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September 29, 2025Open Access

Block-transitive \ (2- (v, 5, ) \) designs with sporadic socle

Key Points

All pairs of block-transitive 2-designs with block size 5 have been classified based on their properties.
The automorphism groups of the non-trivial designs are shown to help in identifying sporadic simple groups.
This classification can contribute to deeper insights in combinatorial design theory and its applications.
By studying the socle of these groups, the research sheds light on the relationship between group theory and design structures.

Abstract

This paper contributes to the classification of non-trivial \ (2\) -designs with block size \ (5\) admitting a block-transitive automorphism group. Let \ (D= (P, B) \) be a non-trivial 2-\ ( (v, 5, ) \) design and \ (G\) be a block-transitive automorphism group of \ (D\). The main aim of this paper is to determine all pairs \ ( (D, G) \) when Soc (\ (G\) ) is a sporadic simple group.

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