Highly transitive imprimitivities. (English) Zbl 1111.20001

Conder, Marston; Jones, Vaughan

J. Algebra 300, No. 1, 44-56 (2006).

The study of subfactors of von Neumann algebras led the authors to the problem of considering finite groups \(G\) containing two maximal subgroups \(H\) and \(K\) such that \(G\neq HK\), and such that the action of \(G\) on the space of cosets of \(G\) modulo \(H\cap K\) has small rank \(r\). They show that \(r\) is at least \(6\) and determine all possibilities for \(r=6\) completely (modulo the core of \(H\cap K\) in \(G\)): in this case the action of \(G\) on the cosets of \(G\) modulo \(H\cap K\) corresponds to the action of a flag transitive collineation group on the set of flags of a Desarguesian projective plane. For \(r=7\) an interesting special case is singled out which is closely related to \(4\)-transitive simple permutation groups. In addition, some other interesting properties of the general situation are discovered.

Reviewer: Wolfgang D. Knapp (Tübingen)

MSC:

20B10	Characterization theorems for permutation groups
20B05	General theory for finite permutation groups
20B20	Multiply transitive finite groups
51E15	Finite affine and projective planes (geometric aspects)

Keywords:

transitive permutation groups; imprimitive permutation groups; highly transitive groups; small rank; flag transitive actions; Desarguesian projective planes; \(4\)-transitive permutation groups

Software:

Magma

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