Given are 3 functions $f,g,h$ of afinite Set $P$, and also given is that $g \circ f$ and $h \circ f$ are bijective.
Need to prove $h \circ g \circ f$ is bijective.
I know that as $g \circ f$ and $h \circ f$ are bijective, so both are injective as well as surjective. So
If $g \circ f$ is surjective then $g$ is surjective
If $h \circ g$ is injective then $g$ is injective
So it means $g$ is bijective
But using this information along with the information that the set is finite, how will I be able to conclude that $h \circ g \circ f$ is bijective.
Can anybody provide a hint to tackle this problem.