Consider the abelian category $\mathsf{finAb}$ of finite abelian groups. It is easy to prove that there is an isomorphism $\mathrm{ord} : K\_0(\mathsf{finAb}) \to \mathbb{Q}^+$. Can you give a reference in the literature for that? More generally:


**Question.** Is there any more concrete description of the Waldhausen's K-theory space $\Omega |w S\_{\bullet} \mathsf{finAb}|$? What is known about its homotopy groups, that is the $K$-theory groups $K\_n(\mathsf{finAb})$ for $n>0$?



K-Theory space of finite abelian groups
Everything is known. In fact as spectra we have canonically $K(\mathsf{finAb}) = \vee\_p K(\mathbb{F}\_p)$, and the spectra $K(\mathbb{F}\_p)$ are identified in the work of Quillen (see e.g. <http://www.math.uiuc.edu/K-theory/1006/>). In particular on $\pi\_0$ we find $K\_0(\mathsf{finAb}) = \oplus\_p \mathbb{Z}$, agreeing with your claim, and on $\pi\_n$ for $n>0$ we find that $K\_n(\mathsf{finAb})$ is $0$ for $n$ even and is $\oplus\_p \mathbb{Z}/(p^k-1)$ (non-canonically) for $n = 2k-1$.


To justify the claimed equality $K(\mathsf{finAb}) = \vee\_p K(\mathbb{F}\_p)$, note first that $\mathsf{finAb}$ is the filtered colimit over increasing finite sets of primes $P$ of the variant $\mathsf{finAb}\_P$ where only products of $p$-groups for $p \in P$ are allowed; since K-theory commutes with filtered colimits, it then suffices to show that each $K(\mathsf{finAb}\_P) = \prod\_{p\in P} K(\mathbb{F}\_p)$ and that for $P \subseteq P'$ this identification intertwines the inclusion $K(\mathsf{finAb}\_P) \to K(\mathsf{finAb}\_{P'})$ with the evident map $\prod\_{p\in P} K(\mathbb{F}\_p) \to \prod\_{p\in P'} K(\mathbb{F}\_p)$ which is zero outside of $P$.


But $\mathsf{finAb}\_P$ is just the product over $p \in P$ of the categories $\mathsf{finAb}\_p$, whose K-theory identifies with that of vector spaces over $\mathbb{F}\_p$ by Quillen's devissage theorem. And K-theory commutes with finite products, so that's that.


Here I guess I was actually arguing using Quillen's Q-construction instead of Waldhausen's $S\_{\bullet}$-construction. Otherwise I'm not sure how to justify the last step, the devissage. Actually I'm sure all of the above is in Quillen's paper on the Q-construction.


