# Math Help - Cardinality of Subgroup H n K

1. ## Cardinality of Subgroup H n K

Hi,

Given that $H \leq G$ and $K \leq G$.How do I show that $\mid H \cap K\mid = gcd(\mid H\mid,\mid K\mid)$?

2. This isn't true. For example, take $G = H \times K$ with $H \cong K$. Then $|(H \times e_K) \cap (e_H \times K)| = 1$ but $gcd(|K|, |H|) = |K| = |H|$...

It is, however, true that $|H \cap K|$ divides $gcd(|H|, |K|)$, and this is because it is a subgroup of both H and of K, so it's order must divide both |H| and |K|.