- #1
happyg1
- 308
- 0
Homework Statement
Let H and K be subgroups of a finite group G with coprime indices. Prove that G=HK
Homework Equations
From a theorem we have, If |G
![Gah! :H :H](/styles/physicsforums/xenforo/smilies/arghh.png)
|G
![Gah! :H :H](/styles/physicsforums/xenforo/smilies/arghh.png)
![Gah! :H :H](/styles/physicsforums/xenforo/smilies/arghh.png)
|G
![Gah! :H :H](/styles/physicsforums/xenforo/smilies/arghh.png)
The Attempt at a Solution
I used the thoerem and I got
|G
![Gah! :H :H](/styles/physicsforums/xenforo/smilies/arghh.png)
![Gah! :H :H](/styles/physicsforums/xenforo/smilies/arghh.png)
but since G and H are of coprime index, (H intersect K=1),
So that I get
|G|=|G
![Gah! :H :H](/styles/physicsforums/xenforo/smilies/arghh.png)
if I let |G
![Gah! :H :H](/styles/physicsforums/xenforo/smilies/arghh.png)
That's where I am and I don't think I'm headed in the right direction.
pointers and clarification will be greatly appreciated.
CC