Pure dense subgroups that are isomorphic to every pure subgroup supported by their socles

1983 ◽  
pp. 519-522 ◽  
Author(s):  
Doyle Cutler
Keyword(s):  
Pure Subgroup ◽  
Dense Subgroups

scholarly journals Free dense subgroups of holomorphic automorphisms

2015 ◽  
Vol 280 (1-2) ◽  
pp. 335-346 ◽  
Author(s):  
Rafael B. Andrist ◽  
Erlend Fornæss Wold
Keyword(s):  
Dense Subgroups ◽  
Holomorphic Automorphisms

Non-Lebesgue measurability of finite unions of Vitali selectors related to different groups

2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Venuste Nyagahakwa ◽  
Gratien Haguma
Keyword(s):  
Topological Group ◽  
Finite Union ◽  
Affine Transformations ◽  
Real Numbers ◽  
Group Isomorphism ◽  
Lebesgue Measurability ◽  
Dense Subgroups

In this paper, we prove that each topological group isomorphism of the additive topological group $(\mathbb{R},+)$ of real numbers onto itself preserves the non-Lebesgue measurability of Vitali selectors of $\mathbb{R}$. Inspired by Kharazishvili's results, we further prove that each finite union of Vitali selectors related to different countable dense subgroups of $(\mathbb{R}, +)$, is not measurable in the Lebesgue sense. From here, we produce a semigroup of sets, for which elements are not measurable in the Lebesgue sense. We finally show that the produced semigroup is invariant under the action of the group of all affine transformations of $\mathbb{R}$ onto itself.

Conjugately Pure Subgroup Problems

1974 ◽  
Vol 81 (2) ◽  
pp. 156-158 ◽  
Author(s):  
Bola O. Balogun
Keyword(s):  
Pure Subgroup

TEST MAP AND DISCRETENESS IN SL(2, ℍ)

2018 ◽  
Vol 61 (03) ◽  
pp. 523-533 ◽  
Author(s):  
KRISHNENDU GONGOPADHYAY ◽  
ABHISHEK MUKHERJEE ◽  
SUJIT KUMAR SARDAR
Keyword(s):  
Hyperbolic Space ◽  
Division Ring ◽  
Discreteness Criteria ◽  
Dense Subgroups ◽  
Real Quaternions ◽  
Real Hyperbolic Space

AbstractLet ℍ be the division ring of real quaternions. Let SL(2, ℍ) be the group of 2 × 2 quaternionic matrices $A={\scriptsize{(\begin{array}{l@{\quad}l} a & b \\ c & d \end{array})}}$ with quaternionic determinant det A = |ad − aca−1b| = 1. This group acts by the orientation-preserving isometries of the five-dimensional real hyperbolic space. We obtain discreteness criteria for Zariski-dense subgroups of SL(2, ℍ).

scholarly journals Some cases of preservation of the Pontryagin dual by taking dense subgroups

2011 ◽  
Vol 158 (14) ◽  
pp. 1836-1843 ◽  
Author(s):  
M.J. Chasco ◽  
X. Domínguez ◽  
F.J. Trigos-Arrieta
Keyword(s):  
Dense Subgroups

scholarly journals When are dense subgroups of LCA groups closed under intersections?

1994 ◽  
Vol 49 (1) ◽  
pp. 59-67
Author(s):  
M.A. Khan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Compact Abelian Group ◽  
Additive Group ◽  
Locally Compact Abelian Group ◽  
Torsion Subgroup ◽  
Dense Subgroup ◽  
Locally Compact ◽  
Dense Subgroups ◽  
Lca Groups

Let G be a nondiscrete locally compact Hausdorff abelian group. It is shown that if G contains an open torsion subgroup, then every proper dense subgroup of G is contained in a maximal subgroup; while if G has no open torsion subgroup, then it has a dense subgroup D such that G/D is algebraically isomorphic to R, the additive group of reals. With each G, containing an open torsion subgroup, we associate the least positive integer n such that the nth multiple of every discontinuous character of G is continuous. The following are proved equivalent for a nondiscrete locally compact abelian group G:(1) The intersection of any two dense subgroups of G is dense in G.(2) The intersection of all dense subgroups of G is dense in G.(3) G contains an open torsion subgroup, and for each prime p dividing the positive integer associated with G, pG is either open or a proper dense subgroup of G.Finally, we construct a locally compact abelian group G with infinitely many dense subgroups satisfying the three equivalent conditions stated above.

scholarly journals Questions and remarks on discrete and dense subgroups of Diff(I)

2014 ◽  
Vol 06 (04) ◽  
pp. 557-571 ◽  
Author(s):  
Azer Akhmedov
Keyword(s):  
Lie Groups ◽  
20Th Century ◽  
Discrete Subgroups ◽  
Infinite Dimensional ◽  
Group Diff ◽  
Dimensional Group ◽  
Recent Developments ◽  
Long Time ◽  
Dense Subgroups ◽  
The 19Th Century

In recent decades, many remarkable papers have appeared which are devoted to the study of finitely generated subgroups of Diff+([0, 1]) (see [8, 15, 16, 19–23, 29, 30, 39, 40] only for some of the most recent developments). In contrast, discrete subgroups of the group Diff+([0, 1]) are much less studied. Very little is known in this area especially in comparison with the very rich theory of discrete subgroups of Lie groups which has started in the works of F. Klein and H. Poincaré in the 19th century, and has experienced enormous growth in the works of A. Selberg, A. Borel, G. Mostow, G. Margulis and many others in the 20th century. Many questions which are either very easy or have been studied a long time ago for (discrete) subgroups of Lie groups remain open in the context of the infinite-dimensional group Diff+([0, 1]) and its relatives.

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