On $(K\sb\ast ({\bf Z}/p\sp{2}{\bf Z})$ and related homology groups

It is shown that, for $p \geqslant 5$,

$$R = {\mathbf{Z}} / {p^2}{\mathbf{Z}},\,{K_3}(R) = {\mathbf{Z}} / {p^2}{\mathbf{Z}} + {\mathbf{Z}} / ({p^2} - 1){\mathbf{Z}}$$

and ${K_4}(R) = 0$. Similar calculations are made for $R$ the ring of dual numbers over ${\mathbf{Z}} / p{\mathbf{Z}}$. The calculation reduces to finding homology groups of $\operatorname{Sl} (R)$. A key tool is the spectral sequence of the group extension of $\operatorname{Sl} (n,\,{p^2})$ over $\operatorname{Sl} (n,\,p)$. The terms of this spectral sequence depend in turn on the homology of $\operatorname{Gl} (n,\,p)$ with coefficients various multilinear modules. Calculation of the differentials uses the Charlap-Vasquez description of ${d^2}$.