proof that all subgroups of a cyclic group are cyclic

The following is a proof that all subgroupsMathworldPlanetmathPlanetmath of a cyclic groupMathworldPlanetmath are cyclic.


Let G be a cyclic group and HG. If G is trivial, then H=G, and H is cyclic. If H is the trivial subgroup, then H={eG}=eG, and H is cyclic. Thus, for the of the proof, it will be assumed that both G and H are nontrivial.

Let g be a generatorPlanetmathPlanetmathPlanetmath of G. Let n be the smallest positive integer such that gnH.

Claim: H=gn

Let agn. Then there exists z with a=(gn)z. Since gnH, we have that (gn)zH. Thus, aH. Hence, gnH.

Let hH. Then hG. Let x with h=gx. By the division algorithmPlanetmathPlanetmath, there exist q,r with 0r<n such that x=qn+r. Thus, h=gx=gqn+r=gqngr=(gn)qgr. Therefore, gr=h(gn)-q. Recall that h,gnH. Hence, grH. By choice of n, r cannot be positive. Thus, r=0. Therefore, h=(gn)qg0=(gn)qeG=(gn)qgn. Hence, Hgn.

This proves the claim. It follows that every subgroup of G is cyclic. ∎

Title proof that all subgroups of a cyclic group are cyclic
Canonical name ProofThatAllSubgroupsOfACyclicGroupAreCyclic
Date of creation 2013-03-22 13:30:47
Last modified on 2013-03-22 13:30:47
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 10
Author Wkbj79 (1863)
Entry type Proof
Classification msc 20A05
Related topic ProofThatEverySubringOfACyclicRingIsACyclicRing