IJPAM: Volume 108, No. 1 (2016)

AUTOMORPHISMS OF THE UNIT GROUPS OF SQUARE
RADICAL ZERO FINITE COMMUTATIVE COMPLETELY
PRIMARY RINGS

Ojiema M. Onyango$^1$, Owino M. Oduor$^2$, Odhiambo P. Oleche$^3$
$^1$Department of Mathematics
Masinde Muliro University of Science and Technology
P.O. Box 190-50100, Kakamega, KENYA
$^2$Department of Mathematics and Computer Science
University of Kabianga
P.O. Box 2030-20200, Kericho, KENYA
$^3$Department of Pure and Applied Mathematics
Maseno University
P.O. Box 333, Maseno, KENYA


Abstract. Let $G$ be a group. The groups $G^{'}$ for which $G$ is an automorphism group have not been fully characterized. Suppose $R$ is a Completely Primary finite Ring with Jacobson Radical $J$ such that $J^{2}=(0)$. In this case, the characteristic of $R$ is $p$ or $p^{2}$ and the group of units $R^{*}=\mathbb{Z}_{p^{r}-1}\times
(I+J)$ . The structure of $R^{*}$ is well known, but its automorphism group is not well documented. Given the group $R^{*}$, let $Aut(R^{*})$ denote the group of isomorphisms $\phi:
R^{*}\rightarrow R^{*}$ with multiplication given by the composition of functions. The structure of the automorphism groups of finite groups is intimately connected to the structure of the finite groups themselves. In this note, we determine the structure of $Aut(R^{*})$ using well known procedures and to this end, extend the results previously obtained in this area of research.

Received: February 4, 2016

AMS Subject Classification: 20K30, 16P10

Key Words and Phrases: automorphism groups, unit groups, square radical zero, completely primary rings

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DOI: 10.12732/ijpam.v108i1.6 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 108
Issue: 1
Pages: 39 - 48


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