Finding a normal subgroup H of Zmn of order m

[Rick Strut]

Homework Statement

Find a normal subgroup H of Z_mn of order m where m and n are positive integers. Show that H is isomorphic to Z_m.

Homework Equations

The Attempt at a Solution

I am honestly not even sure where to start. My initial thoughts were if Z_mn was isomorphic to Z_m x Z_n then I could find a subgroup H from that group. However, I discovered that Z_mn is isomorphic to Z_m x Z_n but the converse is not true. Any help would be appreciated.

Edit: If Z_mn is cyclic has an element of order mn say x. Then nx has order m. Let H=⟨nx⟩.
Now I just need to show that H is isomorphic to Z_m, by constructing an isomorphism.

 

[Dick]

Homework Statement

Find a normal subgroup H of Z_mn of order m where m and n are positive integers. Show that H is isomorphic to Z_m.

Homework Equations

The Attempt at a Solution

I am honestly not even sure where to start. My initial thoughts were if Z_mn was isomorphic to Z_m x Z_n then I could find a subgroup H from that group. However, I discovered that Z_mn is isomorphic to Z_m x Z_n but the converse is not true. Any help would be appreciated.

Edit: If Z_mn is cyclic has an element of order mn say x. Then nx has order m. Let H=⟨nx⟩.
Now I just need to show that H is isomorphic to Z_m, by constructing an isomorphism.

You are just working with numbers mod mn here. Just put x=1. Think about the set of numbers {0,n,2n,3n,...,(m-1)n} like in your edit. Can't you see a correspondence with {0,1,2,3,...,(m-1)}?

 

[Stephen Tashi]

I discovered that Zmn is isomorphic to Zm x Zn but the converse is not true.

What do you mean by that? Shouldn't you make an "If...then..." statement in order to speak of a converse?

Now I just need to show that H is isomorphic to Z_m, by constructing an isomorphism.

Instead of thinking about isomorphisms, you could think about a homomorphism and the "first isomorphism theorem" for groups. http://en.wikipedia.org/wiki/Isomorphism_theorem

 

[Rick Strut]

I do. However, I don't see how to construct a function based on this. I apologize for not being clear but the goal is to construct a bijective homorphic function. Then I can conclude that they are isomorphic.

 

[Dick]

I do. However, I don't see how to construct a function based on this. I apologize for not being clear but the goal is to construct a bijective homorphic function. Then I can conclude that they are isomorphic.

I would say try $\phi(kn \pmod {nm})=k \pmod m$ just from looking at it. Try to prove $\phi$ is an isomorphism.