# 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

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

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.

Title subgroups with coprime orders SubgroupsWithCoprimeOrders 2013-03-22 18:55:58 2013-03-22 18:55:58 pahio (2872) pahio (2872) 8 pahio (2872) Theorem msc 20D99 Gcd CycleNotation