Identity: there exists one unique element e such that for any a it holds that e⋆a = a⋆e = a. This e is the same no matter what a you pick.

Invertibility: given any a, there is b such that a⋆b = b⋆a = e. For two different a's you get two different b's.

As example let G be a set of all strings and the operation is appending. There is element e == "". For any string a it holds that if you append empty string to any side, you get the original string back. However there is no inverse. You can not append two string to form an empty string. Apparently, strings with appending do not form a group.

Group example: let G be a set of all integers and the operation is addition. Then e is 0, as for all a it holds that a + 0 = 0 + a = a, and for each a you can find -a such that a + (-a) = (-a) + a = 0.

The integers >= 0 have 0 as the identity: 0 + n = n + 0 = n. But they don't have inverses, because e.g. there's no positive number n such that n + 1 = 0.

I think you mean the naturals.

People disagree about whether 0 is natural though. Better not to use the word at all :D

I wanted to avoid an argument about whether 0 is a natural number or not.

OP said "integers >= 0". Not much to "think" here.

It's about where "e", the identity element, is placed - in the "identity" rule it is one of the terms of the operation (e * a, a * e) while in the "invertibility" rule it's the result of the operation.

If you want an example, think Z (integers) and "+": there is an element of Z, namely zero (0), where the identity rule tells us that x + 0 = 0 + x = x and the invertibility rule says that for any x in Z, there is an y in z such that x + y = 0

Think of identity as neutral (not good, not bad). Add neutral to good or bad and they remain the same.

Invertibility is existence of Yin for a Yang, superhero for a supervillain, north pole for a south and combining them creates a neutral.