1. ## subgroups

2. Originally Posted by cowgirl123
H and K are finite subgroups of group G. |H| and |K| are relatively prime.
Prove (H intersects K) = {identity}
Let $\displaystyle x\in H \mbox{ and }x\in K$. Let $\displaystyle d = \mbox{ord}(x)$. Then by Lagrange's theorem $\displaystyle d|(|H|)$ and $\displaystyle d|(|K|)$. Since these are relatively prime it means $\displaystyle d=1$. So $\displaystyle x$ must be $\displaystyle \mbox{identity}$.
Q.E.D.

3. i'm not sure how this proves that the intersection of H and K is equal to the identity?
Doesnt this just prove that the identity is in the set?

4. Originally Posted by cowgirl123
i'm not sure how this proves that the intersection of H and K is equal to the identity?
Doesnt this just prove that the identity is in the set?
I have shown that if an element is common to both subgroups then its order must be 1. But the only element with order 1 is the identity element. Thus the identity element is the only possible element common to both.