Let H and K be subgroups of a finite group G such that gcdH

Let H and K be subgroups of a finite group G such that gcd(|H|, |K|) = 1. Show that |H intersection K| = 1.

Solution

Since HK H, hence by Lagrange’s theorem, |HK| divides |H|. Similarly, H K K and hence by Lagrange’s theorem |H K| divides |K|. Thus |H K| is a common divisor of |H| and |K|. Since gcd(|H|, |K|) = 1, it follows that |H K| = 1