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.

A note on the cohomology of metabelian groups

1985 ◽  
Vol 98 (3) ◽  
pp. 437-445 ◽  
Author(s):  
P. H. Kropholler
Keyword(s):  
Finite Rank ◽  
Metabelian Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Soluble Groups

The cohomology of finitely generated metabelian groups has been studied in a series of papers by Bieri, Groves, and Strebel [2, 3, 4]. In particular, Bieri and Groves [2] have shown that every metabelian group of type (FP)∞ is of finite rank, and so is virtually of type (FP). The purpose of the present paper is to provide a generalization of this result and to use our methods to prove the existence of a pathological class of finitely generated soluble groups.

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.

Dehn functions of finitely presented metabelian groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wenhao Wang
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Metabelian Group ◽  
Finitely Generated ◽  
Dehn Function ◽  
Metabelian Groups ◽  
Dehn Functions ◽  
Finitely Presented

Abstract In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.

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.

On the SL(2, C)-representation rings of free abelian groups

2018 ◽  
Vol 167 (02) ◽  
pp. 229-247
Author(s):  
TAKAO SATOH
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Automorphism Groups ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Groups ◽  
The Other ◽  
Subsequent Research ◽  
Representation Rings ◽  
Free Abelian Groups

AbstractIn this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

On the profinite topology on solvable groups

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Khadijeh Alibabaei
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Solvable Group ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finitely Generated Group ◽  
Hnn Extension ◽  
Profinite Topology

AbstractWe show that the wreath product of a finitely generated abelian group with a polycyclic group is a LERF group. This theorem yields as a corollary that finitely generated free metabelian groups are LERF, a result due to Coulbois. We also show that a free solvable group of class 3 and rank at least 2 does not contain a strictly ascending HNN-extension of a finitely generated group. Since such groups are known not to be LERF, this settles, in the negative, a question of J. O. Button.

On certain soluble groups

1969 ◽  
Vol 66 (1) ◽  
pp. 1-4 ◽  
Author(s):  
C. K. Gupta
Keyword(s):  
Normal Subgroups ◽  
Finitely Generated ◽  
Maximal Condition ◽  
Metabelian Groups ◽  
Soluble Groups ◽  
Finitely Generated Groups ◽  
The Law

In (2), Hall considered the question: for what varieties of soluble groups do all finitely generated groups satisfy max-n (the maximal condition for normal subgroups)? He has shown that the variety M of metabelian groups and more generally the variety of Abelian-by-nilpotent-of-class-c (c ≥ 1) groups has this property; whereas on the contrary, there are finitely generated groups in the variety V of centre-by-metabelian groups (i.e. defined by the law [x, y; u, v; z]) which do not satisfy max-n. One naturally raises the question: for what subvarieties of V do all finitely generated groups satisfy max-n?

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 OUTER AUTOMORPHISM GROUPS OF CERTAIN TREE PRODUCTS OF ABELIAN GROUPS

2008 ◽  
Vol 77 (1) ◽  
pp. 9-20 ◽  
Author(s):  
Y. D. CHAI ◽  
YOUNGGI CHOI ◽  
GOANSU KIM ◽  
C. Y. TANG
Keyword(s):  
Automorphism Groups ◽  
Abelian Groups ◽  
Outer Automorphism ◽  
Finitely Generated ◽  
Residually Finite ◽  
Outer Automorphism Groups

AbstractWe prove that certain tree products of finitely generated Abelian groups have Property E. Using this fact, we show that the outer automorphism groups of those tree products of Abelian groups and Brauner’s groups are residually finite.

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