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.

Groups abstract symmetries of a set, that is, bijections $X \to X$ of a set $X$:


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*Function composition is associative.

*Every bijection has an inverse, which is also a bijection.

*The identity map is a bijection.
Closure just allows us to consider sets of bijections that are smaller than the full group of all bijections $X \to X$.
All groups are groups of symmetries of a set: that's Cayley's theorem.
