Flag-transitive $ 2 $-designs with block size 5 and alternating groups

Jiaxin Shen; Yuqing Xia; Jiaxin Shen; Yuqing Xia

doi:10.3934/math.2025469

AIMS Mathematics

2025, Volume 10, Issue 5: 10308-10323. doi: 10.3934/math.2025469

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Research article

Flag-transitive $ 2 $-designs with block size 5 and alternating groups

Jiaxin Shen ^,,
Yuqing Xia

School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China

Received: 15 November 2024 Revised: 24 April 2025 Accepted: 28 April 2025 Published: 06 May 2025
MSC : 05B05, 05B25, 05E18, 20B25

This paper contributes to the classification of flag-transitive 2-designs with block size 5. In a recent paper, the flag-transitive automorphism groups of such designs are reduced to point-primitive groups of affine type and almost simple type, and a classification is given of such automorphism groups with sporadic socle. In the present paper, we classify such designs admitting a flag-transitive automorphism group whose socle is an alternating group. We prove that there are precisely six such designs and determine the corresponding automorphism groups.
- $ 2 $-design,
- alternating group,
- flag-transitive,
- automorphism group,
- primitive group,
- classification
Citation: Jiaxin Shen, Yuqing Xia. Flag-transitive $ 2 $-designs with block size 5 and alternating groups[J]. AIMS Mathematics, 2025, 10(5): 10308-10323. doi: 10.3934/math.2025469

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Abstract

This paper contributes to the classification of flag-transitive 2-designs with block size 5. In a recent paper, the flag-transitive automorphism groups of such designs are reduced to point-primitive groups of affine type and almost simple type, and a classification is given of such automorphism groups with sporadic socle. In the present paper, we classify such designs admitting a flag-transitive automorphism group whose socle is an alternating group. We prove that there are precisely six such designs and determine the corresponding automorphism groups.

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