Geodesic
A Locally Compact Non Divisible Abelian Group Whose Character Group Is Torsion Free and Divisible
Canadian mathematical bulletin, Tome 56 (2013) no. 1, pp. 213-217

Voir la notice de l'article provenant de la source Cambridge University Press

It was claimed by Halmos in 1944 that if $G$ is a Hausdorff locally compact topological abelian group and if the character group of $G$ is torsion free, then $G$ is divisible. We prove that such a claim is false by presenting a family of counterexamples. While other counterexamples are known, we also present a family of stronger counterexamples, showing that even if one assumes that the character group of $G$ is both torsion free and divisible, it does not follow that $G$ is divisible.
DOI : 10.4153/CMB-2011-146-4
Mots-clés : 22B05 
Tausk, Daniel V. A Locally Compact Non Divisible Abelian Group Whose Character Group Is Torsion Free and Divisible. Canadian mathematical bulletin, Tome 56 (2013) no. 1, pp. 213-217. doi: 10.4153/CMB-2011-146-4
@article{10_4153_CMB_2011_146_4,
     author = {Tausk, Daniel V.},
     title = {A {Locally} {Compact} {Non} {Divisible} {Abelian} {Group} {Whose} {Character} {Group} {Is} {Torsion} {Free} and {Divisible}},
     journal = {Canadian mathematical bulletin},
     pages = {213--217},
     year = {2013},
     volume = {56},
     number = {1},
     doi = {10.4153/CMB-2011-146-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-146-4/}
}
TY  - JOUR
AU  - Tausk, Daniel V.
TI  - A Locally Compact Non Divisible Abelian Group Whose Character Group Is Torsion Free and Divisible
JO  - Canadian mathematical bulletin
PY  - 2013
SP  - 213
EP  - 217
VL  - 56
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-146-4/
DO  - 10.4153/CMB-2011-146-4
ID  - 10_4153_CMB_2011_146_4
ER  - 
%0 Journal Article
%A Tausk, Daniel V.
%T A Locally Compact Non Divisible Abelian Group Whose Character Group Is Torsion Free and Divisible
%J Canadian mathematical bulletin
%D 2013
%P 213-217
%V 56
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-146-4/
%R 10.4153/CMB-2011-146-4
%F 10_4153_CMB_2011_146_4

[1] [1] Armacost, D. L., The Structure of Locally Compact Abelian Groups. Monographs and Textbooks in Pure and Applied Mathematics 68. Marcel Dekker, 1981. Google Scholar

[2] [2] Clark, B., Dooley, M., and Schneider, V., Extending topologies from subgroups to groups. Topology Proc. 10 (1985), no. 2, 251–257. Google Scholar

[3] [3] Clark, B. and Schneider, V., The extending topologies. Internat. J. Math. Math. Sci. 7 (1984), no. 3, 621–623. Google Scholar | DOI

[4] [4] Clark, B. and Schneider, V., The normal extensions of subgroup topologiesProc. Amer. Math. Soc. 97 (1986), no. 1, 163–166. Google Scholar | DOI

[5] [5] Halmos, P. R., Comment on the real lineBull. Amer. Math. Soc. 50 (1944), 877–878. Google Scholar | DOI

[6] [6] Harrison, D. K., Infinite abelian groups and homological methods. Ann. of Math. 69 (1959), 366–391. Google Scholar | DOI

[7] [7] Morris, S. A., Pontryagin duality and the structure of locally compact abelian groups. London Mathematical Society Lecture Note Series 29. Cambridge University Press, Cambridge, 1977. Google Scholar

Cité par Sources :