Dense subgroup of torsion elements I need a help to find a counter example (or to prove) the following
Let $G$ be a compact abelian topological group. Assume there exists a dense subgroup $H\leq G$ such that every element of $H$ is of finite order then $G$ is a Lie group (equivalently, is it isomorphic to a direct product of a torus with a direct product of finite groups).
Note that the converse is true, every compact (abelian or non-abelian) Lie groups contains a dense subgroup of torsion elements
 A: There is a lot of counterexamples, but some efforts are needed in order to prove that a candidate group is indeed a counterexample. Also I may ask my teacher about them. The first family consists of zero dimensional candidates, which are closed subgroups of the product $\prod A_\alpha$ of a family $\{A_\alpha\}$ of finite abelian group whose periods are bounded in common. Candidates of bigger dimensions constitute the second family is so-called exotic tori,  introduced by Dikranjan and Prodanov, I guess in [DP], that is compact abelian groups $G$ such that torsion part $t(G)$ has-non zero intersection with each non-zero closed subgroup of $G$. For us is essential that by Proposition 3 from [DP], torsion part of an exotic torus is dense in it. From the other hand, the motivation for introducing the exotic tori in [DP] was the fact that they are precisely the completions of the torsion minimal abelian groups. The structure of exotic tori has been described in detail in [DP]. In particular, Proposition 2.5* describes the structure of $n$-dimensional exotic tori as follows. A compact abelian group $G$ of dimensional $n$ is an exotic torus if and only if there is a representation of $G$ as a projective limit $$G=\lim_{\longleftarrow} G_m$$  such that $G_0=\Bbb T^n$, the continuous homomorphisms $\sigma_m$ in the projective system $$G_0\stackrel{\sigma_1}\longleftarrow G_1\stackrel{\sigma_2}\longleftarrow G_2\cdots \stackrel{\sigma_m} \longleftarrow G_m\longleftarrow\cdots$$ are epimorphisms and $\ker\sigma_m $ is a compact $p_m$-group for each natural $m$. Thus, according to Example 1* of [DP], for each natural $n$ there is a continuum non-isomorphic connected $n$-dimensional exotic tori (so here we have continuum many connected counterexamples). For more details on exotic tore see [DP] or [DPS]. See also [CD], in particular, Theorem 4.1 and Remark 4.2(c). Also you can google for exotic torus Prodanov Dikranjan.
References 
[CD] W. W. Comfort, Dikran Dikranjan, On the poset of totally dense subgroups of compact groups, Topology Proceedings, 24, Summer 1999, 103–127. 
[DP] D. Dikranjan, I. Prodanov, A class of compact Abelian groups, Annuaire Univ. Soﬁa, Fac. Math. Méc. 70 (1975/76), 191–206.
[DPS] D. Dikranjan, I. Prodanov, L. Stoyanov, Topological groups. Characters, dualities and minimal group topologies, Marcel Dekker, Inc., New York, 1990.
