TITLE: Are there injective groups?
QUESTION [16 upvotes]: 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?

REPLY [3 votes]: 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 $(\psi z)^2=1$.