Abstract algebra: elements of fiber writable as

[HJ Farnsworth]

Greetings,

For a homomorphism $\varphi$, I'm trying to show that elements of a fiber, say the fiber above $a$, $X_a$, are writable as a given element of $X_a$ times an element of the kernel $K$. So, if $a\in X_a$ and $b\in X_a$, then $\exists k\in K$ such that $b=ak$.

I want to do this without using the theorem that $\{$left cosets of $K$ in $G\} =G/K$ - in fact, one of my motivations for looking for this is that I want a different proof of this theorem then the ones that I have seen.

Does anyone know of a way to do this?

Thanks for any help that you can give.

-HJ Farnsworth

 

[jbunniii]

By fiber I assume you mean preimage, i.e. "the fiber above $a$" means $\varphi^{-1}(a)$.

Choose $x,y \in \varphi^{-1}(a)$. We can always write $x = y(y^{-1}x)$, so it suffices to show that $y^{-1}x \in \ker \varphi$.

But this is easy: $\varphi(y^{-1}x) = \varphi(y^{-1})\varphi(x) = \varphi(y)^{-1} \varphi(x) = a^{-1}a = 1$.

 

[HJ Farnsworth]

Hi jbunniii,

That is indeed what I mean by fiber. Thanks for the help, that was exactly the kind of thing I was looking for!

-HJ Farnsworth