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: The existence of an inverse (and  of a neutral element or identity with respect to which it is defined) is the fact that guarantees that we can solve an equation of the form: $a*x=b$ and find $x=a^{-1}b$.
This seems a good reason to deserve a special name ( and attention) to such a structure.
In other words: the existence of a neutral element $e$ and an inverse $a^{-1}$ make a group the simpler structure in which we can solve the equation:
$$
a*x=b \Rightarrow (a^{-1}*a)*x=a^{-1}*b \Rightarrow e*x=a^{-1}*b \Rightarrow x=a^{-1}*b
$$
A: Groups abstract symmetries of a set, that is, bijections $X \to X$ of a set $X$:


*

*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.
A: 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.
A: 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.  
A: This page might be worth a read
Think of a group exactly how that page describes, a set $G$ and a function $f:G\times G\rightarrow G$ such that
identity) There exists an $e\in G$ such that for all $g\in G$ we have $f(e,g)=f(g,e)=g$
inverse) Forall $x$ in $G$ there exists a $y\in G$ such that $f(x,y)=f(y,x)=e$
(and the other 2 properties, which you seem to get)
Only now rather than $f$ we have say, $+$, so $a+b:=f(a,b)$, or multiplicative, $xy=f(x,y)$ say. That's all that is going on.
A group is simply a function and a set with those 4 properties.
The integers, $\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$ and $+$ are an example of a group, $0$ is the identity, as $x+0=0+x=x$ for all $x$. The inverse (which is unique, as is the identity) is $-x$, for example $7+(-7)=0$.
Hope this helps.
This structure ensures you have an "element that does nothing (the identity)" and the inverse ensures "you can go back" if you will, this structure occurs in A LOT of places. One not obvious one is in topology with the fundamental group. I leave you to look that up. 
Gist:
Fundamental group - we say two loops (literally loops on a surface say) are homotopic if you can move (deform) one into the other. Two loops are equiv. if they are homotopic. The fundamental group is a group on these homotopic classes! I don't have time to explain it in greater detail. There are far more interesting groups than just number theory would suggest!
