subgroups with coprime orders

If the orders of two subgroupsMathworldPlanetmathPlanetmath of a group are coprime (, the identity elementMathworldPlanetmath 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 GH is a subgroup of both G and H.  By Lagrange’s theorem, |GH| divides both |G| and |H| and consequently it divides also  gcd(|G|,|H|)  which is 1.  Therefore  |GH|=1,  whence the intersection contains only the identity element.

Example.  All subgroups


of order 2 of the symmetric groupMathworldPlanetmathPlanetmath 𝔖3 have only the identity element (1) common with the sole subgroup


of order 3.

Title subgroups with coprime orders
Canonical name SubgroupsWithCoprimeOrders
Date of creation 2013-03-22 18:55:58
Last modified on 2013-03-22 18:55:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 20D99
Related topic Gcd
Related topic CycleNotation