Extremely primitive groups. (English) Zbl 1141.20003
A primitive permutation group is called extremely primitive if a point stabiliser acts primitively on each of its orbits. Examples include cyclic groups of prime order and 2-primitive permutation groups (i.e. a point stabiliser is primitive on the remaining points), a classification of those is known. The authors aim at classifying extremely primitive groups.
The main result states that an extremely primitive group is either of affine type or almost simple. The proof uses O’Nan-Scott Theorem.
Then the authors analyse the affine case in detail. They classify the groups into 3 families: soluble groups (3 families), insoluble 2-transitive groups (2 families and 4 extra examples), and insoluble not 2-transitive groups. For that last case, they find all examples up to a finite number of possibilities and conjecture they found them all. This study relies on the classification of the finite simple groups.
The main result states that an extremely primitive group is either of affine type or almost simple. The proof uses O’Nan-Scott Theorem.
Then the authors analyse the affine case in detail. They classify the groups into 3 families: soluble groups (3 families), insoluble 2-transitive groups (2 families and 4 extra examples), and insoluble not 2-transitive groups. For that last case, they find all examples up to a finite number of possibilities and conjecture they found them all. This study relies on the classification of the finite simple groups.
Reviewer: Alice Devillers (Bruxelles)
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