Representation of a finite group

[syj]

Homework Statement

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

Homework Equations

The Attempt at a Solution

 

[micromass]

Homework Statement

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

Homework Equations

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...

 

[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}={1_G}

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.

 

[micromass]

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}={1_G}

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!

 

[syj]

Ok, so is this enough:

If f is faithful then ker{f}={1_G}
therefore by the first isomorphism theorem, G$\cong$G'

If G$\cong$G' then by the first isomorphism theorem ker{f}={1_G}
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 ;)

 

[micromass]

If G$\cong$G' then by the first isomorphism theorem ker{f}={1_G}

This is true (but only for finite groups), but you might want to explain in some more detail.

The rest is ok!

 

[syj]

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

 

[micromass]

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...

 

[syj]

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($\varphi$)={1_G}

 

[micromass]

Indeed, that works!

 

[syj]

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.