The $p$-gen nature of $M_0(V)$ (I)

Let $V$ be a finite group (not elementary two) and $p\geq 5$ a prime. The question as to when the nearring $M_0(V)$ of all zero–fixing self-maps on $V$ is generated by a unit of order $p$ is difficult. In this paper we show $M_0(V)$ is so generated if and only if $V$ does not belong to one of three finite disjoint families ${\mathcal D}^\#(1,p)$ (=${\mathcal D}(1,p)\cup\{\{0\}\})$, ${\mathcal D}(2,p)$ and ${\mathcal D}(3,p)$ of groups, where ${\mathcal D}(n,p)$ are those groups $G$ (not elementary two) with $|G|\leq np$ and $\delta(G)>(n-1)p$ (see [1] or §.1 for the definition of $\delta(G)$).