Are Faithful Representations the Most Interesting Group Representations?

[BruceW]

Hello everyone!
I've been learning some basic group theory (I'm new to the subject). And I had a (hopefully) fairly simple question. OK, so a 'faithful representation' is defined as an injective homomorphism from some group G to the Automorphism group of some object. Let's call the object S for now. (I'm speaking more generally than just linear representations). So, to use some mathematical notation, we have $\phi : G \rightarrow Aut(S)$ where $\phi$ is the homomorphism.

Now, we can define another representation $\theta$ to be the surjective restriction of $\phi$ (meaning $\theta$ is essentially the same as $\phi$, but with the codomain restricted to the image of $\phi$). Therefore, $\theta$ is an injective surjective homomorphism, meaning it is an isomorphism. So I guess now, my question is: does my logic make sense? To summarize: for every faithful representation, the surjective restriction of that representation is an isomorphism.

Also, as a less concrete follow-up question: does this mean that faithful representations are in a sense somewhat 'boring' ? The image of a faithful representation is isomorphic to the original group, so it seems like we haven't done much by using this representation of the group. It seems to me that the interesting and potentially useful representations are the non-faithful ones... Does that sound about right?

Finally, one last question (sorry so many questions). I've seen the term 'group action' used a few times and it looks like it means the same thing as a representation. Have I understood this correctly? Or are they different things?

Many thanks,
bruce

 

[micromass]

Hello everyone!
I've been learning some basic group theory (I'm new to the subject). And I had a (hopefully) fairly simple question. OK, so a 'faithful representation' is defined as an injective homomorphism from some group G to the Automorphism group of some object. Let's call the object S for now. (I'm speaking more generally than just linear representations). So, to use some mathematical notation, we have $\phi : G \rightarrow Aut(S)$ where $\phi$ is the homomorphism.

Now, we can define another representation $\theta$ to be the surjective restriction of $\phi$ (meaning $\theta$ is essentially the same as $\phi$, but with the codomain restricted to the image of $\phi$). Therefore, $\theta$ is an injective surjective homomorphism, meaning it is an isomorphism. So I guess now, my question is: does my logic make sense? To summarize: for every faithful representation, the surjective restriction of that representation is an isomorphism.

Yes, that is correct.

Also, as a less concrete follow-up question: does this mean that faithful representations are in a sense somewhat 'boring' ? The image of a faithful representation is isomorphic to the original group, so it seems like we haven't done much by using this representation of the group. It seems to me that the interesting and potentially useful representations are the non-faithful ones... Does that sound about right?

I don't see why that makes faithful representations boring. For example, if you can find a faithful linear representation of a group, then you can basically represent the group as matrices. I think this is very interesting because you describe your group in other terminology, while you lose no information. Furthermore, that other description (for example as permutations or matrices) could be interesting to compute things about your group.

But in any case, whether a representation is interesting or not depends on the application you have in mind.

Finally, one last question (sorry so many questions). I've seen the term 'group action' used a few times and it looks like it means the same thing as a representation. Have I understood this correctly? Or are they different things?

Usually, it indeed means a representation like you defined, but one where $S$ is a set. So a group is then represented as bijective functions on a set.

 

[BruceW]

Yes, that is correct.

woah, super-fast reply. thanks micromass :)

I don't see why that makes faithful representations boring. For example, if you can find a faithful linear representation of a group, then you can basically represent the group as matrices. I think this is very interesting because you describe your group in other terminology, while you lose no information. Furthermore, that other description (for example as permutations or matrices) could be interesting to compute things about your group.

hmm I guess. But you could just choose certain matrices to be your group elements in the first place. Maybe using faithful representation is a nice way to acknowledge that the group of all invertible matrices is a 'natural' group, while your choice of a certain subgroup of these matrices (for example when you have finite cyclic group) is not going to be a nice natural choice (i.e. there are many choices which are different, but effectively do the same thing for our purposes).

But in any case, whether a representation is interesting or not depends on the application you have in mind.

yeah, that's true.

Usually, it indeed means a representation like you defined, but one where $S$ is a set. So a group is then represented as bijective functions on a set.

Ah, right. So a group action is a particular example of a representation. cool.

 

[WWGD]

Hello everyone!
I've been learning some basic group theory (I'm new to the subject). And I had a (hopefully) fairly simple question. OK, so a 'faithful representation' is defined as an injective homomorphism from some group G to the Automorphism group of some object. Let's call the object S for now. (I'm speaking more generally than just linear representations). So, to use some mathematical notation, we have $\phi : G \rightarrow Aut(S)$ where $\phi$ is the homomorphism.

Now, we can define another representation $\theta$ to be the surjective restriction of $\phi$ (meaning $\theta$ is essentially the same as $\phi$, but with the codomain restricted to the image of $\phi$). Therefore, $\theta$ is an injective surjective homomorphism, meaning it is an isomorphism. So I guess now, my question is: does my logic make sense? To summarize: for every faithful representation, the surjective restriction of that representation is an isomorphism.

Also, as a less concrete follow-up question: does this mean that faithful representations are in a sense somewhat 'boring' ? The image of a faithful representation is isomorphic to the original group, so it seems like we haven't done much by using this representation of the group. It seems to me that the interesting and potentially useful representations are the non-faithful ones... Does that sound about right?

Finally, one last question (sorry so many questions). I've seen the term 'group action' used a few times and it looks like it means the same thing as a representation. Have I understood this correctly? Or are they different things?

Many thanks,
bruce

For the 1st question:

Notice, by the first isomorphism theorem, if you have f: G-->H with trivial kernel {e}, then G/Ker(f)=G/{e}~ G ~ f(G). So you're right that this is an isomorphism into the image. Not a brilliant comment, but helps dot t's and cross-eyes.