On Intersections of Nilpotent Subgroups in Finite Groups with Simple Socle from the “Atlas of Finite Groups” - Zenkov, V. I. (Proceedings of the Steklov Institute of Mathematics, 2023)
Earlier, the author described up to conjugacy all pairs 𝐴 𝐵 of nilpotent subgroups of a finite group 𝐺 with socle subscript 𝐿 2 𝑞 for which 𝐴 superscript 𝐵 𝑔 1 for any element of 𝐺 . A similar description was obtained by the author later for primary subgroups 𝐴 and 𝐵 of a finite group 𝐺 with socle subscript 𝐿 𝑛 superscript 2 𝑚 . In this paper, we describe up to conjugacy all pairs 𝐴 𝐵 of nilpotent subgroups of a finite group 𝐺 with simple socle from the “Atlas of Finite Groups” for which 𝐴 superscript 𝐵 𝑔 1 for any element 𝑔 of 𝐺 . The results obtained in the considered cases confirm the hypothesis (Problem 15.40 from the “Kourovka Notebook”) that a finite simple nonabelian group 𝐺 for any nilpotent subgroups 𝑁 contains an element 𝑔 such that 𝑁 superscript 𝑁 𝑔 1 .