Understanding the Group Structure A Group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element. The operation satisfies four conditions 


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*Closure

*Associativity

*Existence of Identity

*Existence of Inverse
Intuitively I understand the purpose of Closure and Associativity Property. But I'm not getting the intuition behind Identity and Inverse. Whats the purpose of having these elements in a group.
I searched everywhere and find only common definition which are straight forward. But nobody discuss why it's important. Can anybody explain it to me.

A group describes the symmetries of some other mathematical object. Consider the square:

Associativity: If I rotate the square 90$^{\circ}$ clockwise twice ($RR)$ and then flip it ($F$), I should get the same thing as rotating it 90$^{\circ}$ once, and then performing the second rotation and flip afterwards. That is, $(RR)F=R(RF)$.
Closure: Applying any two symmetries to the square still produces the square (with a potentially different vertex labeling) again. For example applying $RRF$ gives another symmetry of the square above, but the vertices are now labeled (going clockwise) $4,\ 3,\ 2,\ 1$.
Inverses: Simply put, if I apply a symmetry to the square, I should be able to undo that operation to get back to what we had. Following the example above, in order to undo $RRF$ above, we need to flip it again, and then rotate it twice. This is because flipping it across the axis twice does not change the square, and rotating it 90$^{\circ}$ 4 times also doesn't change the labeling of the vertices. So: $(RRF)(FRR) = R^2F^2R^2 = R^4 = I$. This brings us to:
Identity: not changing the vertices of the above is the simplest way to procude a symmetry.
The notion of a group encodes all of these concepts into it's structure.  
