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 


*

*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 huge motivation is the field $\Bbb R$ of real numbers, with which you are probably very familiar. So familiar, in fact, that I bet you completely overlook the importance of identities and inverses. Here, a field is essentially two groups tied together via the distributive property. 
In $\Bbb R$, we have addition and multiplication that get along so nicely that $a\cdot (b + c) = a\cdot b + a\cdot c$. The real numbers form an additive group $(\Bbb R, +)$, while the nonzero real numbers form a multiplicative group $(\Bbb R \setminus \{0\}, \cdot)$.
In $\Bbb R$, we like inverses because they let us solve equations. For example, to solve $2x+3 = 9$, we might start by adding $-3$, the additive inverse of $3$, to both sides:
\begin{align*}2x + \underbrace{3 + (-3)}_\text{inverses} &= 9 + (-3) \\
2x + 0 &= 6.
\end{align*}
We normally suppress writing the $+0$ part, but it's important to realize that additive inverses add up to the additive identity, $0$. Now the property of the additive identity says that adding $0$ doesn't actually change anything, and that our equation is in fact equivalent to $2x = 6$. 
Now we know what $2x$ is, but we really want to know what $x$ is. We need to "undo" multiplying $x$ by $2$. We'll multiply by the multiplicative inverse of $2$ then, $2^{-1} = \frac12$:
\begin{align*}
\underbrace{2^{-1}\cdot 2}_\text{inverses}x &= 2^{-1}\cdot 6 \\
1x &= 3
\end{align*}
where we know, of course, that $1x = x$, since multiplying by $1$ doesn't change anything (just like adding $0$ doesn't either). Another way to phrase this is to say that $1$ is the multiplicative identity.

In general, we like invertible processes. Math is full of operations that change things, and invertible operations - those with an inverse in the group-theoretic sense - are operations that can be undone reliably.
But this brings up the question: Given our vague description of invertible operations and "undoing" things, what does it mean to say that an operation is invertible; how do we know if we really have "undone" a certain operation? This is formalized by having identity elements/operations, those group elements that truly "do nothing" (like multiplying by $1$, or adding $0$). We can easily characterize operations that are inverses as those that, when performed in succession, amount to the identity, or "do nothing" operation.
