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.

 

[blue_raver22]

I would say that whether faithful representations are the most interesting group representations depends on the specific context and purpose for which they are being used. Faithful representations have the advantage of preserving all the group structure and properties, which can be useful for certain applications. However, non-faithful representations can also be interesting and useful in their own right, as they may reveal different aspects of the group or allow for more flexibility in the mathematical manipulations. Therefore, it is not accurate to say that one type of representation is inherently more interesting than the other.

Regarding your logic, it seems sound, but it would be helpful to see a specific example to fully assess it. As for your follow-up question, it is true that the image of a faithful representation is isomorphic to the original group, but this does not necessarily mean that it is uninteresting. It simply means that the representation is preserving all the group structure, which can be useful for certain purposes.

In terms of group actions, they are closely related to representations but are not exactly the same thing. A group action is a generalization of a representation, where the group acts on a set rather than a mathematical object. However, in many cases, group actions and representations are used interchangeably, so it is important to clarify the specific context in which these terms are being used.

Overall, the concept of faithful representations and their usefulness depends on the specific application and context in which they are being used. It is important to consider both faithful and non-faithful representations in order to fully understand and utilize the group structure.