Q: Locally compact topological group is Normal How can I prove directly that a locally compact topological group G is normal? 
I have done this by showing that every locally compact topological group is strongly Paracompact. But I could not prove it directly.

A: Hints
Start with a precompact symmetric open neighbourhood $U$ of the identity and form the subgroup $H = \cup_{n \in \mathbb N} U^n$. Show that $H$ is clopen and $\sigma$-compact and conclude that it's Lindelöf.
Now take an open covering $\mathcal V$ of $G$. Use that $H$ is Lindelöf to find a $\sigma$-locally finite open refinement $\mathcal W$ of $\mathcal V$.
Finishing up the proof of the result, you should know a theorem that says that preregular paracompact spaces are normal.

A: Hausdorfness is not required for the group. Let $H$ be a clopen $\sigma$-compact subgroup of $G$. $H$ exists and is normal. Thus $G$ is normal. This requires the knowledge that given $G$ a topological group and $H \subseteq G$ an open subgroup, $G$ is normal iff $H$ is normal. 
Proof of my last statement is fairly easy as well. $(\Rightarrow)$, $H$ is closed, done. $(\Leftarrow)$ Each coset of $H$ is homeomorphic to $H$ and clopen. Done
