It is known that a compact connected Abelian Lie group is a torus. Is it also true for arbitrary compact connected Abelian group (not necessary a Lie group)?
I was not able to come up with the example of a compact connected Abelian, which is not a torus. All the examples I know are in fact Lie groups (such as real or complex elliptic curves).
There is a structural theorem for locally compact group which states that every locally compact group is a direct sum of $\mathbb R^n$, discrete group and a compact group. Howewer, this theorem has no information about the compact part.