The \(p\)–gen nature of \(M_0(V)\) (I)

Abstract

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 \( {\cal D}^\#(1,p) \) (=\( {\cal D}(1,p)\cup\{\{0\}\}) \), \( {\cal D}(2,p) \) and \( {\cal D}(3,p) \) of groups, where \( {\cal D}(n,p) \) 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 \(\delta(G) \)).