The abelian category of finite abelian groups has no nonzero projective object

The category $\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{A}\mathbf{b}$ is abelian: kernels, cokernels, and finite biproducts of homomorphisms of finite abelian groups are again finite abelian groups. Let $P$ be a nonzero finite abelian group, and fix a prime $p$ for which $P$ has a nonzero $p$-primary quotient. Among all cyclic quotients of $P$ of the form $\mathbb{Z}\mathrm{/}{p}^m$, choose one with maximal $m$, say $u:P\twoheadrightarrow \mathbb{Z}\mathrm{/}{p}^m$.

Let $q:\mathbb{Z}\mathrm{/}{p}^{m+1}\twoheadrightarrow \mathbb{Z}\mathrm{/}{p}^m$ be the canonical quotient map. If $P$ were projective, [L1] would lift $u$ to $g:P\to \mathbb{Z}\mathrm{/}{p}^{m+1}$ with $qg=u$. Since $u$ is surjective, so is $g$. Thus $\mathbb{Z}\mathrm{/}{p}^{m+1}$ would be a quotient of $P$, contradicting maximality of $m$. Therefore no nonzero object of $\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{A}\mathbf{b}$ is projective. So $\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{A}\mathbf{b}$ is an abelian category with no nonzero projective object.