Sylow subgroups for distinct primes and intersection of nilpotent subgroups

2025/05/27 by Lisi, Francesca, Sabatini, Luca · 2 citations
#20B30 #20D20 #20Fxx #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2505.21222

Abstract

Let G be a finite group, and let (P_i)_i=1ⁿ be Sylow subgroups for distinct primes p₁,…,p_n. We conjecture that there exists x ∈ G such that P_i ∩ P_i^x is minimal in \ P_i ∩ P_i^g : g ∈ G\ for all i. As a first step in this direction, we show that a finite group cannot be covered by (proper) Sylow normalizers for distinct primes. Then, we settle the conjecture in two opposite situations: symmetric groups of large degree and metanilpotent groups of odd order. Applications concerning the intersection of nilpotent subgroups are discussed.

Citations

• Intersections of Sylow 2-subgroups in symmetric groups
• On the regularity number of a finite group and other base-related invariants
• Unsolved Problems in Group Theory. The Kourovka Notebook
• On the number and sizes of double cosets of Sylow subgroups of the symmetric group

Cited by

• On the intersections of nilpotent subgroups in simple groups
• Intersections of Sylow 2-subgroups in symmetric groups

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