Let $H$ be an (abstract) group. Does it exist a Lie group $G$ such that $H \leq G$? or equivalently, does it exist a monomorphism from $H$ to a Lie group $G$?
Thanks for the help.
2026-04-04 01:58:24.1775267904
Representation in Lie groups
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To expand on David's comment: no. The question doesn't quite make sense, because mentioning Lie groups means that you want to keep track of differentiable structure, i.e. consider continuous representations w.r.t some topology on $H$.
But OK, let's ignore the topology. Any Lie group embeds into $GL_n(\Bbb{C})$ for some $n$, so, as David said, you're asking for a faithful finite-dimensional representation. Any finite group admits such a representation (Cayley's theorem embeds $H$ into $S_n$ for some $n$, which has a faithful representation by permutation matrices). You don't have to go very far at all to find a counterexample for infinite groups though: it's simple to check that the group $(\Bbb{Z}/2\Bbb{Z})^\Bbb{N}$ of countably infinite sequences of binary digits doesn't have a faithful finite-dimensional representation.