M‐normal subgroups and non‐abelian lower continuous topological groups

Abstract

AbstractIn this paper, we introduce the notion of m‐normal subgroups and show that the m‐normal subgroups of connect locally compact groups and that of compact groups are closely related to Lie groups. As applications, we extend Theorem 3.8 in [D. Peng and W. He, Lower continuous topological groups, Topol. Appl. 265 (2019)] by giving a lower continuity criterion for dense subgroups of arbitrary topological groups. It is shown that lower continuity is preserved under taking topological products. It is also shown that an infinite compact group is hereditarily lower continuous if and only if the normalizer of every non‐trivial finite subgroup is finite.