In 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.

中文翻译：

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M-正态子群和非阿贝尔下连续拓扑群

在本文中，我们引入了m-正规子群的概念，并证明了连接局部紧群的m-正规子群和紧群的m-正规子群与李群密切相关。作为应用，我们扩展了 [D. Peng 和 W. He，下连续拓扑群，Topol。应用程序 265 (2019)] 通过为任意拓扑群的密集子群提供较低的连续性标准。结果表明，在取拓扑积的情况下，保留了较低的连续性。还表明，当且仅当每个非平凡有限子群的归一化器是有限的时，无限紧群是遗传下连续的。