On c#-normal subgroups infinite groups

2018 ◽  
Vol 13 (5) ◽  
pp. 1169-1178
Author(s):  
Huaquan Wei ◽  
Qiao Dai ◽  
Hualian Zhang ◽  
Yubo Lv ◽  
Liying Yang
Keyword(s):  
Normal Subgroups ◽  
Infinite Groups

A Problem of R. H. Fox

1981 ◽  
Vol 24 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Narain Gupta
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Group Rings ◽  
Normal Subgroups ◽  
Infinite Groups ◽  
Expository Article

The purpose of this expository article is to familiarize the reader with one of the fundamental problems in the theory of infinite groups. We give an up-to-date account of the so-called Fox problem which concerns the identification of certain normal subgroups of free groups arising out of certain ideals in the free group rings. We assume that the reader is familiar with the elementary concepts of algebra.

Spotting infinite groups

1999 ◽  
Vol 125 (1) ◽  
pp. 39-42 ◽  
Author(s):  
DANIEL ALLCOCK
Keyword(s):  
Lower Bounds ◽  
Normal Subgroups ◽  
Infinite Groups ◽  
Finitely Presented Group ◽  
Special Case ◽  
Finitely Presented

We generalize a theorem of R. Thomas, which sometimes allows one to tell by inspection that a finitely presented group G is infinite. Groups to which his theorem applies have presentations with not too many more relators than generators, with at least some of the relators being proper powers. Our generalization provides lower bounds for the ranks of the abelianizations of certain normal subgroups of G in terms of their indices. We derive Thomas's theorem as a special case.

Groups

2017 ◽  
Author(s):  
Matt Clay ◽  
Dan Margalit
Keyword(s):  
Group Theory ◽  
Electric Field ◽  
Geometric Group Theory ◽  
Normal Subgroups ◽  
Infinite Groups ◽  
Higher Dimensional ◽  
Group Presentations ◽  
Planar Shape ◽  
Group Of Symmetries

This chapter considers the notion of a group in mathematics. It begins with a discussion of the problem of determining the symmetry of an object such as a planar shape, a higher-dimensional solid, a group, or an electric field. It then describes every group as a group of symmetries of some object and shows what it means for a group to be a group of symmetries of an object. These ideas are at the very heart of geometric group theory, the study of groups, spaces, and the interactions between them. The chapter also examines infinite groups, homomorphisms and normal subgroups, and group presentations. A number of exercises are included.

scholarly journals On the residual and profinite closures of commensurated subgroups

2019 ◽  
Vol 169 (2) ◽  
pp. 411-432
Author(s):  
PIERRE–EMMANUEL CAPRACE ◽  
PETER H. KROPHOLLER ◽  
COLIN D. REID ◽  
PHILLIP WESOLEK
Keyword(s):  
Finite Groups ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Residually Finite ◽  
Finitely Generated Groups ◽  
Residually Finite Groups ◽  
Groups Acting On Trees ◽  
Polycyclic Groups

AbstractThe residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.

scholarly journals Coset posets of infinite groups

2020 ◽  
Vol 23 (4) ◽  
pp. 593-605
Author(s):  
Kai-Uwe Bux ◽  
Cora Welsch
Keyword(s):  
Finite Index ◽  
Normal Subgroups ◽  
Infinite Groups

AbstractWe consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

scholarly journals On F-Normal Subgroups of Finite Soluble Groups

1995 ◽  
Vol 171 (1) ◽  
pp. 189-203 ◽  
Author(s):  
A. Ballesterbolinches ◽  
K. Doerk ◽  
M.D. Perezramos
Keyword(s):  
Normal Subgroups ◽  
Soluble Groups

scholarly journals Normal subgroups of diffeomorphism and homeomorphism groups of ℝn and other open manifolds

2011 ◽  
Vol 31 (6) ◽  
pp. 1835-1847 ◽  
Author(s):  
PAUL A. SCHWEITZER, S. J.
Keyword(s):  
Normal Subgroup ◽  
Compact Manifold ◽  
Normal Subgroups ◽  
Open Manifold ◽  
Open Manifolds ◽  
Group Of Homeomorphisms ◽  
Homeomorphism Groups

AbstractWe determine all the normal subgroups of the group of Cr diffeomorphisms of ℝn, 1≤r≤∞, except when r=n+1 or n=4, and also of the group of homeomorphisms of ℝn ( r=0). We also study the group A0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with non-empty boundary, then the quotient of A0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.

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