# Thread: subgroups and normal subgroups

1. ## 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)

2. Originally Posted by fuzzy topology
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 $\,\leq G$ , then for any $b\in G\,,\,\,b^{-1}ab\in $ (why?), so...

Tonio

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