The difference between vector space and group

Solution 1:

As you may know, a vector space is a set $V$ together with operations $+:V \times V \to V$ and $\cdot:K \times V \to V$ that satisfy certain conditions, where $K$ is a field (take $K = \mathbb{R}$ for instance). Turns out that these conditions makes $(V,+)$ into an abelian group, a fancy term for a commutative group. This means that if you take $V$ and remove the scalar multiplication operator, the elements of $V$ forms a group and commute with each others.

Conversely, you can take an abelian group and try to turn it into a vector space by adding scalar multiplication on it. This additional structure comes in handy when you want to reason about lengths and angles of vectors in $V$. A geometric interpretation of this is that it stretches, or contracts, vectors $v \in V$ by a constant factor $\alpha \in K$. In fact, scalars scale vectors.

Without scalar multiplication, it is not possible think of any way of constructing a basis in a group $G$. If you think back of the definition of a basis, you will see that it involves a field. The definition of a vector space encapsulates the notion of basis in some sense.