subgroups with coprime orders

If the orders of two subgroups of a group are coprime (http://planetmath.org/Coprime), the identity element is the only common element of the subgroups.

Proof.  Let $G$ and $H$ be such subgroups and $|G|$ and $|H|$ their orders.  Then the intersection $G\cap H$ is a subgroup of both $G$ and $H$.  By Lagrange’s theorem, $|G\cap H|$ divides both $|G|$ and $|H|$ and consequently it divides also  $\gcd (|G|,|H|)$  which is 1.  Therefore  $|G\cap H|=1$,  whence the intersection contains only the identity element.

Example.  All subgroups

$$\{(1),(12)\},\{(1),(13)\},\{(1),(23)\}$$

of order 2 of the symmetric group ${𝔖}_3$ have only the identity element $(1)$ common with the sole subgroup

$$\{(1),(123),(132)\}$$

of order 3.