This example is from Tapp's book "Matrix Groups for Undergraduates":
I'm trying to show that $G$ does not have the structure of a manifold, and thus cannot be a Lie group.
Any neighborhood of $I$ contains matrices of the form $g_{2\pi n}$ for arbitrarily large $n$. My idea is to try to show that this means that $I$ does not have a neighborhood diffeomorphic to a subset of the Euclidean space, but I cannot seem to make the argument work. Thanks!