**The background**: We recall/define the following:


* $\Omega\_n=\{1,\dots,n\}$.
* $M\_n$ is the Mathieu group of degree $n$. We follow the Wikipedia article "Mathieu group" and define these groups for each of the 10 $n$ values 8-12 and 20-24. In each of these two series, $M\_{k-1}$ is the point stabilizer in $M\_k$ of $k\in\Omega\_k$.
* Given an action of a group $G$ on a set $\Omega$ with a subset $\Delta\subseteq\Omega$, ${\mathbf C}\_G(\Delta)$ denotes the pointwise stabilizer of $\Delta$ in $G$ (with set braces omitted when $\Delta$ is a singleton $\{\delta\}$).
* An action (of $G$ on $\Omega$) is *half-transitive* when either (a) $|G|>1$ and every orbit has the same nontrivial cardinality or (b) [the degenerate case] $|G|=|\Omega|=1$.


**The question**: Given the Mathieu group $M\_{20}$ as a permutation group on $\Omega\_{20}$, is the induced action of ${\mathbf C}\_{M\_{20}}(20)$ on $\Omega\_{19}$ half-transitive?


**The motivation**: Just like transitivity and primitivity, half-transitivity has higher-fold gradations $(n+\frac 12)$-transitivity for each integer $n\geq 1$. Indeed, like the way $n$-transitivity and $n$-primitivity interleave, it is true that $(n+\frac 12)$-transitive implies $n$-transitive, which then implies $(n-\frac12)$-transitive. It is also known that an $n$-transitive action is $(n+\frac 12)$-transitive if and only if for each $\Delta\subseteq\Omega$ with $|\Delta|=n$, the induced action of ${\mathbf C}\_G(\Delta)$ on $\Omega\setminus\Delta$ is half-transitive. The parallel statement involving a reduction by one on $n$ holds when points $\delta\in\Omega$ replace the cardinality $n$ subsets $\Delta\subseteq\Omega$.


In particular, these lead to the following statements being equivalent; either they are all true or they are all false.


* The action of $M\_{24}$ on $\Omega\_{24}$ is $\frac{11}2$-transitive.
* The action of $M\_{23}$ on $\Omega\_{23}$ is $\frac 92$-transitive.
* The action of $M\_{22}$ on $\Omega\_{22}$ is $\frac 72$-transitive.
* The action of $M\_{21}$ on $\Omega\_{21}$ is $\frac 52$-transitive.
* The action of $M\_{20}$ on $\Omega\_{20}$ is $\frac 32$-transitive.
* The action of ${\mathbf C}\_{M\_{20}}(20)$ on $\Omega\_{19}$ is $\frac 12$-transitive.


My personal curiosity revolves around the $4$- and $5$-transitive actions interacting with higher-fold half-transitivity at the upper end of this list.


The Wikipedia article goes on to say that $M\_{20}$ has the form $2^4:A\_5$, but I have no idea what action this entails which leaves as unknown the form of the point stabilizer in this group.


As an aside, the corresponding question applied to the lower 5 Mathieu groups $M\_8$ through $M\_{12}$ similarly reduces to whether or not ${\mathbf C}\_{M\_8}(8)$ is half-transitive on $\Omega\_7$. In that case, $M\_8$ acts regularly on $\Omega\_8$ (indeed, the action is the regular action of the quaternion group on itself), guaranteeing ${\mathbf C}\_{M\_8}(8)$ is trivial. As $|\Omega\_7|=7>1$ is not trivial, this action is not half-transitive.


**EDIT** (9 Sep 2022) I previously commented on not knowing what the action of $A\_5$ on $2^4$ was. I now understand it to be $SL\_2(\mathbb F\_4)\cong A\_5$ in its natural action on $\mathbb F\_4{\!}^2$.



Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive?
No it is not: half-transitive means that all orbits have equal size (and the groups acts non-trivially), but this is not the case here. One can verify this e.g. using GAP:



```
gap> M21:=MathieuGroup(21);
Group([ (1,4,5,9,3)(2,8,10,7,6)(12,15,16,20,14)(13,19,21,18,17), 
  (1,21,5,12,20)(2,16,3,4,17)(6,18,7,19,15)(8,13,9,14,11) ])
gap> M20:=Stabilizer(M21,21);
Group([ (1,5,18,16,17)(2,20,14,7,8)(3,12,11,19,10)(4,15,6,9,13), 
  (2,18,10)(3,4,5)(6,16,14)(7,15,11)(8,20,17)(12,19,13) ])
gap> M19:=Stabilizer(M20,20);
Group([ (1,18,19)(2,8,7)(3,13,16)(4,12,17)(5,15,9)(6,11,10), 
  (1,9,8)(2,5,11)(3,13,16)(4,14,12)(6,15,18)(7,10,19), (1,19,14)
  (2,10,9)(3,13,16)(4,7,5)(6,17,15)(8,12,11) ])
gap> Orbits(M19);
[ [ 1, 18, 9, 19, 6, 5, 8, 2, 7, 14, 11, 15, 17, 4, 12, 10 ], 
  [ 3, 13, 16 ] ]
gap> OrbitLengths(M19);
[ 16, 3 ]

```

One can also try to see this geometrically: note that $M\_{21}$ is in fact $PSL\_3(\mathbb{F}\_4)$, which I will denote as $PSL(3,4)$, acting on the projective space $\mathbb{P}\_3(\mathbb{F}\_4)$. This space contains $\frac{4^3-1}{4-1}=21$ projective points; any two points span a unique projective line with 5 points on it; there are 21 such lines, inducing a block system on the 21 points. Stabilizing two points then stabilizes the unique line through them. The stabilizer of this projective line can be thought of as the stabilizer in $SL(3,4)$ of a 2-dimensional subspace, which induces the action of $PSL(2,4)$ on that subspace, which acts 3-transitively on the 5 points. But we did not just fix the line, we also fixed two points on it; so this stabilizer is 1-transitive on the remaining three points. This is exactly the orbit `[3, 13, 16]` in the GAP computation above.


I am too rusty in this stuff, so I don't immediately see a simple argument why it acts transitively on the remaining 21-5 = 16 points (perhaps someone can fill that in). However, that's not even necessary: clearly, it is impossible to divide these remaining points into orbits of size 3. Thus the action can't be half-transitive.


