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Math Help - subgroups and normal subgroups

  1. #1
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    subgroups and normal subgroups

    Hi :

    I need your help in proving these two statements:

    1) Let G be a finite group and let H and K be two subgroups of G such that [G:H]=p and [G:k]=q, where p and q are disinct primes. Prove that pq divides [G:H(intersection)k]

    2) Let G be a group all of whose subgroups are normal. If a,b are in G, prove that there is an integer k such that ab=b(a^k)
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  2. #2
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    Quote Originally Posted by fuzzy topology View Post
    Hi :

    I need your help in proving these two statements:

    1) Let G be a finite group and let H and K be two subgroups of G such that [G:H]=p and [G:k]=q, where p and q are disinct primes. Prove that pq divides [G:H(intersection)k]


    We have that H\cap K\subset H\,,\,K , so [G:H]\mid[G:H\cap K]\,\,\,and\,\,\,also\,\,\,[G:K]\mid[G:H\cap K] , so...

    2) Let G be a group all of whose subgroups are normal. If a,b are in G, prove that there is an integer k such that ab=b(a^k)


    Take the subgroup <a>\,\leq G , then for any b\in G\,,\,\,b^{-1}ab\in <a> (why?), so...

    Tonio

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