Group algebras of free products of finite groups which are CS-rings

2018 ◽  
pp. 1850224
Author(s):  
Shoichi Kondo
Keyword(s):  
Free Product ◽  
Group Algebra ◽  
Finite Groups ◽  
Dihedral Group ◽  
Group Algebras ◽  
Free Products ◽  
Cyclic Groups ◽  
Algebra Over A Field ◽  
Cs Rings

This paper proves that the infinite dihedral group is only such a free product of two finite groups that its group algebra over a field [Formula: see text] is a CS-ring in case the orders of two groups are not zero in [Formula: see text]. Furthermore, it is shown that the group algebra of any free product of two finite cyclic groups does not satisfy the condition [Formula: see text].

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.

A Universal Coefficient Decomposition for Subgroups Induced by Submodules of Group Algebras

1997 ◽  
Vol 40 (1) ◽  
pp. 47-53 ◽  
Author(s):  
Manfred Hartl
Keyword(s):  
Group Algebra ◽  
Prime Power ◽  
Decomposition Theorem ◽  
Group Algebras ◽  
Group Rings ◽  
Prime Power Order ◽  
Relative Dimension ◽  
Theoretical Methods ◽  
Cyclic Groups ◽  
Coefficient Ring

AbstractDimension subgroups and Lie dimension subgroups are known to satisfy a ‘universal coefficient decomposition’, i.e. their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the ‘universal’ coefficient rings given by the cyclic groups of infinite and prime power order. Here this fact is generalized to much more general types of induced subgroups, notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary N-series, as well as certain common generalisations of these which occur in the study of the former. This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi, Parmenter and Seghal), to all additive subgroups of group rings. This is possible by using homological instead of ring theoretical methods.

A note on commutators in free products

1954 ◽  
Vol 50 (2) ◽  
pp. 178-188 ◽  
Author(s):  
H. B. Griffiths
Keyword(s):  
Topological Group ◽  
Free Product ◽  
Commutator Subgroup ◽  
Infinite Sequence ◽  
Natural Topology ◽  
Free Products ◽  
General Sequence ◽  
Cyclic Groups ◽  
The Continuum

In (1), Higman introduces the unrestricted free product of a set of groups, and gives it a natural topology. When this set is an infinite sequence of free cyclic groups he denotes by F their unrestricted free product; we shall denote by K the product for a general sequence {Gn} and, throughout the paper, we assume that each Gn is non-trivial and countable. Higman proves as an incidental result that the commutator subgroup [F, F] is not closed in the topological group F, and the first object of this note is to generalize this result to K. From this, the more interesting deduction immediately follows that K is never equal to L = [K, K]. Indeed, we prove in fact that the cardinal of L and its index in K are both c, the cardinal of the continuum.

scholarly journals The palindromic width of a free product of groups

2006 ◽  
Vol 81 (2) ◽  
pp. 199-208 ◽  
Author(s):  
Valery Bardakov ◽  
Vladimir Tolstykh
Keyword(s):  
Free Product ◽  
Free Groups ◽  
Uniform Bound ◽  
Free Products ◽  
Cyclic Groups ◽  
Similar Fact ◽  
Products Of Groups ◽  
Free Product Of Groups ◽  
Verbal Subgroups ◽  
Product Of Groups

AbstractPalindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups G has infinite palindromic width, provided that G is not the free product of two cyclic groups of order two (Theorem 2.4). This means that there is no uniform bound k such that every element of G is a product of at most k palindromes. Earlier, the similar fact was established for non-abelian free groups. The proof of Theorem 2.4 makes use of the ideas by Rhemtulla developed for the study of the widths of verbal subgroups of free products.

scholarly journals Anisotropic random walks on free products of cyclic groups, irreducible representations and idempotents of C*reg(G)

1992 ◽  
Vol 128 ◽  
pp. 95-120 ◽  
Author(s):  
Gabriella Kuhn
Keyword(s):  
Harmonic Analysis ◽  
Homogeneous Space ◽  
Free Product ◽  
Dual Space ◽  
Von Neumann Algebra ◽  
Point Of View ◽  
Irreducible Representations ◽  
Free Products ◽  
Von Neumann ◽  
Cyclic Groups

Let be the free product of q + 1 copies of Zn+1 and let denote its Cayley graph (with respect to aj, 1 ≤ j ≤ q + 1). We may think of G as a group acting on the “homogeneous space” , This point of view is inspired by the case of SL2(R) acting on the hyperbolic disk and is developed in [FT-P] [I-P] [FT-S] [S] (but see also [C]).Since G is a group we may investigate some classical topics: the full (reductive) C* algebra, its dual space, the regular Von Neumann algebra and so on. See [B] [P] [L] [V] and also [H]. These approaches give results pointing up the analogy between harmonic analysis on these groups and harmonic analysis on more classical objects.

