If $f\colon B \to C$ is a bijective function and $g\colon A \to B$ is injective, what can we say about $fg\colon A \to C$?

I think $fg\colon A \to C$ is bijective as well but I am not sure. What do you guys think?

Any help would be really appreciated.

Thanks!

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If $fg$ is a bijection, then $g=f^{-1}fg$ is also a bijection. –  wj32 Feb 23 '13 at 21:44
Try breaking it up: Can you prove $fg$ is surjective? Injective? Since $f$ has an inverse, what can you say about the existence of $(fg)^{-1}$? –  andybenji Feb 23 '13 at 21:44
i understand why it should only be injective. What do you think will happen if $f$ was reduced to being surjective only? –  uh1 Feb 23 '13 at 22:34

• An example of a bijective function $f:B\to C$ is when $B=C$ and $f$ is the identity map.
Hint The composition of injective functions is always injective. Think what happens when $B=C$ and $A$ has a strictly smaller size.