scholarly journals A GENERATING SET FOR THE AUTOMORPHISM GROUP OF A GRAPH PRODUCT OF ABELIAN GROUPS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250003 ◽  
Author(s):  
L. J. CORREDOR ◽  
M. A. GUTIERREZ
Keyword(s):  
Automorphism Group ◽  
London Math ◽  
Abelian Groups ◽  
Graph Products ◽  
Graph Product ◽  
Finitely Generated ◽  
Generating Set ◽  
Labeled Graph

We find a set of generators for the automorphism group Aut G of a graph product G of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup Aut ⋆ G defined in [Automorphisms of graph products of abelian groups, to appear in Groups, Geometry and Dynamics]. We follow closely the plan of M. Laurence's paper [A generating set for the automorphism group of a graph group, J. London Math. Soc. (2)52(2) (1995) 318–334].

scholarly journals RIGIDITY OF GRAPH PRODUCTS OF ABELIAN GROUPS

2008 ◽  
Vol 77 (2) ◽  
pp. 187-196 ◽  
Author(s):  
MAURICIO GUTIERREZ ◽  
ADAM PIGGOTT
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Graph Products ◽  
Vertex Group ◽  
Graph Product ◽  
Finitely Generated ◽  
Product Decomposition ◽  
Unique Decomposition ◽  
Product Of Groups

AbstractWe show that if G is a group and G has a graph-product decomposition with finitely generated abelian vertex groups, then G has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely generated abelian group and the graph satisfies the T0 property. Our results build on results by Droms, Laurence and Radcliffe.

On the existence of ergodic automorphisms in ergodic ℤd-actions on compact groups

2009 ◽  
Vol 30 (6) ◽  
pp. 1803-1816 ◽  
Author(s):  
C. R. E. RAJA
Keyword(s):  
London Math ◽  
Abelian Groups ◽  
Chain Condition ◽  
Compact Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Metrizable Group ◽  
Compact Abelian Groups ◽  
Descending Chain Condition

AbstractLet K be a compact metrizable group and Γ be a finitely generated group of commuting automorphisms of K. We show that ergodicity of Γ implies Γ contains ergodic automorphisms if center of the action, Z(Γ)={α∈Aut(K)∣α commutes with elements of Γ} has descending chain condition. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which, in particular, provide new proofs to the theorems of Berend [Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc.289(1) (1985), 393–407] and Schmidt [Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480–496].

Representations of holomorphs of group extensions with Abelian kernels

1975 ◽  
Vol 78 (3) ◽  
pp. 357-368 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Finite Rank ◽  
Automorphism Groups ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Group Extensions

This paper is devoted to the construction of faithful representations of the automorphism group and the holomorph of an extension of an abelian group by some other group, the representations here being homomorphisms into certain restricted parts of the automorphism groups of smallish abelian groups. We apply these to two very specific cases, namely to finitely generated metabelian groups and to certain soluble groups of finite rank. We describe the applications first.

scholarly journals Fixing Numbers of Graphs and Groups

10.37236/128 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Courtney R. Gibbons ◽  
Joshua D. Laison
Keyword(s):  
Automorphism Group ◽  
Abelian Groups ◽  
Symmetric Groups ◽  
Finite Abelian Groups ◽  
Finite Graphs ◽  
Labeled Graph ◽  
Distinguishing Number

The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $S$ such that only the trivial automorphism of $G$ fixes every vertex in $S$. The fixing set of a group $\Gamma$ is the set of all fixing numbers of finite graphs with automorphism group $\Gamma$. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label $G$ so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups.

On autocommutators and the autocommutator subgroup in infinite abelian groups

2018 ◽  
Vol 30 (4) ◽  
pp. 877-885
Author(s):  
Luise-Charlotte Kappe ◽  
Patrizia Longobardi ◽  
Mercede Maj
Keyword(s):  
Automorphism Group ◽  
Abelian Groups ◽  
Torsion Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Minimal Order ◽  
Autocommutator Subgroup ◽  
Free Abelian Groups ◽  
Torsion Free

Abstract It is well known that the set of commutators in a group usually does not form a subgroup. A similar phenomenon occurs for the set of autocommutators. There exists a group of order 64 and nilpotency class 2, where the set of autocommutators does not form a subgroup, and this group is of minimal order with this property. However, for finite abelian groups, the set of autocommutators is always a subgroup. We will show in this paper that this is no longer true for infinite abelian groups. We characterize finitely generated infinite abelian groups in which the set of autocommutators does not form a subgroup and show that in an infinite abelian torsion group the set of commutators is a subgroup. Lastly, we investigate torsion-free abelian groups with finite automorphism group and we study whether the set of autocommutators forms a subgroup in those groups.

scholarly journals Separability properties of automorphisms of graph products of groups

2016 ◽  
Vol 26 (01) ◽  
pp. 1-27
Author(s):  
Michal Ferov
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Technical Condition ◽  
Graph Products ◽  
Graph Product ◽  
Finitely Generated ◽  
Residually Finite ◽  
Products Of Groups ◽  
Text Version ◽  
Inner Automorphisms

We study properties of automorphisms of graph products of groups. We show that graph product [Formula: see text] has nontrivial pointwise inner automorphisms if and only if some vertex group corresponding to a central vertex has nontrivial pointwise inner automorphisms. We use this result to study residual finiteness of [Formula: see text]. We show that if all vertex groups are finitely generated residually finite and the vertex groups corresponding to central vertices satisfy certain technical (yet natural) condition, then [Formula: see text] is residually finite. Finally, we generalize this result to graph products of residually [Formula: see text]-finite groups to show that if [Formula: see text] is a graph product of finitely generated residually [Formula: see text]-finite groups such that the vertex groups corresponding to central vertices satisfy the [Formula: see text]-version of the technical condition then [Formula: see text] is virtually residually [Formula: see text]-finite. We use this result to prove bi-orderability of Torreli groups of some graph products of finitely generated residually torsion-free nilpotent groups.

Some Remarks on IA Automorphisms of Free Groups

1988 ◽  
Vol 40 (5) ◽  
pp. 1144-1155 ◽  
Author(s):  
J. McCool
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Free Group ◽  
Quotient Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Generating Set ◽  
Automorphisms Of Free Groups ◽  
Identity Automorphism ◽  
Image Position

Let An be the automorphism group of the free group Fn of rank n, and let Kn be the normal subgroup of An consisting of those elements which induce the identity automorphism in the commutator quotient group . The group Kn has been called the group of IA automorphisms of Fn (see e.g. [1]). It was shown by Magnus [7] using earlier work of Nielsen [11] that Kn is finitely generated, with generating set the automorphismsandwhere x1, x2, …, xn, is a chosen basis of Fn.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

2021 ◽  
pp. 1-36
Author(s):  
ARIE LEVIT ◽  
ALEXANDER LUBOTZKY
Keyword(s):  
Ergodic Theorem ◽  
Abelian Groups ◽  
Wreath Products ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Amenable Groups ◽  
Pointwise Ergodic Theorem ◽  
Lamplighter Group ◽  
Invariant Random Subgroups

Abstract We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

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