# 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 $\displaystyle H\cap K\subset H\,,\,K$ , so $\displaystyle [G:H]\mid[G:H\cap K]\,\,\,and\,\,\,also\,\,\,[G:K]\mid[G:H\cap K]$ , 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 $\displaystyle <a>\,\leq G$ , then for any $\displaystyle b\in G\,,\,\,b^{-1}ab\in <a>$ (why?), so...

Tonio

.

#### Search Tags

normal, subgroups 