# Homology of group rings

Let $\mathbb Z G$ be the group ring of $G$. Denote by $\mathbb Z G ^{gp}$ the universal enveloping group of the monoid $(\mathbb Z G,*)$, i.e. the fundamental group of the classyfiyng space of $(\mathbb Z G,*)$. How to compute $H_* (\mathbb Z G ^{gp},\mathbb Z )=H_* (B (\mathbb Z G,*),\mathbb Z )$ in terms of $H_* (G,\mathbb Z )$?

• Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. – José Carlos Santos Apr 29 '18 at 9:56
• @JoséCarlosSantos I want to obtain in some canonical way a group H from $\mathbb Z G$ with the same homology groups as G has – Fat ninja Apr 29 '18 at 12:54
• $(\Bbb ZG, *)$ has a zero element ($0 * g = 0$ for all $g \in \Bbb ZG$). That relation forces the universal enveloping monoid to be zero. – user98602 May 5 '18 at 11:39