Question #17576

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}.

Expert's answer

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.

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.

## Comments

## Leave a comment