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The permutation groups $A = PSL(2,7)$ with its natural action on the projective line $\mathbb{P}^1(\mathbb{F}_7)$ and $B = A\Gamma L(1,8)$ with its natural action on the affine line $\mathbb{F}_8$ have the interesting property that $A$ and $B$ are non-isomorphic, but the point stabilizers $A_x$ and $B_y$ are isomorphic permutation groups (namely a Frobenius group of order 21 acting on 7 points).

Are there other examples of non-isomorphic two-transitive permutation groups with isomorphic point stabilizers?

Note that I really require the point stabilizers to be isomorphic as permutation groups, not just as abstract groups.

(My original motivation for this question is the fact that from every such pair of groups, one can construct an example of two non-isomorphic totally disconnected locally compact groups with isomorphic compact open subgroups. In the meantime, it is already known that there are many examples of this phenomenon, but at that time, this was unknown. I nevertheless believe that the question might still be interesting in its own right.)

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Actually, your pair is the only example acting on $\le 30$ points (by going through Magma's transitive group database), so it does look like these are very hard to come by. –  Tim Dokchitser Feb 16 '11 at 15:04
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up vote 7 down vote accepted

There are lists of known finite 2-transitive groups, and for examples other than those of affine type (i.e. those with a regular normal elementary abelian subgroup), it should not be hard to show that there are no more examples.

But I would expect there to be examples of affine type, and by searching through the Magma database of primitive permutation groups I found an example of degree 343. These are

PrimitiveGroup(343,49) and PrimitiveGroup(343,50) in Magma. Irritatingly, the numbering is different in GAP, and (hoping I have got this right), they are numbers 67 and 68 in the GAP list.

The two groups in question have order $343 \times 342$ - so they are sharply 2-transitive. Their stabilizers are isomorphic nonabelian groups of order 342, with centres of order 6.

I would conjecture that there are infinitely many examples of a similar type.

Added: The group ${\rm A}\Gamma{\rm L}(1,7^3)$ has structure $7^3:(7^3-1):3$, and has four subgroups of index 3. The examples here are two of those subgroups. One of the other two is ${\rm AGL}(1,7^3)$, which is also sharply 2-transitive, but has cyclic point stabilizer. The fourth does not act 2-transitively. There are corresponding subgroups of ${\rm A}\Gamma{\rm L}(1,p^3)$ for all prime $p \equiv 1 \bmod 3$, so there are indeed infinitely many examples. More generally, for $p,q$ primes with $p \equiv 1 \bmod q$, there will be $q-1$ non-isomorphic sharply 2-transitive groups of degree $p^q$ with isomorphic stabilizers.

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Wonderful! :) A natural next question would be: are there other examples that are 2-transitive but not sharply 2-transitive? (This is just a question out of curiosity now, I'm perfectly happy with your examples!) –  Tom De Medts Feb 17 '11 at 10:31
I would guess that two of the 2-transitive subgroups of $\operatorname{A\Gamma L}(1,7^6)$ of index $3$ provide an example, or more generally, subgroups of index $q$ of $\operatorname{A\Gamma L}(1, p^{kq})$, with $p$ and $q$ as above, and $k>1$. These are 2-transitive but not sharply 2-transitive, their order being $p^{kq}(p^{kq}-1)k$. –  Frieder Ladisch Feb 17 '11 at 16:20
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