Question

Let G be a group and H and K be subgroups of G of orders p and q, respectively. Prove that if p and q are relatively prime, then H intersection K = {e}.

Answer 1

Suppose x belongs to the intersection H and K.
Let k be order of x in G, so x^k=e.
Since x belongs to H and K, we have that k should dividep and q, and so k divides the greatest common divisor (p,q)=1.
Hence k=1, and so x = x^1 =e.