Isomorphisms between cyclic groups? (stupid question)

[TaylorWatts]

Why is Z mod 2 x Z mod 3 isomorphic to Z mod 6 but Z mod 2 x Z mod 2 not isomorphic to Z mod 4?

 

[matt grime]

On what level do you want the 'why' explained? I suppose the 'real' answer is because 2 and 3 are coprime, but 2 is not coprime to 2.

This is either a consequence of the structure theorem for abelian groups, or an explanation of that theorem, depending on how you look at it.

We could look a bit deeper: suppose that G and H are cyclic, when is GxH cyclic? Well, suppose that we claim (g,h) is a cyclic generator of GxH. Well, g must be a generator of G with order p, say, and h a generator of H with order q, and (g,h) must have order pq. If I raise (g,h) to the power p, then it is (1,k) for some k in H, and in order for (g,h) to have order pq k must have order q, i.e. also be a generator for H.

Pulling out the important thing: H must have an element h of order so that h^p still has order q. But this is if and only if p is coprime to q (this is an elementary fact you may have learned already).