if we have Groups G, and H, and homomorphism F between them

and left cosets g*Ker(F)

Why is it that **"there is one left coset [g*Ker(F)] for each element of Im(F)"**

(This is part of the answer to part (ii) of the attached question).

Image of full question

Thanks!

## Best Answer

Because $S_{h'}=S_h\iff h'=h$, and, by (i), $S_h$ is a left coset of the kernel of $\phi$. (Symbols are as in the linked excerpt.)