Vector Space as a Group I'm beginning to learn Abstract Algebra and my professor was talking of groups.
He said that Vector Spaces are Groups but not with Scalar Multiplication.
In Vector Spaces
1.The operation of Addition if associative
2.It has an identity element namely the zero vector
3.It has an inverse, the negative of the vector.
and plus it is commutative A+B=B+A which makes it an abelian group.
But these conditions do not change with the inclusion of scalar multiplication then why does he say forget scalar multiplication.
Is there something I'm missing.
 A: It's not that scalar multiplication makes a vector space less of a group - rather, it makes a vector space more than just a group. There's a pair of general terms (coming from logic, specifically model theory) which are useful here and which I'll define informally:

*

*A reduct of a structure is what you get by "forgetting" some of the original structure. For example, the integers with just addition is a reduct of the integers with both addition and multiplication.


*An expansion is the opposite: we take a structure and "add" some additional operations/relations/"stuff" to it. "$\mathcal{A}$ is an expansion of $\mathcal{B}$" is the same as "$\mathcal{B}$ is a reduct of $\mathcal{A}$."
Expanding a structure doesn't change any of its existing properties, it just broadens the range of properties we consider. For example, distributivity doesn't make sense if all we have is addition, but once we also have multiplication it's a thing we can meaningfully consider.
Sometimes taking a reduct doesn't actually lose any information. For example, consider the real numbers with addition, multiplication, and the ordering. The ordering is in fact definable from addition and multiplication alone: $x\le y$ iff $x+z^2=y$ for some $z$. So if I "forget" the ordering, I haven't really lost any information. By contrast, the scalar multiplication in a vector space is generally not redundant. So a vector space really is more than "just" a group.
