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then $G$ (with the operation $*$) is called an \emph{abelian group}.  \begin{remark}  The adjective "abelian"only  indicates that ii) holds, i.e. that the operation $*$ is commutative. If only i), iii) and iv) hold, $G$ is only \underline{a group}. \end{remark}  Our previous observations about the sum in $\ZZ$ and the sum in $\Fb$ lead us to claim that both $\ZZ$ and $\Fb$ are \emph{abelian groups w.r.t. their sum}.  %