= Extension of a faithful representation from an open normal subgroup
Let $H$ be an <open normal subgroup> of a <compact topological group> $G$, and let $\tau:H\to\operatorname{GL}_m(\mathbb Z_p)$ be a continuous <faithful representation>. Choose representatives $t_j$ for the finite left <cosets> $G/H$. Write $gt_j=t_i h_{ij}$ and act on the $j$-th summand of $\bigoplus_j\mathbb Z_p^m$ by $\tau(h_{ij})$ into the $i$-th summand. The cocycle identity proves this is a continuous <group representation>. An element outside $H$ has a nontrivial coset permutation, while an element of $H$ acting trivially has $\tau(g)=1$ on the identity-coset summand. Hence the representation of $G$ is faithful and integral.
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