On local near-rings with Miller?Moreno multiplicative group

Abstract

A near-ring $R$ with identity is local if the set $L$ of all its noninvertible elements is a subgroup of the additive group $R^{+}$. We study the local near-rings of order $2^n$ whose multiplicative group $R^{*}$ is a Miller-Moreno group, i.e., a non-abelian group all proper subgroups of which are abelian. In particular, it is proved that if $L$ is a subgroup of index $2^m$ in $R^{+}$, then either $m$ is a prime for which $2^m - 1$ is a Mersenna prime or $m = 1$. In the first case $n = 2m$, the subgroup $L$ is elementary abelian, the exponent of $R^{+}$ does not exceed 4, and $R^{*}$ is of order $2^m(2^m - 1)$. In the second case either $n < 7$ or the subgroup $L$ is abelian and $R^{*}$ is a nonmetacyclic group of order $2^{n−1}$ and of exponent at most $2^{n−4}$.