Independence of Vector Space Axioms

[jgens]

Homework Statement

Determine whether the commutativity of (V,+) is independent from the remaining vector space axioms.

Homework Equations

N/A

The Attempt at a Solution

I am having a really hard time with this problem. Off the top of my head I could not think of any way to prove commutativity using the other axioms. On the other hand, I cannot think of any non-abelian groups (V,+) with a map R x V -> V that satisfies all the desired properties.

If someone could get me pointed in the right direction, it would be appreciated.

Edit: I should clarify that the underlying field is R

 

[micromass]

What can you deduce from

$$2\cdot (x+y)=2\cdot x+2\cdot y$$

which is one of the axioms.