How can I prove that there are no finite groups $G$ where there exists an endomorphism $G\rightarrow G$ that is injective but not surjective, or other way round?
If $X$ is a finite set and $f:X \to X$ is any function, then the conditions below are equivalent:
$f$ is bijective
$f$ is injective
$f$ is surjective
It is not a problem about groups but a general fact about sets:
Let $E, F$ be finite sets such that $\lvert E\rvert=\lvert F\rvert$, and $f\colon E\rightarrow F$. The following are equivalent:
- $f$ is injective;
- $f$ is surjective;
$f$ is bijective.
(1) $\Rightarrow$ (2): If $f$ is injective, $\lvert f(E)\rvert=\lvert E\rvert=\lvert F\rvert$, hence $f(E)=F$, which means $f$ is surjective. (2) $\Rightarrow$ (1) by contraposition: if $f$ is not injective, it can't be surjective since $\lvert f(E)\rvert<\lvert E\rvert=\lvert F\rvert$.
Since (1) $\Leftrightarrow$ (2), it is trivial both are equivalent to (3).