On existence of compact open normal subgroups of 0-dimensional groups

Lee, D. H.; Wu, T. S.

doi:10.1090/S0002-9939-1970-0268325-1

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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On existence of compact open normal subgroups of $0$-dimensional groups
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by D. H. Lee and T. S. Wu

Proc. Amer. Math. Soc. 26 (1970), 526-528

DOI: https://doi.org/10.1090/S0002-9939-1970-0268325-1

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Abstract:

Let $G$ be a locally compact $0$-dimensional group, and let $B(G)$ denote the set of all elements of $G$ whose conjugacy classes are relatively compact. We proved that the group $G$ has an open compact normal subgroup if and only if $B(G)$ is open in $G$.

References

Kenkichi Iwasawa, Topological groups with invariant compact neighborhoods of the identity, Ann. of Math. (2) 54 (1951), 345–348. MR 43106, DOI 10.2307/1969536
B. H. Neumann, Groups with finite classes of conjugate elements, Proc. London Math. Soc. (3) 1 (1951), 178–187. MR 43779, DOI 10.1112/plms/s3-1.1.178