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For a project for one of our subjects we have to write about a certain topic. Our topic is very transitive groups. This means a permutation group on the natural integers, that is $k$-transitive for every $k$.

One of the questions we have to try to answer about this subject is, if there is a list with all the examples for this special kind of group. Since we are having a hard time finding an answer on the internet we hope that you guys can help us.

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  • $\begingroup$ It is not a group that is highly transitive. It is a group action that is. But sometimes one says that a group is highly transitive when it has a natural action. Free groups of at most countable rank admit an action which is highly transitive. Burger and Mozes constructed a natural action of certain 'universal groups' on regular trees in 2000, which they prove is highly transitive. I think you'll have a hard time listing 'all' examples. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Nov 14, 2020 at 10:07
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    $\begingroup$ @Carl-FredrikNybergBrodda 50 years before being defined as abstract objects, groups were defined as groups of permutations of a set. This language hasn't completely disappeared. $\endgroup$
    – YCor
    Commented Nov 14, 2020 at 12:41
  • $\begingroup$ No, you won't find any list because it is hopeless to classify such groups. In the literature there are, however, some results giving some information about the class of groups that admit a highly transitive (=$k$-transitive for every $k$) faithful action on an infinite countable set. The keyword "highly transitive" might be helpful to find references. $\endgroup$
    – YCor
    Commented Nov 14, 2020 at 12:44
  • $\begingroup$ @YCor Well, picking apart the history of defining a group abstractly (or indeed at all) is non-trivial, but in any case, as I mentioned, I'm aware of this usage. OP's phrasing made it seem like they possibly weren't, so I wanted to make it clear. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Nov 14, 2020 at 12:50
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    $\begingroup$ @Carl-FredrikNybergBrodda how could an action on a tree (by tree automorphisms) be highly transitive? it should preserve the distance, which for a 2-transitive action implies the distance of all distinct pairs is the same. $\endgroup$
    – YCor
    Commented Nov 14, 2020 at 20:19

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