If the orders of two subgroups of a group are 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 $$\{(1),\,(12)\},\quad \{(1),\,(13)\}, \quad \{(1),\,(23)\}$$ of order 2 of the symmetric group $\mathfrak{S}_3$ have only the identity element $(1)$ common with the sole subgroup $$\{(1),\,(123),\,(132)\}$$ of order 3.