TITLE: Why are abelian groups of interest? What is their usefulness?
QUESTION [16 upvotes]: I am reading about Abelian groups
So apparently it is a set, with an associative binary operation, and identity element, an inverse operation and the binary operation must also be symmetric.
But it is not clear to me how they are useful. Trying to find why they are important it seems they arise as "additive structures" in various systems but this is too abstract for me.
Could someone give some less formal/more practical application or usages showing what is important about abelian groups (layman's terms basically)?

REPLY [9 votes]: The same question could really be asked about any abstract algebraic structure: why do we study fields or rings or groups or semigroups, etc.?  For that matter, why do we study abstract algebra at all?
As other answers have noted, algebraic structures are studied because they are useful abstractions.  For example, the reason why we study groups is because the definition of what counts as a group is:

broad enough that a lot of things in mathematics satisfy it, but
specific enough that we can prove quite a few useful theorems using only the group axioms, which means that these theorems automatically apply to everything that satisfies those axioms.

For example, the integers (and rationals and reals and even complex numbers) under addition, positive rational (and real) numbers under multiplication and invertible $n \times n$ matrices under matrix multiplication all satisfy the definition of a group.  This means that any theorems that we prove about groups in general apply to all of those systems, as different as they may seem on the surface.
And we study abelian groups because, as useful as groups are, it turns out that there are also quite a few useful theorems that we cannot prove just from the group axioms unless we also assume commutativity.  Adding the commutativity axiom thus allows us to prove more useful results — the tradeoff, of course, being that those results won't apply to groups that are not commutative, such as the group of $n \times n$ invertible matrices mentioned above.
This is a common situation in abstract algebra (and in many other areas of "abstract" mathematics): the more specific we make our abstractions, the more results we can prove, but the fewer things those results will apply to.  The trick here is to find abstractions that happen to hit a sweet spot where there are just enough axioms to prove useful results, but no unnecessary ones that we could remove and still prove essentially the same results.  Of course, in practice, there are many such "sweet spots" of varying specificity.  Experience has shown that groups are certainly one of them, while abelian groups are another.