1. ## Help using lagrange....

Hi I have this problem, and I cant seem to get it. It is not for a grade just for study practice. Let me know if you can do it. Thanks...

Let G be a group, |G|= $p^{t}m$ for a p a prime and (p,m) =1 and H $\le$ G with |H|= $p^{t}$. If K $\le$ G with |K|= $p^{s}$ and HK = KH, then show K $\le$ H.

Need to show K is a subgroup of H. Thanks again.

2. Originally Posted by ElieWiesel
Hi I have this problem, and I cant seem to get it. It is not for a grade just for study practice. Let me know if you can do it. Thanks...

Let G be a group, |G|= $p^{t}m$ for a p a prime and (p,m) =1 and H $\le$ G with |H|= $p^{t}$. If K $\le$ G with |K|= $p^{s}$ and HK = KH, then show K $\le$ H.

Need to show K is a subgroup of H. Thanks again.
What are you having trouble with exactly? What points aren't clear? Do you see why it is significant that $(p,m)=1$?

3. Okay well maybe if I can explain what I have so far, maybe It will help.

HK= HK implies that HK $\le$ G

I also know that the |HK| = |H||K|/|H $\cap$ K|

I also know that |G| = |H| [G:H] which seems to imply that [G:H] = m by lagrange

Also I know that |HK| divides |G| by lagrange

Im thinking I have all the pieces put I dont know how to put them together right.