The Structure of the Unit Group of the Group Algebra

2011 ◽  
Vol 54 (2) ◽  
pp. 237-243 ◽  
Author(s):  
Leo Creedon ◽  
Joe Gildea
Keyword(s):  
Finite Field ◽  
Group Ring ◽  
Group Algebra ◽  
Dihedral Group ◽  
Unit Group ◽  
Cyclic Groups ◽  
Split Extensions

AbstractLet RG denote the group ring of the group G over the ring R. Using an isomorphism between RG and a certain ring of n×n matrices in conjunction with other techniques, the structure of the unit group of the group algebra of the dihedral group of order 8 over any finite field of chracteristic 2 is determined in terms of split extensions of cyclic groups.

scholarly journals PROPER ACTIONS OF AUTOMORPHISM GROUPS OF FREE PRODUCTS OF FINITE GROUPS

2005 ◽  
Vol 15 (02) ◽  
pp. 255-272 ◽  
Author(s):  
YUQING CHEN ◽  
HENRY H. GLOVER ◽  
CRAIG A. JENSEN
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Free Products ◽  
Proper Actions

If G is a free product of finite groups, let Σ Aut 1(G) denote all (necessarily symmetric) automorphisms of G that do not permute factors in the free product. We show that a McCullough–Miller and Gutiérrez–Krstić derived (also see Bogley–Krstić) space of pointed trees is an [Formula: see text]-space for these groups.

PROFINITE TOPOLOGIES IN FREE PRODUCTS OF GROUPS

2004 ◽  
Vol 14 (05n06) ◽  
pp. 751-772 ◽  
Author(s):  
LUIS RIBES ◽  
PAVEL ZALESSKII
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Finitely Generated ◽  
Free Products ◽  
Products Of Groups

Let [Formula: see text] be a nonempty class of finite groups closed under taking subgroups, quotients and extensions. We consider groups G endowed with their pro-[Formula: see text] topology, and say that G is 2-subgroup separable if whenever H and K are finitely generated closed subgroups of G, then the subset HK is closed. We prove that if the groups G1 and G2 are 2-subgroup separable, then so is their free product G1*G2. This extends a result to T. Coulbois. The proof uses actions of groups on abstract and profinite trees.

scholarly journals On free decompositions of verbally closed subgroups in free products of finite groups

2017 ◽  
Vol 20 (5) ◽  
Author(s):  
Andrey M. Mazhuga
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Necessary Conditions ◽  
Free Products

AbstractFor a subgroup of a free product of finite groups, we obtain necessary conditions (on its Kurosh decomposition) for it to be verbally closed.

scholarly journals Length functions on hypercentral groups

1985 ◽  
Vol 28 (3) ◽  
pp. 303-304
Author(s):  
David L. Wilkens
Keyword(s):  
Abelian Group ◽  
Free Product ◽  
Dihedral Group ◽  
Value Function ◽  
Length Function ◽  
Soluble Groups ◽  
Absolute Value ◽  
Easy Consequence ◽  
Cyclic Groups ◽  
A Chain

In [6] the structure of any real valued length function on an abelian group G is determined. It is shown there, in Theorem 6.1., that such a length function is an extension of a non-Archimedean length function l1 on N by an Archimedean length function l2 on H=G/N. Any non-Archimedean length function is given by a chain of subgroups, as described in [5], and following from results of Nancy Harrison [2], the length l2 is essentially the absolute value function on a subgroup of R. In the situation above if N≠G then N is a subgroup of G whose elements have bounded lengths. In this paper we show that it is an easy consequence of techniques developed in [1] that this result can be extended to hypercentral groups, thus determining the structure of any length function in this case. We point out that the result does not extend to soluble groups. The infinite dihedral group D∞ is soluble. However if D∞ is regarded as a free product of two cyclic groups of order 2 and is given the length function associated with a free product, as described by Lyndon [3], then N is not a subgroup of D∞, and the lengths of its elements are unbounded.

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