Homogeneous Functions and Algebraic $K$--theory

In this paper we develop the theory of homogeneous functions between finite abelian groups. Here, a function $f:G⟶H$ between finite abelian groups is homogeneous of degree $d$ if $f(nx)=n^{d}f(x)$ for all $x\in G$ and all $n$ which are relatively prime to the order of $x$. We show that the group of homogeneous functions of degree one from a group $G$ of odd order to $\mathbb{Q}\mathrm{/}\mathbb{Z}$ maps onto $S{K}_1(\mathbb{Z}[G])$, generalizing a result of R. Oliver for $p$-groups.