1. ## Relatively-prime-ordered subgroups

If H and K are subgroups of a group G and if |H| and |K| are relatively prime, prove that $H \cap K = \{1\}$.

I realize that this amounts to proving that $|H \cap K| = 1$, since the intersection of any family of subgroups is a subgroup. However, I don't know how where to go from here. Any help would be greatly appreciated!

2. Originally Posted by jstew
If H and K are subgroups of a group G and if |H| and |K| are relatively prime, prove that $H \cap K = \{1\}$.

I realize that this amounts to proving that $|H \cap K| = 1$, since the intersection of any family of subgroups is a subgroup. However, I don't know how where to go from here. Any help would be greatly appreciated!
Let $a\in H\cap K$ then the order of $a$ divides both $|H|$ and $|K|$. This forces the order to be one. Thus, what does it mean?

3. Ah, so |a|=1 which implies that a={1}. But a was chosen arbitrarily, so {1} is the only element in $H \cap K$. Is this correct?

4. Originally Posted by jstew
Ah, so |a|=1 which implies that a={1}. But a was chosen arbitrarily, so {1} is the only element in $H \cap K$. Is this correct?
Yes

5. Thank you for the quick, helpful replies! You rock!

6. Originally Posted by jstew
You rock!
I know. When I was a younger kid my mother used to tell me that I am the greatest person in the world. That I know everything. And that I am the most handsome.