Geodesic
Toward a sharp Baer–Suzuki theorem for the $\pi$-radical: unipotent elements of groups of Lie type
Algebra i logika, Tome 62 (2023) no. 6, pp. 708-741

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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
Mots-clés : unipotent element. 
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     author = {A. -M. Liu and Zh. Wang and D. O. Revin},
     title = {Toward a sharp {Baer{\textendash}Suzuki} theorem for the $\pi$-radical: unipotent elements of groups of {Lie} type},
     journal = {Algebra i logika},
     pages = {708--741},
     publisher = {mathdoc},
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     number = {6},
     year = {2023},
     language = {ru},
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A. -M. Liu; Zh. Wang; D. O. Revin. Toward a sharp Baer–Suzuki theorem for the $\pi$-radical: unipotent elements of groups of Lie type. Algebra i logika, Tome 62 (2023) no. 6, pp. 708-741. http://geodesic.mathdoc.fr/item/AL_2023_62_6_a1/