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August 26, 2025Open Access

Block-Transitive 3-(v, 5, λ) Designs with Sporadic or Alternating Socle

Key Points

All classified 3-designs feature a block size of 5, enriching the understanding of their properties.
Sporadic simple groups and alternating groups play crucial roles in the classification of these designs.
Using group theory, we explored transitive actions on blocks, defining their socle structures.
Highlights the complexities of block-transitive designs, suggesting future research directions into group actions.

Abstract

This paper presents a systematic classification of 3-designs with block size 5. Let be a nontrivial 3-(v, 5, λ) design, and let G ≤ Aut act transitively on blocks. We classify all such pairs (, G) where the socle of G is either: one of the 26 sporadic simple groups, or an alternating group An (n ≥ 5).

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