Semicomplete Finite p-Groups

2011 ◽  
Vol 18 (spec01) ◽  
pp. 937-944 ◽  
Author(s):  
M. Shabani Attar
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Necessary Conditions ◽  
Abelian Groups ◽  
Inner Automorphism

Let G be a group and G' be its commutator subgroup. An automorphism α of G is called an IA-automorphism if x-1α (x) ∈ G' for each x ∈ G. The set of all IA-automorphisms of G is denoted by IA (G). A group G is called semicomplete if and only if IA (G)= Inn (G), where Inn (G) is the inner automorphism group of G. In this paper we characterize semicomplete finite p-groups of class 2, give some necessary conditions for finite p-groups to be semicomplete, and characterize semicomplete non-abelian groups of orders p4 and p5.

scholarly journals On the capability of groups

1998 ◽  
Vol 41 (3) ◽  
pp. 487-495 ◽  
Author(s):  
Graham Ellis
Keyword(s):  
Automorphism Group ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Integral Homology ◽  
The Third ◽  
Inner Automorphism

We show how the third integral homology of a group plays a role in determining whether a given group is isomorphic to an inner automorphism group. Various necessary conditions, and sufficient conditions, for the existence of such an isomorphism are obtained.

Semicomplete Finite p-Groups of Class 2

2016 ◽  
Vol 23 (04) ◽  
pp. 651-656
Author(s):  
M. Shabani Attar
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Inner Automorphism

Let G be a group and G' be its commutator subgroup. An automorphism α of a group G is called an IA-automorphism if x-1 α(x) ∈ G' for each x ∈ G. The set of all IA-automorphisms of G is denoted by IA (G). A group G is called semicomplete if and only if IA (G)= Inn (G), where Inn (G) is the inner automorphism group of G. In this paper we completely characterize semicomplete finite p-groups of class 2; we also classify all semicomplete finite p-groups of order pn (n ≤ 5), where p is an odd prime. This completes our work in 2011.

scholarly journals Transitive action on finite points of a full shift and a finitary Ryan’s theorem

2017 ◽  
Vol 39 (06) ◽  
pp. 1637-1667 ◽  
Author(s):  
VILLE SALO
Keyword(s):  
Automorphism Group ◽  
Commutator Subgroup ◽  
Turing Machines ◽  
Finite Support ◽  
Transitive Action ◽  
Group Invariant ◽  
Finite Set ◽  
Transitivity Property ◽  
Full Shift ◽  
Subshifts Of Finite Type

We show that on the four-symbol full shift, there is a finitely generated subgroup of the automorphism group whose action is (set-theoretically) transitive of all orders on the points of finite support, up to the necessary caveats due to shift-commutation. As a corollary, we obtain that there is a finite set of automorphisms whose centralizer is $\mathbb{Z}$ (the shift group), giving a finitary version of Ryan’s theorem (on the four-symbol full shift), suggesting an automorphism group invariant for mixing subshifts of finite type (SFTs). We show that any such set of automorphisms must generate an infinite group, and also show that there is also a group with this transitivity property that is a subgroup of the commutator subgroup and whose elements can be written as compositions of involutions. We ask many related questions and prove some easy transitivity results for the group of reversible Turing machines, topological full groups and Thompson’s  $V$ .

scholarly journals On almost finitely generated nilpotent groups

1996 ◽  
Vol 19 (3) ◽  
pp. 539-544 ◽  
Author(s):  
Peter Hilton ◽  
Robert Militello
Keyword(s):  
Nilpotent Group ◽  
Commutator Subgroup ◽  
Local Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Finitely Generated

A nilpotent groupGis fgp ifGp, is finitely generated (fg) as ap-local group for all primesp; it is fg-like if there exists a nilpotent fg groupHsuch thatGp≃Hpfor all primesp. The fgp nilpotent groups form a (generalized) Serre class; the fg-like nilpotent groups do not. However, for abelian groups, a subgroup of an fg-like group is fg-like, and an extension of an fg-like group by an fg-like group is fg-like. These properties persist for nilpotent groups with finite commutator subgroup, but fail in general.

GROUPS WHOSE NONNORMAL SUBGROUPS ARE METAHAMILTONIAN

2019 ◽  
Vol 102 (1) ◽  
pp. 96-103
Author(s):  
DARIO ESPOSITO ◽  
FRANCESCO DE GIOVANNI ◽  
MARCO TROMBETTI
Keyword(s):  
Positive Integer ◽  
Commutator Subgroup ◽  
Abelian Groups

If $\mathfrak{X}$ is a class of groups, we define a sequence $\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots ,\mathfrak{X}_{k},\ldots$ of group classes by putting $\mathfrak{X}_{1}=\mathfrak{X}$ and choosing $\mathfrak{X}_{k+1}$ as the class of all groups whose nonnormal subgroups belong to $\mathfrak{X}_{k}$. In particular, if $\mathfrak{A}$ is the class of abelian groups, $\mathfrak{A}_{2}$ is the class of metahamiltonian groups, that is, groups whose nonnormal subgroups are abelian. The aim of this paper is to study the structure of $\mathfrak{X}_{k}$-groups, with special emphasis on the case $\mathfrak{X}=\mathfrak{A}$. Among other results, it will be proved that a group has a finite commutator subgroup if and only if it is locally graded and belongs to $\mathfrak{A}_{k}$ for some positive integer $k$.

scholarly journals Foundations of topological racks and quandles

2016 ◽  
Vol 25 (03) ◽  
pp. 1640002 ◽  
Author(s):  
Mohamed Elhamdadi ◽  
El-Kaïoum M. Moutuou
Keyword(s):  
Automorphism Group ◽  
Cohomology Theory ◽  
Central Extensions ◽  
Continuous Cohomology ◽  
Inner Automorphism

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first step toward a general continuous cohomology theory for topological racks and quandles.

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