## Highly transitive subgroups of the symmetric group on the natural numbers

### Volume 112 / 2008

#### Abstract

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.