One exercise in Robinson's A Course in the Theory of Groups is to prove that the groups which have the projective property are necessarily free. I'm not able to prove that because I haven't gotten to that chapter yet, but I tried to find in the book any mention of groups that have the injective property. I know that the injective abelian groups are exactly the divisible groups. But nothing is said about injectivity in the class of all groups. Why is that?
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"The only injective object in the category of groups is the trivial group," is the statement of the theorem in M. Nogin's "A short proof of Eilenberg and Moore's theorem," 2007. The cited work of S. Eilenberg and J.C. Moore is "Foundations of relative homological algebra," 1965.
Another approach is to consider again the free group $F$ on two letters $x,y$ and the automorphism $x\leftrightarrow y$ that gives a semidirect product $F\rtimes C_2$ where the generator $\sigma $ of $C_2$ acts by $\sigma x\sigma =y$ and $\sigma y\sigma =x$. Then consider the canonical injection $\iota : F\to F\rtimes C_2$ and given $g\in G$, where $G$ is injective, the morphism $F\to G$ with $x\to 1 $ and $y\to g$. An extension $\psi: F\rtimes C_2\to G$ gives that $\psi z g\psi z=1$ so that $g=1$ since $(\psi z)^2=1$.