Is there a group $G$ of mappings $X \to X$ that has a non-bijective map in it? I mean, for each element of G, it must has its inverses at right and left, and those must be the same, so the element is necessarily bijective, right? What am I missing?
2026-03-25 19:02:17.1774465337
Group of mappings
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Just to expand on Dietrich Burde's comment, if you take an idempotent $e$ in any transformation semigroup, then $\{e\}$ is a trivial group. Moreover an idempotent transformation is not necessarily a permutation.
Actually, if you take the transformation semigroup $T_n$ on $n$ elements, it contains several groups apart from the permutation group $S_n$, including non-trivial ones.