What Does Up to Isomorphism Really Mean?

[Bachelier]

what does it really mean?

for instance if asked to list all abelian grps of order 12 up to iso, then do we include Z₁₂ or not?

 

[jgens]

When you are asked to classify all groups of order 12 up to isomorphism, this means two things:

• You need to create a list of groups such that every group of order 12 is isomorphic to one of these groups.
• None of you the groups you listed are isomorphic to each other.

 

[micromass]

Specifically: yes, you need to include $\mathbb{Z}_{12}$ or some group isomorphic to it. There are more groups of order 12 though...

 

[Bachelier]

Yes thanks I get it now.

Hence for ABELIAN groups of order 12 we have $\mathbb{Z}_{12}$ ≈ $\mathbb{Z}_{3}$ X $\mathbb{Z}_{4}$, and $\mathbb{Z}_{2}$ X $\mathbb{Z}_{2}$ X $\mathbb{Z}_{3}$

what would be $\mathbb{Z}_{2}$ X $\mathbb{Z}_{6}$ isomorphic to?

 

[micromass]

Yes thanks I get it now.

Hence for ABELIAN groups of order 12 we have $\mathbb{Z}_{12}$ ≈ $\mathbb{Z}_{3}$ X $\mathbb{Z}_{4}$, and $\mathbb{Z}_{2}$ X $\mathbb{Z}_{2}$ X $\mathbb{Z}_{3}$

what would be $\mathbb{Z}_{2}$ X $\mathbb{Z}_{6}$ isomorphic to?

In general: if gcd(a,b)=1, then $\mathbb{Z}_{ab}\cong \mathbb{Z}_a\times \mathbb{Z}_b$ (try to prove this!).

So we would have $\mathbb{Z}_2\times \mathbb{Z}_6\cong \mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_3$.

 

[Bachelier]

In general: if gcd(a,b)=1, then $\mathbb{Z}_{ab}\cong \mathbb{Z}_a\times \mathbb{Z}_b$ (try to prove this!).

So we would have $\mathbb{Z}_2\times \mathbb{Z}_6\cong \mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_3$.

we can construct an isomorphism ψ: from $\mathbb{Z}_{ab}$ to $\mathbb{Z}_a\times \mathbb{Z}_b$

such that ψ(x)= (x mod a, x mod b), it is a homomorphism, a surjection(for any y in the range, there exists an x congruent to z(mod ab) "the solution to the congruence system, and finally ψ is an injection because domain and codomain are finite sets with equal cardinality or order .

 

[micromass]

Yes, but where did you use that gcd(a,b)=1?? You do need this!

 

[Bachelier]

my computer just died.
we used the fact that a and b are coprime to show that the system of equations has a solution by CRT.

 

[micromass]

my computer just died.
we used the fact that a and b are coprime to show that the system of equations has a solution by CRT.

That seems alright!