text: 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 $\displaystyle H \cap K = \{1\}$.
  
  I realize that this amounts to proving that $\displaystyle |H \cap K| = 1$, 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 $\displaystyle H \cap K = \{1\}$.
  
  I realize that this amounts to proving that $\displaystyle |H \cap K| = 1$, 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 $\displaystyle a\in H\cap K$ then the order of $\displaystyle a$ divides both $\displaystyle |H|$ and $\displaystyle |K|$. 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 $\displaystyle H \cap K$. 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 $\displaystyle H \cap K$. 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.