):  
Jiangtao Shi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Subgroup ◽  
Subnormal Subgroup ◽  
Cyclic Subgroups ◽  
Abelian Subgroups ◽  
A Finite Group

It is known that a TI-subgroup of a finite group may not be a subnormal subgroup and a subnormal subgroup of a finite group may also not be a TI-subgroup. For the non-abelian subgroups, we prove that if every non-abelian subgroup of a finite group [Formula: see text] is a TI-subgroup or a subnormal subgroup, then every non-abelian subgroup of [Formula: see text] must be subnormal in [Formula: see text]. We also show that the non-cyclic subgroups have the same property.

scholarly journals Centralizers of abelian subgroups in locally finite simple groups

1997 ◽  
Vol 40 (2) ◽  
pp. 217-225
Author(s):  
M. Kuzucuoǧlu
Keyword(s):  
Simple Group ◽  
Finite Length ◽  
Finite Simple Group ◽  
Abelian Subgroup ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Locally Finite ◽  
Odd Order ◽  
Non Linear ◽  
Abelian Subgroups

It is shown that, if a non-linear locally finite simple group is a union of finite simple groups, then the centralizer of every element of odd order has a series of finite length with factors which are either locally solvable or non-abelian simple. Moreover, at least one of the factors is non-linear simple. This is also extended to abelian subgroup of odd orders.

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