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Let $G$ be a finite permutation group acting transitively, but not regularly, on a set $V$. Let $H$ be the stabilizer of some point $v\in V$, and suppose that $H$ acts 2-transitively on one of its (non-trivial) orbits, i.e. $H$ acts 2-transitively on the (left) cosets of $H\cap H^g$ for an appropriate choice of $g\in G$.

General question: What can I conclude about the permutation character of $G$ acting on $V$?

Specifically:

  • Can I give a lower bound for the multiplicities of the non-trivial irreducible components of the permutation character of $G$ on $V$?
  • What about the dimensions of these components?
  • What about if I weaken '2-transitively' to 'primitively' or something similar?
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    $\begingroup$ Look at the case where the action of G is 3-transitive. Then a point stabilizer is 2-transitive on its (unique) nontrivial orbit. In this case the permutation character of G is the sum of the principal character and one other irreducible, so the (unique) nontrivial irreducible constituent has multiplicity 1. Thus 1 is the best possible lower bound on the multiplicities. That's not very interesting, so I wonder if the first question contains an error, and "upper bound" is intended in place of "lower bound" $\endgroup$ Commented Apr 19, 2013 at 20:39

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