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

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 $\mathrm{\Gamma }$ be the maximal $\mathbb{Z}$-order of $\mathbb{Q}[G]$, and let $\tilde{G}$ denote the sum of all elements of $G$ in the group ring. By employing the framework of Kähler differentials, we first determine that the relative $K$-group $K_2({\mathbb{F}}_p[G],(\tilde{G}))$ is an elementary abelian $p$-group of rank $r$ when $\mathrm{\mid }G\mathrm{\mid }>2$. Building upon this result, we establish an explicit isomorphism for $r>1$: $$K_2(\mathbb{Z}[G]\mathrm{/}(\mathrm{\mid }G\mathrm{\mid }\mathrm{\Gamma }\cap p\mathbb{Z}[G]))\cong K_2(\mathbb{Z}[G]\mathrm{/}\mathrm{\mid }G\mathrm{\mid }\mathrm{\Gamma })\oplus K_2({\mathbb{F}}_p[G],(\tilde{G}))\mathrm{.}$$