subgroups of finite cyclic group

Let n be the order of a finite cyclic groupMathworldPlanetmath G.  For every positive divisor ( m of n, there exists one and only one subgroupMathworldPlanetmathPlanetmath of order m of G.  The group G has no other subgroups.

Proof.  If g is a generatorPlanetmathPlanetmathPlanetmath of G and  n=mk,  then gk generates the subgroup gk, the order of which is equal to the order of gk, i.e. equal to m.  Any subgroup H of G is cyclic (see entry).  If  |H|=m,  then H must have a generator of order m; thus apparently  H=g±k=gk.

Title subgroups of finite cyclic group
Canonical name SubgroupsOfFiniteCyclicGroup
Date of creation 2013-03-22 18:57:13
Last modified on 2013-03-22 18:57:13
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Theorem
Classification msc 20A05