On local near-rings with Miller?Moreno multiplicative group
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...
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| Date: | 2012 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2012
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/2618 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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 $2^{n−4}$. |
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