Show that G is profinite

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Let G be a compact group, H be an open subgroup of G. Show that if H is profinite, then G is also profinite.




Lemma to use as a hint is this:




Let G be a compact group and $ i in I$ be directed family of closed normal subgroups of G of finite index such that $cap N_i=1.$(i.e. intersection of them is 1). Then G is profinite.




I know that if H is open subgroup, then H is closed of finite index and since it is profinite, H is inverse limit of inverse limit systen of finite groups. Somehow I have to construct these $N_i$‘s from the closed subgroups that construct H. I also know that intersection of all open normal sungroups is 1. But I cant see the way to combine al of these. Any hint is welcomed.




topological-groups profinite-groups




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    up vote
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    down vote

    favorite













    Let G be a compact group, H be an open subgroup of G. Show that if H is profinite, then G is also profinite.




    Lemma to use as a hint is this:




    Let G be a compact group and $ i in I$ be directed family of closed normal subgroups of G of finite index such that $cap N_i=1.$(i.e. intersection of them is 1). Then G is profinite.




    I know that if H is open subgroup, then H is closed of finite index and since it is profinite, H is inverse limit of inverse limit systen of finite groups. Somehow I have to construct these $N_i$‘s from the closed subgroups that construct H. I also know that intersection of all open normal sungroups is 1. But I cant see the way to combine al of these. Any hint is welcomed.




    topological-groups profinite-groups




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite












      Let G be a compact group, H be an open subgroup of G. Show that if H is profinite, then G is also profinite.




      Lemma to use as a hint is this:




      Let G be a compact group and $ i in I$ be directed family of closed normal subgroups of G of finite index such that $cap N_i=1.$(i.e. intersection of them is 1). Then G is profinite.




      I know that if H is open subgroup, then H is closed of finite index and since it is profinite, H is inverse limit of inverse limit systen of finite groups. Somehow I have to construct these $N_i$‘s from the closed subgroups that construct H. I also know that intersection of all open normal sungroups is 1. But I cant see the way to combine al of these. Any hint is welcomed.




      topological-groups profinite-groups




      share|cite|improve this question












      Let G be a compact group, H be an open subgroup of G. Show that if H is profinite, then G is also profinite.




      Lemma to use as a hint is this:




      Let G be a compact group and $ i in I$ be directed family of closed normal subgroups of G of finite index such that $cap N_i=1.$(i.e. intersection of them is 1). Then G is profinite.




      I know that if H is open subgroup, then H is closed of finite index and since it is profinite, H is inverse limit of inverse limit systen of finite groups. Somehow I have to construct these $N_i$‘s from the closed subgroups that construct H. I also know that intersection of all open normal sungroups is 1. But I cant see the way to combine al of these. Any hint is welcomed.





      topological-groups profinite-groups





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Aug 1 at 8:06









      Burak

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