In this MO question it was mentioned that the following fact seems to be true:

If $G$ is doubly transitive on $X$ and the one-point stabilizer $G_x$ has a non-trivial center, then $G$ is of affine type, that is, the socle is elementary abelian.

Does anyone know if it is true?

I checked it with GAP for all 2-transitive groups up to degree 2499 and I no counterexamples appeared. Here is the code for groups with maximun degree of transitivity equal to two:

```
for gr in AllPrimitiveGroups(NrMovedPoints, [1..2499], Transitivity, 2) do
stab := Stabilizer(gr, 1);
if not IsTrivial(Center(stab)) then
soc := Socle(gr);
if not IsAbelian(soc) then
Print("Counterexample of degree ", NrMovedPoints(gr), "\n");
fi;
fi;
od;
```

I am mainly interested in the following question:

Is it true that if $G$ is a 2-transitive group with simple socle then $Z(G_x)=1$?

I need only to consider groups with maximum degree of transitivity equal to two. Further, many of the groups appearing in the classification of 2-transitive groups with simple socle are easy to handle. However, I cannot prove the fact in the following cases:

- Socle $PSL(d,q)$, degree $\frac{q^d-1}{q-1}$, and $d\geq3$ (two actions).
- Socle $PSU(3,q)$, degree $q^3+1$ and $q\geq3$.
- Socle $Sz(q)$, degree $q^2+1$ and $q=2^{2d+1}>2$.
- Socle $Ree(q)$, degree $q^3+1$ and $q=3^{2d+1}>3$.

Is it true that if $G$ is a 2-transitive almost simple group with simple socle isomorphic to $PSL(d,q)$, $PSU(3,q)$, $Sz(q)$ or $Ree(q)$, then $Z(G_x)=1$?

References:

- Cameron, Peter J. Finite permutation groups and finite simple groups. Bull. London Math. Soc. 13 (1981), no. 1, 1--22. MR0599634 (83m:20008). The table with the classification of 2-transitive groups with simple socle appears in page 157, see google books)