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A class of polynomial permutations on groups

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

G. Kowol
G. Kowol
Affiliation:
University of Vienna1090 Vienna, Austria
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Abstract

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Let G be a finite and u(G) the group of all invertible transformations (polynomial permutations) of the form x→a1 x1→ xk a2⃛ar xkr ar+1 (aiε G, x runs through G). Continuing investigations of H. Lausch of groups satisfying u(G) = {X→axk b} we show here that this condition implies that G is the direct product of its {2, 3}-Hall subgroup and its {2, 3}′-Hall subgroup H where H is nilpoint of class ≤2. Essentially all non-nilpoint groups G of order 2m 3n are described having the property u(G)= {x→axk b}

MSC classification

Secondary: 20B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 2 , March 1979 , pp. 189 - 198
Copyright
Copyright © Australian Mathematical Society 1979

References

Baer, R. (1951/1952), ‘Endlichkeitskriterien für Kommutatorgruppen’, Math. Ann. 124, 161–177.Google Scholar
Baer, R. (1957), ‘Classes of finite groups and their properties’, Illinois J. Math. 1, 115–187.Google Scholar
Huppert, B. (1967), Endliche Gruppen I (Springer, Berlin-Heidelberg-New York).CrossRefGoogle Scholar
Kowol, G. (1977), ‘Fast-n-abelsche Gruppen’, Arch. Math. 29, 55–66.CrossRefGoogle Scholar
Kowol, G. (1978), ‘Polynomfunktionen auf 3-Gruppen’, Contributions to General Algebra, Klagenfurt, edited by Kautschitsch, H., Müller, W. and Nöbauer, W. (J. Heyn, Klagenfurt), in print.Google Scholar
Kurosch, A. G. (1970), Gruppentheorie I, 2nd ed. (Akademic-Verlag, Berlin).Google Scholar
Lausch, H. (1966), ‘Eine Charakterisierung nilpotenter Gruppen der Klasse 2’, Math. Z. 93, 206–209.CrossRefGoogle Scholar
Lausch, H. and Nöbauer, W. (1973), Algebra of polynomials (North-Holland, Amsterdam–London; American Elsevier, New York).Google Scholar
Lausch, H., Nöbauer, W. and Schweiger, F. (1965), ‘Polynompermutationen auf Gruppen I’, Monatsh. Math. 69, 410–423.Google Scholar
Lautsch, H., Nöbauer, W. and Schweiger, F. (1966), ‘Polynompermutationen auf Gruppen II’, Monatsh. Math. 70, 118–126.Google Scholar
Schumacher, F. (1970), Über die polynompermutationen der endlichen Gruppen (Thesis, Wien).Google Scholar
Scott, S. D. (1969), ‘The arithmetic of polynomial maps over a group and the structure of certain permutational polynomial groups I’, Monatsh. Math. 73, 250–267.CrossRefGoogle Scholar