We know Lie groups are manifolds. Lie groups are governed by local Lie algebra generators and their commutation relations.
All Lie groups are manifolds.
But not all manifolds are Lie groups.
One can also consider Lie groups with additional finite group structures.
We can consider reductive Lie algebra and reductive Lie groups such as $U(1)^n=T^n$.
The above are familiar facts.
However, do we have any groups that have continuous structures but they are NOT Lie groups? If so, what are some examples?
For example, it is natural to consider some failed examples from manifolds, $S^1$ as $U(1)$, or $S^3$ as $SU(2)$, etc -- these are famous Lie groups as manifolds. But we also have more exotic manifolds (like $\mathbb{RP}^n$, $\mathbb{CP}^m$, $\mathbb{HP}^d$. Are there any continuous manifolds that are groups but not Lie groups? )