Definition. An action of a group $G$ on a set $X$ is strongly $n$-transitive if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is $n$-transitive if $G$ acts transitively on unordered subsets of size $n$ in $X$.

The automorphism group of a connected topological manifold of dimension at least two acts strongly $n$-transitively for all $n$. On the other hand, for actions of groups that are smaller, the situation is different:

Theorem. (Kramer, following Tits) There is no locally compact, $\sigma$-compact topological transformation group $G$ which acts effectively and $4$-transitively on a Hausdorff non-totally-disconnected space $X$.

(As a side note, by stating the theorem like this, I am butchering a beautiful and intricate classification of all $2$-transitive group actions satisfying the other hypotheses)

My question is whether there is any room between these extremes by weakening Kramer's hypotheses:

Question. Does there exist a group $G$ that acts (strongly?) $n$-transitively but not $(n+1)$-transitively on a space $X$ for $n>3$?

For finite discrete or indiscrete $X$, the symmetric and alternating groups are examples, along with some of the Mathieu groups, but let's exclude those. My preference would be for either $X$ or $G/G_x$ to be a connected manifold but I'll take what I can get.

This question was inspired in part by this question.

Linus Kramer, MR 2009240 Two-transitive Lie groups, J. Reine Angew. Math. 563 (2003), 83--113.

Definition. An action of a group $G$ on a set $X$ is strongly $n$-transitive if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is $n$-transitive if $G$ acts transitively on unordered subsets of size $n$ in $X$.

The automorphism group of a connected topological manifold of dimension at least two acts strongly $n$-transitively for all $n$. On the other hand, for actions of groups that are smaller, the situation is different:

Theorem. (Kramer, following Tits) There is no locally compact, $\sigma$-compact topological transformation group $G$ which acts effectively and $4$-transitively on a Hausdorff non-totally-disconnected space $X$.

(As a side note, by stating the theorem like this, I am butchering a beautiful and intricate classification of all $2$-transitive group actions satisfying the other hypotheses)

My question is whether there is any room between these extremes by weakening Kramer's hypotheses:

Question. Does there exist a group $G$ that acts (strongly?) $n$-transitively but not $(n+1)$-transitively on a space $X$ for $n>3$?

For finite discrete or indiscrete $X$, the symmetric and alternating groups are examples, along with some of the Mathieu groups, but let's exclude those. My preference would be for either $X$ or $G/G_x$ to be a connected manifold but I'll take what I can get.

This question was inspired in part by this question.

Linus Kramer, MR 2009240 Two-transitive Lie groups, J. Reine Angew. Math. 563 (2003), 83--113.

Definition. An action of a group $G$ on a set $X$ is strongly $n$-transitive if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is $n$-transitive if $G$ acts transitively on unordered subsets of size $n$ in $X$.

The automorphism group of a connected topological manifold of dimension at least two acts strongly $n$-transitively for all $n$. On the other hand, for actions of groups that are smaller, the situation is different:

Theorem. (Kramer, following Tits) There is no locally compact, $\sigma$-compact topological transformation group $G$ which acts effectively and $4$-transitively on a Hausdorff non-totally-disconnected space $X$.

(As a side note, by stating the theorem like this, I am butchering a beautiful and intricate classification of all $2$-transitive group actions satisfying the other hypotheses)

My question is whether there is any room between these extremes by weakening Kramer's hypotheses:

Question. Does there exist a group $G$ that acts (strongly?) $n$-transitively but not $(n+1)$-transitively on a space $X$ for $n>3$?

For finite discrete or indiscrete $X$, the symmetric and alternating groups are examples, along with some of the Mathieu groups, but let's exclude those. My preference would be for either $X$ or $X/G$ to be a connected manifold but I'll take what I can get.

This question was inspired in part by this question.

Linus Kramer, MR 2009240 Two-transitive Lie groups, J. Reine Angew. Math. 563 (2003), 83--113.

Definition. An action of a group $G$ on a set $X$ is strongly $n$-transitive if $G$ acts transitively on $n$-tuples of distinct elements in $X$ (via the diagonal action), and is $n$-transitive if $G$ acts transitively on unordered subsets of size $n$ in $X$.

The automorphism group of a connected topological manifold of dimension at least two acts strongly $n$-transitively for all $n$. On the other hand, for actions of groups that are smaller, the situation is different:

Theorem. (Kramer, following Tits) There is no locally compact, $\sigma$-compact topological transformation group $G$ which acts effectively and $4$-transitively on a Hausdorff non-totally-disconnected space $X$.

(As a side note, by stating the theorem like this, I am butchering a beautiful and intricate classification of all $2$-transitive group actions satisfying the other hypotheses)

My question is whether there is any room between these extremes by weakening Kramer's hypotheses:

Question. Does there exist a group $G$ that acts (strongly?) $n$-transitively but not $(n+1)$-transitively on a space $X$ for $n>3$?

For finite discrete or indiscrete $X$, the symmetric and alternating groups are examples, along with some of the Mathieu groups, but let's exclude those. My preference would be for either $X$ or $G/G_x$ to be a connected manifold but I'll take what I can get.

This question was inspired in part by this question.

Linus Kramer, MR 2009240 Two-transitive Lie groups, J. Reine Angew. Math. 563 (2003), 83--113.