What is the difference between a vector space and a group?

[adartsesirhc]

I've taken a course in Linear Algebra, so I'm used to working with vector spaces. But now, I'm reading Griffith's Introduction to Elementary Particles, and it talks about groups having closure, an identity, an inverse, and being associative.

With the exception of commutativity (unless the group is abelian), and scalar multiplication, is a group the same thing as a vector space? If not, what's the difference?

 

[Phrak]

A group has axioms:

1. Closure
2. Associativity
3. An identity element
4. An inverse element

A vector space has axioms:

1. Associativity of addition u + (v + w) = (u + v) + w
2. Commutativity of addition v + w = w + v
3. Identity element of addition, v + 0 = v for all v ∈ V.
4. Inverse elements of addition w ∈ V, v + w = 0.

5. Distributivity of scalar multiplication with respect to vector addition a (v + w) = a v + a w
6. Distributivity of scalar multiplication with respect to field addition (a + b) v = a v + b v
7. Compatibility of scalar multiplication with field multiplication a (b v) = (ab) v [nb 3]

8. Identity element of scalar multiplication 1v = v, where 1 is the multiplicative identity in F

 

[malawi_glenn]

Axiom 4 on groups, clarification:

There exists ONE inverse element for EACH group element.

And one has:
The group multiplication can be "anything", e.g. number addition, number multiplication, matrix multiplication etc etc.

 

[Hurkyl]

There is an algebraic structure called a module. It has the same axioms as a vector space, except for one: a vector space requires that it's scalars form a field, whereas a module merely requires a ring. The definitional difference between the two is that fields have division, but rings don't. (e.g. the rational numbers are a field, but the integers are only a ring)

It turns out that every Abelian group is an integer module if you define multiplication by an integer in the obvious way:

0 * g = 0
(n+1) * g = (n * g) + g
(-n) * g = -(n * g)​

One can still do linear algebra with modules (and thus with Abelian groups), but it's more complex -- things aren't always as simple as with vector spaces.