Highly transitive subgroups of the symmetric group on the natural numbers

U. B. Darji ¹   ; J. D. Mitchell ²

¹ Mathematics Department University of Louisville Louisville, KY 40292, U.S.A.
² Mathematics Institute University of St Andrews North Haugh St Andrews, Fife, KY16 9SS, UK

Colloquium Mathematicum, Tome 112 (2008) no. 1, pp. 163-173

Résumé

Highly transitive subgroups of the symmetric group on the natural numbers are studied using combinatorics and the Baire category method. In particular, elementary combinatorial arguments are used to prove that given any nonidentity permutation $\alpha$ on $\mathbb{N}$ there is another permutation $\beta$ on $\mathbb{N}$ such that the subgroup generated by $\alpha$ and $\beta$ is highly transitive. The Baire category method is used to prove that for certain types of permutation $\alpha$ there are many such possibilities for $\beta$. As a simple corollary, if $2 \leq \kappa \leq 2 ^{\aleph _0}$, then the free group of rank $\kappa$ has a highly transitive faithful representation as permutations on the natural numbers.

Détail

DOI : 10.4064/cm112-1-9

Keywords: highly transitive subgroups symmetric group natural numbers studied using combinatorics baire category method particular elementary combinatorial arguments prove given nonidentity permutation alpha mathbb there another permutation beta mathbb subgroup generated alpha beta highly transitive baire category method prove certain types permutation alpha there many possibilities beta simple corollary leq kappa leq aleph group rank kappa has highly transitive faithful representation permutations natural numbers