When do objects factorize uniquely?

[Gerenuk]

Integer number and groups have unique factorizations into irreducible parts.

In general, what are the abstract requirements for mathematical objects to factorize uniquely?

I suppose for this question to ask a binary operation has to be defined. So is this question only possible for elements of groups?

 

[CRGreathouse]

The specific term for this (more specific than group) is UFD: a unique factorization domain. I don't know of any easy characterizations of UFDs.

 

[robert Ihnot]

Isn't it basically just a case of unique factorization in an integral domain/communitive ring, similar to what is found with the integers, and called, "The Fundamental Theorem of Arithmetic"? A prime being a number, not 1, divisible only by 1 or by itself.

The main property of a prime being if p/ab, then p/a or p/b. This can be shown with the Euclidean Algorithm.
Proof: if p does not divide a, then there exists integers such that: 1 = pm+an. Thus b=bpm+ban. Thus p divides b.

Gerenuk: I suppose for this question to ask a binary operation has to be defined. So is this question only possible for elements of groups?

Every element of a group has an inverse. In such a case a divides b gives b times a inverse. What we want is a group under addition and multiplicatively an Integral Domain.