Is a vector space a subset of an abelian group?

Question

As we know that given a vector space $E$, we can remove the scalar multiplication operator, then the elements of $E$ forms a group and commute with each others (we refer this commutative group to abelian group).

I am not 100% sure that if a vector space $E$ is a subset of the abelian group? Could anyone give me an answer and explanation please? Thanks in advance!

Answer 1

A vector space is composed of 4 things, a set $V$, a field $\Bbb F$, an operation $+$ and a scalar multiplication $\cdot$, written as $(V,\Bbb F,+,\cdot)$ here. It satisfies the following axioms

1. $+$ is closed
2. $+$ is commutative
3. $+$ has inverses
4. $+$ has an identity element
5. $\cdot$ is a function from $\Bbb F\times V$ to $V$
6. $\cdot$ is distributive over $+$
7. $\cdot$ has an identity
8. $\cdot$ is associative

The first 4 axioms means that $(V,+)$ is an abelian group, but notice that it's the same set, $V$ that we are using both for the abelian group and the vector space. The vector space adds a field which gives scalar multiplication that results in another element in $V$ and not in something else.