Geodesic
Finite generability of some groups of recursive permutations
Diskretnaya Matematika, Tome 20 (2008) no. 4, pp. 61-78
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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. 
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     author = {S. A. Volkov},
     title = {Finite generability of some groups of recursive permutations},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2008_20_4_a4/}
}
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S. A. Volkov. Finite generability of some groups of recursive permutations. Diskretnaya Matematika, Tome 20 (2008) no. 4, pp. 61-78. http://geodesic.mathdoc.fr/item/DM_2008_20_4_a4/

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