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We can define two topologies on a group $G$ by considering all normal subgroups of finite index (resp. of index a finite power of $p$ - where $p$ is a prime) as basis of $1\in G$.

My questions: Under what conditions these two topologies coincide? What are the usual approaches to check whether they coincide in a given f.g. group?

To coincide is equivalent to the following. For each normal subgroup $N$ of finite index, there exists a normal subgroup $P$ of index a finite power of $p$ such that $P\subseteq N$.

So, a trivial situation is when each finite quotient of $G$ is a $p$-group.

Thanks in advance for any ideas! Explicit examples are very welcome too.

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Clearly, if $P\subseteq N$, then the index of $P$ in $G$ is divisible by that of $N$ in $G$. So your equivalent condition is satisfied if and only if each finite quotient of $G$ is a $p$-group. –  Alex B. Nov 20 '10 at 5:52
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