Some small Sylow numbers

Wolfgang Knapp; Peter Schmid

doi:10.1515/jgth-2025-0052

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Article

Some small Sylow numbers

Wolfgang Knapp and Peter Schmid

Published/Copyright: June 19, 2025

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From the journal Journal of Group Theory

Abstract

Given a prime 𝑝 and a finite group 𝐺, let s p ⁢ ( G ) denote the number of Sylow 𝑝-subgroups of 𝐺. Suppose that this Sylow number is small in terms of the prime 𝑝. What can be said, then, about the structure of 𝐺, the order of a Sylow 𝑝-subgroup, or the union of the Sylow 𝑝-subgroups? In answering these questions, it is appropriate to assume that 𝐺 is generated by its Sylow 𝑝-subgroups and that the intersection of the normalizers of the Sylow 𝑝-subgroups is trivial (which does not alter s p ⁢ ( G ) ). If 𝐺 is such a 𝑝-reduced group, it is a transitive permutation group of degree s p ⁢ ( G ) . In this paper, we shall treat cases where s p ⁢ ( G ) < 1 + 3 ⁢ p + 2 ⁢ p 2 , and we shall determine the structure of 𝐺 when s p ⁢ ( G ) ≤ 1 + p + p 2 .

Communicated by: Andrea Lucchini

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Received: 2025-03-26

Revised: 2025-05-20

Published Online: 2025-06-19

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https://doi.org/10.1515/jgth-2025-0052