Are vector spaces important? And integers, real numbers, complex numbers?

@Bernard: I am sorry if the way I pose the question seems dismissive. I am just trying to understand what is the intuition we get using the concept

One of the most prominent things in math is the idea of combining two things to make a new thing with similar properties. This gives a notion of a group. However, when this "combination" operation is commutative, i.e. $a$ combined with $b$ is the same as $b$ combined with $a$, we can talk about an $abelian$ group. This idea shows up in almost every field of math.

It is a structure found almost everywhere in mathematics, why, by the way, are not conceived to be useful, but for mathematicians to play with.

There are many reasons to study abelian groups, one reason you may wish to study them, or study their properties, is to understand the decomposition of more sophisticated objects, which have an underlying abelian group structure, eg vector spaces, modules, fields, division algebras, and the list goes on and on and on . . .

All of number theory can be thought of as the study of certain discrete abelian groups.

Would you find it unsettling if the bottom line of your bank statement was dependent on the order of the credits and debits? I would. So I am glad that the additive group of integers is abelian.

@MichaelMorrow: Why is the identity and inverse property important for defining the process of combining two things and make a new thing with similar properties?

@RobArthan It can depend on the order, though. At least in the US, banks can reorder your payments within a day. If your chronologically last payment was a large payment that overdrafted your account, the bank can reorder them so that that happened first, and so your other, smaller payments that day also incurs overdraft fees.

@Jim It isn't important, per se. But many combination processes turn out to have those concepts. So it makes sense to study combination processes that have them.

A point of notation: We talk about operations being commutative, not symmetric. Relations are symmetric.

@user4894 Yet its customary to use a small a in abelian. It just is. I have no idea why this is an exception.

@Arthur: in the scenario you describe the entries in the bank statement would differ in other respects than the order of the credits and debits.

@RobertShore A function $X^2\to X$ for some set $X$ is called symmetric, though, if swapping its arguments doesn't change the value. And binary operations (such as found in groups) are such functions. So it's not wrong. Just a little uncommon.

@user4894: I disagree with your algebra professor. If you become really famous, then your name becomes absorbed into the vocabulary.

This question seems very naive, but it seems a pity ot close it as being opinion based. I bet if soemone asked why residually finite groups were important then it would not be closed for that reason.

One of the reasons my degree is in mathematics is because it’s cool, without regard to usefulness. So one reason why we might study Abelian groups a lot is because they are fascinating.

Many questions about groups have much easier and better to remember answers , if the groups are abelian.

It's not clear to me if you're asking about groups, or abelian groups in particular.

We wouldn't have one of the most memorable math riddles: "What's purple and commutes?"