Let $G$ be a non-Abelian infinite group. Can $G$ admit more than one (inequivalent) non-compact locally compact metrizable second countable topologies that make it a topological group?


Thank you.



Different locally compact metrizable second countable topologies on the same group
Turning my comment into an aswer: yes, it is possible for such a $G$ to have more than one topology with your requirements.


Consider $(\Bbb R,+)$ with its usual topology. As a group it is isomorphic to $(\Bbb C,+)$, so there is also a topology in which $(\Bbb R,+)$ is a topological group homeomorphic to $\Bbb R^2$, both topologies obviously satisfy your requirements.


The only issue is that $\Bbb R$ is Abelian, but that can be fixed by taking a direct product with any locally compact, second countable, metrizable, nonabelian topological group, since all of this properties are preserved under finite products.


