The automorphism group of a finitely generated virtually abelian group

2016 ◽  
Vol 8 (1) ◽  
Author(s):  
Bettina Eick
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Automorphism Groups ◽  
Finitely Generated ◽  
Crystallographic Groups ◽  
Practical Algorithm

AbstractWe describe a practical algorithm to compute the automorphism group of a finitely generated virtually abelian group. As application, we describe the automorphism groups of some small-dimensional crystallographic groups.

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 Free Steiner triple systems and their automorphism groups

2014 ◽  
Vol 14 (02) ◽  
pp. 1550025 ◽  
Author(s):  
Alexander Grishkov ◽  
Diana Rasskazova ◽  
Marina Rasskazova ◽  
Izabella Stuhl
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Steiner Triple Systems ◽  
Finitely Generated ◽  
Combinatorial Structures ◽  
Triple Systems ◽  
Free Objects

The paper is devoted to the study of free objects in the variety of Steiner loops and of the combinatorial structures behind them, focusing on their automorphism groups. We prove that all automorphisms are tame and the automorphism group is not finitely generated if the loop is more than 3-generated. For the free Steiner loop with three generators we describe the generator elements of the automorphism group and some relations between them.

scholarly journals On automorphisms of finite Abelian p-groups

2008 ◽  
Vol 58 (4) ◽  
Author(s):  
Marek Golasiński ◽  
Daciberg Gonçalves
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Type K ◽  
Group A ◽  
Necessary And Sufficient ◽  
Torsion Part

AbstractLet A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part A p.For a finite abelian p-group A of type (k 1, ..., k n), simple necessary and sufficient conditions for an n × n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k 1, k 2) is analyzed.

scholarly journals Bounds on the Distinguishing Chromatic Number

10.37236/177 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Karen L. Collins ◽  
Mark Hovey ◽  
Ann N. Trenk
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Automorphism Group ◽  
Chromatic Number ◽  
Automorphism Groups ◽  
Upper Bounds ◽  
Underlying Graph ◽  
Minimum Number

Collins and Trenk define the distinguishing chromatic number $\chi_D(G)$ of a graph $G$ to be the minimum number of colors needed to properly color the vertices of $G$ so that the only automorphism of $G$ that preserves colors is the identity. They prove results about $\chi_D(G)$ based on the underlying graph $G$. In this paper we prove results that relate $\chi_D(G)$ to the automorphism group of $G$. We prove two upper bounds for $\chi_D(G)$ in terms of the chromatic number $\chi(G)$ and show that each result is tight: (1) if Aut$(G)$ is any finite group of order $p_1^{i_1} p_2^{i_2} \cdots p_k^{i_k}$ then $\chi_D(G) \le \chi(G) + i_1 + i_2 \cdots + i_k$, and (2) if Aut$(G)$ is a finite and abelian group written Aut$(G) = {\Bbb Z}_{p_{1}^{i_{1}}}\times \cdots \times {\Bbb Z}_{p_{k}^{i_{k}}}$ then we get the improved bound $\chi_D(G) \le \chi(G) + k$. In addition, we characterize automorphism groups of graphs with $\chi_D(G) = 2$ and discuss similar results for graphs with $\chi_D(G)=3$.

scholarly journals Finitely generated groups with virtually free automorphism groups

1995 ◽  
Vol 38 (3) ◽  
pp. 475-484 ◽  
Author(s):  
Martin R. Pettet
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Finitely Generated ◽  
Full Automorphism Group ◽  
Finitely Generated Groups ◽  
Special Cases ◽  
Central Quotient ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

It is shown that the full automorphism group of a finitely generated group G is virtually free if and only if the center Z(G) is finitely generated of torsion-free rank r at most two and, depending on the value of r, the central quotient G/Z(G) belongs to one of three precisely defined classes of virtually free groups. Some consequences and special cases are also discussed.

scholarly journals FREE GROUPS AND AUTOMORPHISM GROUPS OF INFINITE STRUCTURES

2014 ◽  
Vol 2 ◽  
Author(s):  
PHILIPP LÜCKE ◽  
SAHARON SHELAH
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Set Theory ◽  
Free Group ◽  
Automorphism Groups ◽  
Free Abelian Group ◽  
Free Groups ◽  
Large Cardinals ◽  
Free Objects ◽  
Models Of Set Theory

AbstractGiven a cardinal $\lambda $ with $\lambda =\lambda ^{\aleph _0}$, we show that there is a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank $2^\lambda $ is the automorphism group of a field of cardinality $\lambda $ whenever $\lambda $ is a cardinal with $\lambda =\lambda ^{\aleph _0}$. Moreover, we apply these techniques to show that consistently the assumption that $\lambda =\lambda ^{\aleph _0}$ is not necessary for the existence of a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. Finally, we use them to prove that the existence of a cardinal $\lambda $ of uncountable cofinality with the property that there is no field of cardinality $\lambda $ whose automorphism group is a free group of rank greater than $\lambda $ implies the existence of large cardinals in certain inner models of set theory.

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

scholarly journals HEYDE’S CHARACTERIZATION THEOREM FOR DISCRETE ABELIAN GROUPS

2010 ◽  
Vol 88 (1) ◽  
pp. 93-102 ◽  
Author(s):  
MARGARYTA MYRONYUK
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Real Line ◽  
Linear Form ◽  
Conditional Distribution ◽  
Random Variables ◽  
Characterization Theorem ◽  
Gaussian Distributions ◽  
Discrete Abelian Group

AbstractLet X be a countable discrete abelian group with automorphism group Aut(X). Let ξ1 and ξ2 be independent X-valued random variables with distributions μ1 and μ2, respectively. Suppose that α1,α2,β1,β2∈Aut(X) and β1α−11±β2α−12∈Aut(X). Assuming that the conditional distribution of the linear form L2 given L1 is symmetric, where L2=β1ξ1+β2ξ2 and L1=α1ξ1+α2ξ2, we describe all possibilities for the μj. This is a group-theoretic analogue of Heyde’s characterization of Gaussian distributions on the real line.

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.

On Commutator Length and Square Length of the Wreath Product of a Group by a Finitely Generated Abelian Group

2010 ◽  
Vol 17 (spec01) ◽  
pp. 799-802 ◽  
Author(s):  
Mehri Akhavan-Malayeri
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Finitely Generated ◽  
Commutator Length

Let W = G ≀ H be the wreath product of G by an n-generator abelian group H. We prove that every element of W′ is a product of at most n+2 commutators, and every element of W2 is a product of at most 3n+4 squares in W. This generalizes our previous result.

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