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# Forschungseinheiten

Aufsatz / Paper in SCI-Expanded-Zeitschrift

## Polynomial functions and endomorphism near-rings on certain linear groups

Aichinger E., Mayr P.: Polynomial functions and endomorphism near-rings on certain linear groups, in: Communications in Algebra, Volume 31 (11), Page(s) 5627-5651, 2003.

BibTeX

@ARTICLE{
title = {Polynomial functions and endomorphism near-rings on certain linear groups},
type = {Aufsatz / Paper in SCI-Expanded-Zeitschrift},
author = {Aichinger, Erhard and Mayr, Peter},
language = {EN},
abstract = {We describe the unary polynomial functions on the non-solvable groups \$G\$ with \$\SL(n,q) \le G \le \GL(n,q)\$ and on their quotients \$G/Y\$ with \$Y \le Z(G)\$, and we compute the size of the inner automorphism near-ring \$I(G/Y)\$. We compare this near-ring to the endomorphism near-ring \$E(G/Y)\$, and we obtain a full characterization of those \$G\$ and \$Y\$ for which \$I(G/Y) = E(G/Y)\$ holds. For the case \$Y = \{1\}\$, this characterization yields that we have \$E(G) = I(G)\$ if and only if \$G = \SL(n,q)\$. We investigate the automorphism near-ring \$A(G)\$, and we show that for all non-solvable groups \$G\$ with \$\SL(n,q) \le G \le \GL(n,q)\$, we have \$I(G) = A(G)\$. Our results are based on a description of the polynomial functions on those non-abelian finite groups \$G\$ that satisfy the following conditions: \$G' = G''\$, \$G/Z(G)\$ is centerless, and there is no normal subgroup \$N\$ of \$G\$ with \$G' \cap Z(G) < N < G'\$.},
pages = {5627-5651},
journal = {Communications in Algebra},
volume = {31 (11)},
issn = {0092-7872},
year = {2003},
url = {http://www.dekker.com/servlet/product/productid/AGB},
}

### Details

Zusammenfassung: We describe the unary polynomial functions on the non-solvable groups \$G\$ with \$\SL(n,q) \le G \le \GL(n,q)\$ and on their quotients \$G/Y\$ with \$Y \le Z(G)\$, and we compute the size of the inner automorphism near-ring \$I(G/Y)\$. We compare this near-ring to the endomorphism near-ring \$E(G/Y)\$, and we obtain a full characterization of those \$G\$ and \$Y\$ for which \$I(G/Y) = E(G/Y)\$ holds. For the case \$Y = \{1\}\$, this characterization yields that we have \$E(G) = I(G)\$ if and only if \$G = \SL(n,q)\$. We investigate the automorphism near-ring \$A(G)\$, and we show that for all non-solvable groups \$G\$ with \$\SL(n,q) \le G \le \GL(n,q)\$, we have \$I(G) = A(G)\$. Our results are based on a description of the polynomial functions on those non-abelian finite groups \$G\$ that satisfy the following conditions: \$G' = G''\$, \$G/Z(G)\$ is centerless, and there is no normal subgroup \$N\$ of \$G\$ with \$G' \cap Z(G) < N < G'\$.

Journal: Communications in Algebra
Volume: 31 (11)
Erscheinungsjahr: 2003
Seitenreferenz: 5627-5651
Anzahl Seiten: 25
Web: http://www.dekker.com/servlet/product/productid/AGB (Communications in Algebra)
ISSN: 0092-7872

### Beteiligte

AutorInnen / HerausgeberInnen: Assoz.Univprof. DI Dr. Erhard Aichinger, Priv.-Doz. DI Dr. Peter Mayr

Forschungseinheiten der JKU:

Wissenschaftszweige: 101 Mathematik | 101001 Algebra | 101005 Computeralgebra | 101013 Mathematische Logik | 102031 Theoretische Informatik