Why is this group a Lie group?

Question

I'm reading an article in which the author proves a Theorem about topological groups (this is not an important theorem and it appears in the Appendix, however I would still like to understand it). The author claim (without proving) the following

Let $G_1,...,G_5$ be five compact abelian groups such that $G_1/G_2$ is finite, $G_2/G_3$ is a torus, $G_3/G_4$ is either finite or the circle group and $G_4/G_5$ is finite then $G_1/G_5$ is a Lie group.

Any Ideas why this is true?

Thanks!

Answer 1

Note that you can replace all of your groups by $H_i=G_i/G_5$.

By Hilbert's fifth problem a locally compact topological manifold group is necessarily a Lie group (a finite extension or an extension by a circle certainly gives a topological group). The abelian case was solved considerably earlier; see there.