Published online by Cambridge University Press: 20 November 2018
Let $G$ be a finitely generated, infinite group, let $p\,>\,1$, and let ${{L}^{p}}\left( G \right)$ denote the Banach space $\left\{ \sum{_{x\in G}{{a}_{x}}x}|\sum{_{x\in G}|{{a}_{x}}{{|}^{p}}<\infty } \right\}$. In this paper we will study the first cohomology group of $G$ with coefficients in ${{L}^{p}}\left( G \right)$, and the first reduced ${{L}^{p}}$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.