One Decomposition of $K_2$-Group for Certain Quotients over $\mathbb{Z}[G]$ with $G$ a Finite Abelian $p$-Group

Abstract

This paper investigates the structure of $K_2$-groups for certain quotient rings of the integral group ring $\mathbb{Z}[G]$. Let $G$ be a finite abelian $p$-group with $p$-rank $r$, let $\Gamma$ be the maximal $\mathbb{Z}$-order of $\mathbb{Q}[G]$, and let $\widetilde{G}$ denote the sum of all elements of $G$ in the group ring. By employing the framework of K\"{a}hler differentials, we first determine that the relative $K$-group $K_2(\mathbb{F}_p[G], (\widetilde{G}))$ is an elementary abelian $p$-group of rank $r$ when $|G|>2$. Building upon this result, we establish an explicit isomorphism for $r > 1$: $$ K_2(\mathbb{Z}[G]/(|G|\Gamma \cap p\mathbb{Z}[G])) \cong K_2(\mathbb{Z}[G]/|G|\Gamma) \oplus K_2(\mathbb{F}_p[G], (\widetilde{G})). $$