Abstract
We refer to \( d(G) \) as the minimal size of a generating set of a finite group \( G \), and say that \( G \) is \( d \)-generated if \( d(G)\leq d \). A transitive permutation group \( G \) is called \( \frac{3}{2} \)-transitive if the point stabilizer \( G_{\alpha} \) is nontrivial and its orbits distinct from \( \{\alpha\} \) are of the same size. We prove that \( d(G)\leq 4 \) for every primitive \( \frac{3}{2} \)-transitive permutation group \( G \) and, moreover, \( G \) is \( 2 \)-generated except for the rather particular solvable affine groups that we describe completely. In particular, all finite \( 2 \)-transitive and \( 2 \)-homogeneous groups are \( 2 \)-generated. We also show that every finite group whose abelian subgroups are cyclic is \( 2 \)-generated, and so is every Frobenius complement.
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The authors were supported by the Ministry of Science and Higher Education of the Russian Federation (Project FWNF–2022–0002).
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 6, pp. 1213–1223. https://doi.org/10.33048/smzh.2022.63.603
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Vasil’ev, A.V., Zvezdina, M.A. & Churikov, D.V. The Size of a Minimal Generating Set for Primitive \( \frac{3}{2} \)-Transitive Groups. Sib Math J 63, 1041–1048 (2022). https://doi.org/10.1134/S0037446622060039
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DOI: https://doi.org/10.1134/S0037446622060039
Keywords
- minimal generating set
- primitive permutation group
- \( \frac{3}{2} \)-transitive group
- \( 2 \)-transitive group
- \( 2 \)-homogeneous group
- Frobenius complement