How can I prove that there exists only one subgroup of A4 of order 4 Size of A4 =12
So, there are either 1 or 3 2-sylow subgroups 
and there are either 1 or 4 3-sylow subgroups.
Suppose that there are 4 3-sylow subgroups. Then, there are 8 elements of order 3
Thus, 2-sylow subgroup must be unique.
However, this is case when I suppose that there are 3-sylow subgroups.
What If I suppose that there are 2-sylow subgroups and derive the result that there can't be 4 3-sylow subgroups?
Also, How can I find the subgroup of order 4? 
Do I have to look at every element in A4 and consider possible cases?

You know that the Klein 4 group V=(1,(12)(34),(13)(24),(14)(23))is a subgroup of order 4 of A4. It's a normal subgroup of order 4 of A4. Any automorphism of A4 must map an element of order 2 to an element of order 2, and since V contains only the 3 elements of order 2 and the idendity, then V is stable under any automorphism of A4,, and hence is a normal subgroup of A4. V is a normal 2 sylow subgroup of A4, and hence is a unique 2 sylow subgroup (order 4) of A4.
