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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

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# Faithful representation

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