2
$\begingroup$

I am currently doing self-study on profinite groups and I'm stuck trying to prove the following lemma.

If a topological group $G$ is compact and totally disconnected, then the open normal subgroups of $G$ intersect in the trivial subgroup $\{\,1_{G}\,\}$.

While I hope that mere hints will allow me to see how to prove this, I think I may need someone to just spell-it-out for me. I've been racking my brain as to how I could prove this and I've made little progress. Either way, any help would be much appreciated.

$\endgroup$
  • $\begingroup$ See An equivalent definition of the profinite group $\endgroup$ – Martin Brandenburg Mar 19 '13 at 0:26
  • 1
    $\begingroup$ @MartinBrandenburg I have read that post several times, and while I see that it is related to my question, I still don't see how it helps me. $\endgroup$ – David K. Mar 19 '13 at 0:30
  • $\begingroup$ It gives you immediately the answer since Hausdorff spaces are accessible. $\endgroup$ – Damien L Mar 19 '13 at 1:05
2
$\begingroup$

Hint: In a compact totally disconnected group $G$, the family of open invariant subgroups forms a local base at the neutral element of $G$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.