Why is $\prod_{p\in\mathbb{P}}\mathbb{Z}/p\mathbb{Z}$ not an open subgroup of a compactly generated simple group In my lecture we were given the following theorem and corollary.

Theorem: Let $G$ be a compactly generated, topologically simple group. Then the $lpc(G)≠\emptyset$ and $lpc(G)$ is finite. (where $lpc$ is the local prime content)
Corollary: $\prod_{n≥1}\mathbb{Z}/n\mathbb{Z}$ and $\prod_{p\in\mathbb{P}}\mathbb{Z}/p\mathbb{Z}$ can never be an open subgroup of a compactly generated simple group.

Unfortunately, we were not given a proof for this corollary. I would appreciate if anyone could help me understand why this is true.
Thanks in advance.
 A: It is a theorem of G. Willis that in a nondiscrete, compactly generated topologically simple locally compact group, no open subgroup is solvable. 
This is Theorem 2.2 in [W]. The proof is short and seems to be self-contained. The result about local prime content, obtained in the same paper, seems to be significantly harder.
Let me give the proof with open abelian subgroups as it is a bit simpler. Let $G$ be such a group, with an open abelian subgroup $U$.
Since $G$ is not discrete, $U\neq 1$, and hence $U$ normally generates $G$. Since $U$ is open and $G$ is compactly generated, this implies that there is $n$ and elements $g_1,\dots,g_n\in G$ such that $G$ is generated by $\bigcup_{i=1}^ng_iUg_i^{-1}$. Define $V=\bigcap_{i=1}^ng_iUg_i^{-1}$: it is open in $U$. Then $V$ being a subgroup of the abelian subgroup $g_iUg_i^{-1}$, it is centralized by it. Since this holds for each $i$, the generating assumption implies that $V$ is central in $G$. Since $G$ is topologically simple and non-abelian, $V=1$, and since $V$ is open and $G$ is nondiscrete, we have a contradiction. $\Box$
Actually, this proves that whenever a locally compact group has an open abelian subgroup, then either it has a normal open abelian subgroup, or it has an open subgroup that is normal and not compactly generated (and every open subgroup has an open subgroup of this form).
[W] G. Willis. Compact open subgroups in simple totally disconnected groups. Journal of Algebra Volume 312, Issue 1, 1 June 2007, Pages 405-417. Sciencedirect link.
