A simple question about representation of a group

I am taking a course that mentions that sometime we would like to look at a group $G$ as a group of matrices. From another course I took a while ago I remember that this is called a representation.

I tried looking on Wikipedia, but didn't found a definition, the course I'm taking now is somewhat unformal and said that a representation of a group $G$ is a homomorphism of groups: $\phi:G \to GL_{n}(\mathbb{R})$ or $\phi:G \to GL_{n}(\mathbb{C})$.

Is this the general case ? or maybe we can take any ring ? (any field ?)

This course also claimed (with no proof) :"..then $Im(\phi)\simeq GL_{n}(\mathbb{R})$ " (or $GL_{n}(\mathbb{C}$ in the case of representation to $\mathbb{C}$). Can someone explain this ? (what if $\phi\equiv0$ ?

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\sim produces $\sim$; \equiv produces $\equiv$. – joriki Apr 16 '12 at 20:45
@joriki thanks, can I produce "=" with $\sim$ above ? – Belgi Apr 16 '12 at 20:46
I don't think I've ever seen that used. You can use \simeq to produce $\simeq$, though. – joriki Apr 16 '12 at 20:47
its cong $\cong$ – Steven-Owen Apr 16 '12 at 20:49
How do you mean you tried looking on Wikipedia and didn't find a definition? Representation leads directly to Group representation, which has a section Definitions -- how is that not what you were looking for? – joriki Apr 16 '12 at 20:50

A linear representation of a group $G$ over a field $k$ is a group homomorphism $\rho : G \to GL(E)$ where $E$ is a $k$-vector space and $GL(E)$ is the group of invertible $k$-linear maps $E \to E$.

Now if the dimension of $E$ is finite, $GL(E)$ and $GL_n(E)$ are isomorphic.

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Typically representations are over vector spaces, so we have to use fields, usually $\mathbb{C}$. In fact its an interesting study to look at real rep'n that are restrictions of complex ones and more interestingly are the complex rep'n induced by real rep'n.

I am not sure what is meant by "..then Im(ϕ)=GLn(R)" because that is most certainly not the case with finite groups. If you are taking a more advanced class on Lie Algebra then sure that can be the case but it's certainly not always true. As you point out, you can have the trivial rep'n where $\phi$ is the zero-map.

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en.wikipedia.org/wiki/Modular_representation_theory But of course, as mathematicians, everything must generalize and thus you could do things over rings. – Steven-Owen Apr 16 '12 at 21:10

What you've said above does generalize. For a finite group $G$ we take the group ring $k[G]$ and look at modules over that group. Maschke's Theorem tells us that if $k$ is algebraically closed the characteristic of $k$ does not divide the order of the group, then $k[G]$ is semisimple; that is, it is a direct sum of matrix rings over $k$, and the modules of such a ring $k[G]$ are precisely direct sums of vector spaces these matrices act on. So if we look at how $g\in k[G]$ acts on a module, we get a linear transformation of a vector space, ie. a matrix (which depends on the choice of basis, of course). This is equivalent to specifying a homomorphism from $G$ into the endomorphism ring (ring of matrices) of that vector space.

If $G$ is not finite or the characteristic of your field divides the order of the group, you can still define a representation as a homomorphism of a group into a matrix group. There is also a close generalization of what I have written above for compact topological groups.

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