## representation of a finite group

1. The problem statement, all variables and given/known data

Prove that a representation of a finite group G is faithful if and only if its image is isomorphic to G.

2. Relevant equations

3. The attempt at a solution

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 Quote by syj 1. The problem statement, all variables and given/known data Prove that a representation of a finite group G is faithful if and only if its image is isomorphic to G. 2. Relevant equations 3. The attempt at a solution
What did you try already?? If you show us where you're stuck, then we'll know where to help...

 I am not very eloquent when it comes to proofs. So I am just going to lay out what I know. Let the representation be noted as F, and the image of G' if F is a faithful representation then ker{F}={1G} Can I conclude then by the first isomorphism theorem that G is isomorphic to G'? I know that for an "if and only if" proof there are two directions. If I can get the first direction of the proof, I can easily get the other direction.

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## representation of a finite group

 Quote by syj I am not very eloquent when it comes to proofs. So I am just going to lay out what I know. Let the representation be noted as F, and the image of G' if F is a faithful representation then ker{F}={1G} Can I conclude then by the first isomorphism theorem that G is isomorphic to G'? I know that for an "if and only if" proof there are two directions. If I can get the first direction of the proof, I can easily get the other direction.
Indeed, the first isomorphism theorem does the trick!!

 Ok, so is this enough: If f is faithful then ker{f}={1G} therefore by the first isomorphism theorem, G$\cong$G' If G$\cong$G' then by the first isomorphism theorem ker{f}={1G} therefore by the definition of a faithful representataion, f is faithful. it seems so plain. lol. too plain to be complete. but if it is, i am one happy girl ;)

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 Quote by syj If G$\cong$G' then by the first isomorphism theorem ker{f}={1G}
This is true (but only for finite groups), but you might want to explain in some more detail.

The rest is ok!

 can you please explain how i should expand further? I am told that G is finite in the question. thanks

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 Quote by syj can you please explain how i should expand further? I am told that G is finite in the question. thanks
Well, you know that

$$G\cong G/\ker(\phi)$$

Why does that imply that $\ker(\phi)=\{1\}$ ??

Think of the order...

 ok, am i making sense here: a corollary to the first isomorphism theorem says: |G:ker($\varphi$|=|$\varphi$(G)| from this can I conclude: |$\frac{G}{ker(\varphi)}$|=|G'| and then conclude: ker($\varphi$)={1G}
 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus Indeed, that works!!
 wooo hoooo !!!! i am the happiest girl in the world!! until the next proof comes my way ... at which time I shall bug u some more! thanks so much.

 Tags abstract algebra, group representation