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 Diskr. Mat., 2008, Volume 20, Issue 4, Pages 61–78 (Mi dm1026)

Finite generability of some groups of recursive permutations

S. A. Volkov

Abstract: Let a class $\mathcal Q$ of functions of natural argument be closed with respect to a superposition and contain the identity function. The set of permutations $f$ such that $f,f^{-1}\in\mathcal Q$ forms a group (with respect to the operation of composition) which we denote by $Gr(\mathcal Q)$. We prove the finite generability of $Gr(\mathcal Q)$ for a large family of classes $\mathcal Q$ satisfying some conditions. As an example, we consider the class $\mathrm{FP}$ of functions which are computable in polynomial time by a Turing machine. The obtained result is generalised to the classes $\mathscr E^n$ of the Grzegorczyk system, $n\ge2$.
It is proved that for the considered classes $\mathcal Q$ the minimum number of permutations generating the group $Gr(\mathcal Q)$ is equal to two. More exactly, there exist two permutations of the given group such that any permutation of this group can be obtained by compositions of these permutations.

DOI: https://doi.org/10.4213/dm1026

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English version:
Discrete Mathematics and Applications, 2008, 18:6, 607–624

Bibliographic databases:

UDC: 519.7

Citation: S. A. Volkov, “Finite generability of some groups of recursive permutations”, Diskr. Mat., 20:4 (2008), 61–78; Discrete Math. Appl., 18:6 (2008), 607–624

Citation in format AMSBIB
\Bibitem{Vol08} \by S.~A.~Volkov \paper Finite generability of some groups of recursive permutations \jour Diskr. Mat. \yr 2008 \vol 20 \issue 4 \pages 61--78 \mathnet{http://mi.mathnet.ru/dm1026} \crossref{https://doi.org/10.4213/dm1026} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2500604} \zmath{https://zbmath.org/?q=an:1177.20011} \elib{http://elibrary.ru/item.asp?id=20730266} \transl \jour Discrete Math. Appl. \yr 2008 \vol 18 \issue 6 \pages 607--624 \crossref{https://doi.org/10.1515/DMA.2008.046} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-57349180541}