Groups that have a definition analogous to projective modules.

Question

I am reading a paper (link provided) which studies certain groups with definitions analogous to projective, injective, and semisimple modules. In that, they split (or half split) all short exact sequences.

In the section on projective (and half-projective) groups we get the definition that a group $G$ is projective if every short exact sequence $0\rightarrow A\overset{f}{\rightarrow} B \overset{g}{\rightarrow} G\rightarrow 0$ splits (half splits). Then follows a theorem (and proof) that states that a group $G$ is projective if and only if $G$ is the trivial group.

I am having a problem with this being true because I know that $\mathbb{Z}$ is a projective $\mathbb{Z}$-module. So, since $\mathbb{Z}$-modules are just abelian groups, should we not have that $\mathbb{Z}$ is a projective group. i.e. if $\mathbb{Z}$ splits all short exact sequences of modules with $\mathbb{Z}$ as the right side $\mathbb{Z}$-module, should it not split all short exact sequences of groups where $\mathbb{Z}$ is the right side group?

Link: https://www.universiteitleiden.nl/binaries/content/assets/science/mi/scripties/aponbach.pdf

Answer 1

When $G$ is abelian, considering exact sequences with $A,B$ possibly non-abelian groups, is different from considering exact sequences with $A,B$ being abelian.

Another case where this happens: extensions of an abelian group $G$ by an abelian group $B$, in the category of abelian groups, are classified by $\mathrm{Ext}^1_{\Bbb Z}(G, B)$. On the other hand, central extensions of an an abelian group $G$ by an abelian group $B$, in the category of (possibly non-abelian) groups, are classified by $H^2(G, B) = \mathrm{Ext}^2_{\Bbb Z[G]}(\Bbb Z, B)$. See here for relevant information.

Answer 2

Definition: a group $G$ is projective if every exact sequence $1\to A\to B\to G\to 1$ splits.

Prop: $G$ is projective iff it's free (as a group).

Indeed clearly free groups are projective. The converse holds because every subgroup of a free group is free (Schreier).

An abelian group is projective if the same holds for exact sequences of abelian group (hence this means projective as $\mathbf{Z}$-module). Then an abelian group is projective (as an abelian group) iff it's free abelian. Thus note that free abelian groups of rank $\ge 2$ are projective as abelian groups, but not as groups.