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The classification of almost simple \(\frac 32\)-transitive groups. (English) Zbl 1281.20002

A finite transitive permutation group \(G\) is called \(\frac 32\)-transitive if \(G\) is not regular and its nontrivial subdegrees are all equal (the subdegrees of \(G\) are the sizes of the orbits of its one-point stabilizers \(G_\alpha\)). The term was introduced by Helmut Wielandt in his book [H. Wielandt, Finite permutation groups. New York and London: Academic Press (1964; Zbl 0138.02501)] where he showed that each \(\frac 32\)-transitive group is either primitive or is a Frobenius group.
The authors of the present paper define a transitive permutation group \(G\) to be strongly \(\frac 32\)-transitive when the nonprincipal constituents of the permutation character of \(G\) are all distinct and have the same degree. They note that Wielandt also showed that a strongly \(\frac 32\)-transitive group is either \(\frac 32\)-transitive or a regular Abelian group. The present paper classifies the primitive \(\frac 32\)-transitive groups which are not affine.
Theorem 1.1. A primitive \(\frac 32\)-transitive group is either affine or almost simple.
Theorem 1.2. An almost simple \(\frac 32\)-transitive group \(G\) is either (i) \(2\)-transitive; (ii) of degree \(21\) with \(G\cong A_7\) or \(S_7\); or (iii) of degree \(2^{f-1}(2^f-1)\) with \(f\) an odd prime and \(G\cong\text{PSL}_2(2^f)\) or \(\text{P}\Gamma\)L

MSC:

20B20 Multiply transitive finite groups
20B05 General theory for finite permutation groups
20B15 Primitive groups

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GAP; Magma
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