Explicit $K_2$ of certain quotient rings over $\mathbb{Z}[G]$ for $G$ an elementary abelian $p$-group

Abstract

Let $G$ be an elementary abelian $p$-group. In this paper, we calculate the $K_2$-groups of some quotient rings $\mathbb{Z}[G]/I$ for certain ideals $I \subseteq \mathbb{Z}[G]$ of finite $p$-power index. These results are established through the explicit computation of Dennis-Stein symbols. As an application, we provide a definitive characterization of the relative group $SK_1(\mathbb{Z}[G], p^k\mathbb{Z}[G])$ for any odd prime $p$ and $k \ge 1$.