On Intersections of Nilpotent Subgroups in Finite Groups with Simple Socle from the “Atlas of Finite Groups”

• Published in: Proceedings of the Steklov Institute of Mathematics Vol. 323; no. Suppl 1; pp. S321 - S332
• Main Author: Zenkov, V. I.
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Moscow Pleiades Publishing 01.12.2023; Springer Nature B.V
• Subjects: Group theory; Mathematics; Mathematics and Statistics; Subgroups; nilpotent subgroup; finite group; intersection of subgroups; Fitting subgroup
• ISSN: 0081-5438; 1531-8605
• DOI: 10.1134/S0081543823060251

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 .