concept between group and vector space, compare G/N with V/W When we considered factor groups G/N, we need N to be normal,but in vector space V/W, why W only be subspace?
 A: Vector addition is commutative, and so a vector space $V$ is an abelian group under addition. It follows that every subgroup of $V$ is a normal subgroup. In particular, every subspace $W\subset V$ is a normal subgroup of $V$ under addition, and so we can construct the quotient group $V/W$ (which also inherits an operation of scalar multiplication from $V$, making it a vector space).
A: We need the quotient object to inherit the properties of the objects we're looking at (if we want an interesting object to result at least), so we want quotient groups to be groups and quotient spaces to be subspaces.
In order for the set of cosets of a subgroup of a group to have a well-defined natural group structure which corresponds to the relation $[a][b]=[ab]$ we require that the subgroup be normal (as spelled at in any proof that the quotient by a normal subgroup is itself a group).
For vector spaces, if we consider the class of all cosets of a subspace, then not only is this a normal subgroup (normal because every vector space is an abelian group under addition) but it also inherits a corresponding linear structure from the original vector space.
There is a deeper answer to this question when you consider these objects to be objects in a suitable abelian category and then the quotients defined turn out to precisely be the quotient objects in this category, but this is probably beyond the scope of your background.
