Proving Cyclic Property in Factor Groups

[Benzoate]

Homework Statement

Prove that a factor group of a cyclic group is cyclic

Homework Equations

The Attempt at a Solution

For a group to be cyclic, the cyclic group must contain elements that are generators which prodduced all the elements within that group . A factor group is based on the definition that G be a group and let H be a normal group subgroup of G: (aH)(bH).

I have no idea how to proved that a factor group is cyclic

 

[HallsofIvy]

First, if G is cyclic it is Abelian and so every subgroup is a normal subgroup. That means that, given any subgroup H of G, the set of all left cosets is a group, G/H, the "factor group".

To show that a group is cylic, you must show that any member of the group, other than the identity, e, is a generator of the group. Let a be a member of G that is NOT in H (if a is in H, then its left coset, AH, correspond to the identity in G/H). Let b be any other member of G. You need to show that a^nH= bH for some positive integer n. That will certainly be true if a^n= b.