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) $).