Original:  While developing rearrangement operations on the natural numbers, we discovered that the bi-immune rearrangements generate an uncountable subgroup of the infinite symmetric group  } on the natural numbers . } We show that this new subgroup contains the bounded symmetric group on the natural numbers, and consequently is highly transitive.
Revised:  While developing rearrangement operations on the natural numbers, we discovered that the bi-immune rearrangements generate an uncountable subgroup of the infinite symmetric group } on the natural numbers . } We show that the bi-immune symmetric group contains the finitary symmetric group on the natural numbers, and consequently is highly transitive.
Edit intent: meaning-changed