Abstract
A group G is called functionally complete if for an arbitrary natural number n every mapping f: Gn → G can be realized by a “polynomial” in at most n variables over the group G. We know that a group G is functionally complete if and only if it is either trivial or a finite simple non-Abelian group [Ref. Zh. Mat. 9A174 (1975)]. In this article the “degree” of a polynomial and the connected notions of n-functional completeness, (n; k1, ..., kn)-functional completeness, and strong functional completeness are introduced. It is shown that for n > 1 these notions and the notion of functional completeness are equivalent, and apart from all finite simple non-Abelian groups, only the trivial group and groups of second order are 1-functionally complete.
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Literature cited
H. Werner, “Finite simple non-Abelian groups are functionally complete,” Bull. Soc. Roy. Sci. Liège,43, 400 (1974).
W. D. Maurer and J. L. Rhodes, “A property of finite simple non-Abelian groups,” Proc. Am. Math. Soc.,16, 522–554 (1965).
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Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 147–151, July, 1977.
In conclusion, the author expresses deep gratitude to M. M. Glukhov for assistance with the article.
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Anashin, V.S. Functionally complete groups. Mathematical Notes of the Academy of Sciences of the USSR 22, 571–574 (1977). https://doi.org/10.1007/BF01147703
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DOI: https://doi.org/10.1007/BF01147703