scholarly journals Residual finiteness of infinite amalgamated products of cyclic groups

2007 ◽  
Vol 208 (3) ◽  
pp. 1091-1097
Author(s):  
V. Metaftsis ◽  
E. Raptis
Keyword(s):  
Residual Finiteness ◽  
Cyclic Groups ◽  
Amalgamated Products

scholarly journals On the centre and residual finiteness of the automorphism group of a group ring

1987 ◽  
Vol 30 (2) ◽  
pp. 207-213 ◽  
Author(s):  
D. A. R. Wallace
Keyword(s):  
Automorphism Group ◽  
Group Ring ◽  
Infinite Order ◽  
The Other ◽  
Group Rings ◽  
Residual Finiteness ◽  
Original Proof ◽  
Residually Finite ◽  
Cyclic Groups ◽  
Trivial Centre

Let G be a group and let Aut(G) be its automorphism group. It is notorious that the properties of Aut (G) do not relate well to the properties of G, perhaps the only twogeneral results being that if G has a trivial centre then the same is true of Aut (G) [2, p.89] and Baumslag's theorem that if G is finitely generated and residually finite then Aut (G) is also residually finite [1, Theorem 1, p. 117]. In the paper we shall attempt tofind analogues of these results for therelationship between the properties of R(G), the group ring of G over a ring R, and the properties of Aut R(G), the automorphism of R(G). We prove that if R(G) has a trivial centre then Aut R(G) has a trivial centre. We establish the analogue, Theorem 2.3, of Baumslag's theorem by ring-theoretic methods; our original proof used properties of group rings, the present simplified proof we owe to the referee. As an example we calculate Aut ℤ(G) in the case that G is the direct product of two cyclic groups, one of infinite order and the other of order 5. This calculation will, it is hoped, give some indication of the difficulties in determining automorphisms of the group ring of an infinite group.

scholarly journals Residual finiteness of cyclic extensions of certain free products

1978 ◽  
Vol 19 (2) ◽  
pp. 197-203
Author(s):  
Benjamin Baumslag ◽  
Narain Gupta ◽  
Frank Levin
Keyword(s):  
Free Product ◽  
Cyclic Extension ◽  
Residual Finiteness ◽  
Free Products ◽  
Residually Finite ◽  
Cyclic Groups

We prove that a cyclic extension of a free product of finitep-groups is residually finite,pa fixed prime. The question of cyclic extensions of free products of arbitrary cyclic groups remains open.

On the minimal varieties of PI-algebras graded by finite cyclic groups

2021 ◽  
pp. 1-25
Author(s):  
Marcos Antônio da Silva Pinto ◽  
Viviane Ribeiro Tomaz da Silva
Keyword(s):  
Cyclic Groups

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 The residual finiteness of HNN-extensions and generalized free products of nilpotent groups: a characterization

1992 ◽  
Vol 53 (3) ◽  
pp. 408-420 ◽  
Author(s):  
E. Raptis ◽  
D. Varsos
Keyword(s):  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Free Products ◽  
Hnn Extensions ◽  
Base Group ◽  
Generalized Free Products

AbstractWe study the residual finiteness of free products with amalgamations and HNN-extensions of finitely generated nilpotent groups. We give a characterization in terms of certain conditions satisfied by the associated subgroups. In particular the residual finiteness of these groups implies the possibility of extending the isomorphism of the associated subgroups to an isomorphism of their isolated closures in suitable overgroups of the factors (or the base group in case of HNN-extensions).

scholarly journals Cubulating hyperbolic free-by-cyclic groups: The irreducible case

2016 ◽  
Vol 165 (9) ◽  
pp. 1753-1813 ◽  
Author(s):  
Mark F. Hagen ◽  
Daniel T. Wise
Keyword(s):  
Cyclic Groups

scholarly journals The Segal Conjecture for Cyclic Groups

1981 ◽  
Vol 13 (1) ◽  
pp. 42-44 ◽  
Author(s):  
Douglas C. Ravenel
Keyword(s):  
Cyclic Groups

scholarly journals On connectedness of power graphs of finite groups

2018 ◽  
Vol 17 (10) ◽  
pp. 1850184 ◽  
Author(s):  
Ramesh Prasad Panda ◽  
K. V. Krishna
Keyword(s):  
Finite Groups ◽  
Upper Bound ◽  
The Other ◽  
Number Of Components ◽  
Cyclic Groups ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Power

The power graph of a group [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of [Formula: see text]-groups are studied. In particular, the number of components of that of abelian [Formula: see text]-groups are determined.

scholarly journals Higher K -theory of group-rings of virtually infinite cyclic groups

2003 ◽  
Vol 325 (4) ◽  
pp. 711-726 ◽  
Author(s):  
Aderemi O. Kuku ◽  
Guoping Tang
Keyword(s):  
Group Rings ◽  
Cyclic Groups ◽  
K Theory
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