Question

Prove that z is isomorphic to nz where n is not 0.

Answer 1

The answer depends a bit on what exactly do you mean by Z. Do you mean integers? Integers with addition? Integers with addition and multiplication? The more properties you add, the longer your proof needs to be. (I'm assuming that nZ is the set 0,±n,±2n,... plus whatever additional properties

In general, to show that two things are isomorphic, you need to show that there's an invertible function from one to the other that preserves the properties you are looking for. If you are treating Z,nZ as sets, there's no additional properties, so you just need to show that an invertible function F:Z↦nZ exists. If you want it to preserve addition, then you need to show that F(a+b)=F(a)+F(b), for all a,b∈Z as well. Similarly for multiplication, but for completeness you should also probably show it distributes over addition as well.

For a given n, it's easy to show that F(a)=na meets the bill for invertibility. na∈nZ, and F−1(na)=a∈Z clearly exists.

Showing that addition is preserved is almost as easy:

F(a+b)=n(a+b)=na+nb=F(a)+F(b)

Unfortunately, this F does not seem to work for multiplication:

F(a)F(b)=nanb=n(nab)=nF(ab)≠F(ab)