A. -M. Liu, Zh. Wang, D. O. Revin, “Toward a sharp Baer–Suzuki theorem for the <nobr>$\pi$</nobr>-radical: unipotent elements of groups of Lie type”, Algebra Logika, 2023, Volume 62, Number 6,Pages <nobr>708

This article is cited in 1 paper

Toward a sharp Baer–Suzuki theorem for the $\pi$-radical: unipotent elements of groups of Lie type

A. -M. Liu^a, Zh. Wang^a, D. O. Revin^b

^a School of Math. Stat., Hainan Univ., Haikou, CHINA
^b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: We will look into the following conjecture, which, if valid, would allow us to formulate an unimprovable analog of the Baer–Suzuki theorem for the $\pi$-radical of a finite group $($here $\pi$ is an arbitrary set of primes$)$. For an odd prime number $r$, put $m=r$, if $r=3$, and $m=r-1$ if $r\geqslant 5$. Let $L$ be a simple non-Abelian group whose order has a prime divisor $s$ such that $s=r$ if $r$ divides $|L|$, and $s>r$ otherwise. Suppose also that $x$ is an automorphism of prime order of $L$. Then some $m$ conjugates of $x$ in the group $\langle L,x\rangle$ generate a subgroup of order divisible by $s$. The conjecture is confirmed for the case where $L$ is a group of Lie type and $x$ is an automorphism induced by a unipotent element.

Keywords: $\pi$-radical, Baer–Suzuki $\pi$-theorem, group of Lie type, unipotent element.

UDC: 512.542

Received: 06.12.2023
Revised: 02.12.2024

DOI: 10.33048/alglog.2023.62.602