Minimality of the inner automorphism group
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911935321079808 |
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| author | Peng, Dekui Shlossberg, Menachem |
| author_facet | Peng, Dekui Shlossberg, Menachem |
| contents | By [6], a minimal group $G$ is called $z$-minimal if $G/Z(G)$ is minimal. In this paper, we present the $z$-Minimality Criterion for dense subgroups with some applications to topological matrix groups. For a locally compact group $G$, let $\operatorname{Inn}(G)$ be the group of all inner automorphisms of $G,$ endowed with the Birkhoff topology. Using a theorem by Goto [14], we obtain our main result which asserts that if $G$ is a connected Lie group and $H\in\{G/Z(G), \operatorname{Inn}(G)\},$ then $H$ is minimal if and only if it is centre-free and topologically isomorphic to $\operatorname{Inn}(G/Z(G)).$ In particular, if $G$ is a connected Lie group with discrete centre, then $\operatorname{Inn}(G)$ is minimal. We prove that a connected locally compact nilpotent group is $z$-minimal if and only if it is compact abelian. In contrast, we show that there exists a connected metabelian $z$-minimal Lie group that is neither compact nor abelian. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_17065 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Minimality of the inner automorphism group Peng, Dekui Shlossberg, Menachem General Topology Group Theory Number Theory By [6], a minimal group $G$ is called $z$-minimal if $G/Z(G)$ is minimal. In this paper, we present the $z$-Minimality Criterion for dense subgroups with some applications to topological matrix groups. For a locally compact group $G$, let $\operatorname{Inn}(G)$ be the group of all inner automorphisms of $G,$ endowed with the Birkhoff topology. Using a theorem by Goto [14], we obtain our main result which asserts that if $G$ is a connected Lie group and $H\in\{G/Z(G), \operatorname{Inn}(G)\},$ then $H$ is minimal if and only if it is centre-free and topologically isomorphic to $\operatorname{Inn}(G/Z(G)).$ In particular, if $G$ is a connected Lie group with discrete centre, then $\operatorname{Inn}(G)$ is minimal. We prove that a connected locally compact nilpotent group is $z$-minimal if and only if it is compact abelian. In contrast, we show that there exists a connected metabelian $z$-minimal Lie group that is neither compact nor abelian. |
| title | Minimality of the inner automorphism group |
| topic | General Topology Group Theory Number Theory |
| url | https://arxiv.org/abs/2309.17065 |