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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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    Representations can be defined over any ring, but the complex case is simpler in many ways. There,'s absolutely no reason to think the morphism should be surjective in general (for example that is impossible for a finite group).2012-04-16

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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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    http://en.wikipedia.org/wiki/Modular_representation_theory But of course, as mathematicians, everything must generalize and thus you could do things over rings.2012-04-16
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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.