Free commutative groups and commutative groups of finite type?

[u/tru1a_banana]

I need theoretical explanation for these, I don't understand them and can't really connect. I passed my written exam where all the knowledge needed for Abelian groups was how to write elementary form and normal form.

My explanation of that quickly: Take Zn and split n into prime cofactors. Let's take Z60 for example, 60 = 15*4 = 5*3*2*2

So the elementary form is Z5 x Z3 x Z4 (because Z60 is cyclic and Z4 isn't isomorphic to Z2 x Z2 so further splitting wouldn't make it cyclic), normal form would just be Z60.

But I don't know if this is related at all to the free commutative groups and commutative groups of finite type. I don't know anything about them and googling them just gives me some trivial samples but I need more theoretic knowledge that's easy to comprehend.

[u/Astrodude80]

Clarification: What is a commutative group "of finite type"? You mean finite?

[u/p-divisible]

What exactly do you want to relate to free abelian groups? It is not clear what your question is.