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.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{AL_2023_62_6_a1,
     author = {A. -M. Liu and Zh. Wang and D. O. Revin},
     title = {Toward a sharp {Baer--Suzuki} theorem for the $\pi$-radical: unipotent elements of groups of {Lie} type},
     journal = {Algebra i logika},
     pages = {708--741},
     publisher = {mathdoc},
     volume = {62},
     number = {6},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2023_62_6_a1/}
}
TY  - JOUR
AU  - A. -M. Liu
AU  - Zh. Wang
AU  - D. O. Revin
TI  - Toward a sharp Baer--Suzuki theorem for the $\pi$-radical: unipotent elements of groups of Lie type
JO  - Algebra i logika
PY  - 2023
SP  - 708
EP  - 741
VL  - 62
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/AL_2023_62_6_a1/
LA  - ru
ID  - AL_2023_62_6_a1
ER  - 
%0 Journal Article
%A A. -M. Liu
%A Zh. Wang
%A D. O. Revin
%T Toward a sharp Baer--Suzuki theorem for the $\pi$-radical: unipotent elements of groups of Lie type
%J Algebra i logika
%D 2023
%P 708-741
%V 62
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/AL_2023_62_6_a1/
%G ru
%F AL_2023_62_6_a1
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/

[1] N. Yan, Chzh. U, D. O. Revin, E. P. Vdovin, “O tochnoi teoreme Bera–Suzuki dlya $\pi$-radikala konechnoi gruppy”, Matem. sb., 214:1 (2023), 113–154 | DOI | MR | Zbl

[2] N. Yang, D. O. Revin, E. P. Vdovin, “Baer–Suzuki theorem for the $\pi$-radical”, Isr. J. Math., 245:1 (2021), 173–207 | DOI | MR | Zbl

[3] R. Baer, “Engelsche Elemente Noetherscher Gruppen”, Math. Ann., 133:3 (1957), 256–270 | DOI | MR | Zbl

[4] M. Suzuki, “Finite groups in which the centralizer of any element of order 2 is 2-closed”, Ann. Math. (2), 82 (1965), 191–212 | DOI | MR | Zbl

[5] J. Alperin, R. Lyons, “On conjugacy classes of $p$-elements”, J. Algebra, 19:2 (1971), 536–537 | DOI | MR | Zbl

[6] D. O. Revin, “O $\pi$-teoremakh Bera–Sudzuki”, Sib. matem. zh., 52:2 (2011), 430–440 | MR | Zbl

[7] P. Flavell, S. Guest, R. Guralnick, “Characterizations of the solvable radical”, Proc. Am. Math. Soc., 138:4 (2010), 1161–1170 | DOI | MR | Zbl

[8] N. Gordeev, F. Grunewald, B. Kunyavskii, E. Plotkin, “A description of Baer–Suzuki type of the solvable radical of a finite group”, J. Pure Appl. Algebra, 213:2 (2009), 250–258 | DOI | MR | Zbl

[9] N. Gordeev, F. Grunewald, B. Kunyavski, E. Plotkin, “From Thompson to Baer–Suzuki: a sharp characterization of the solvable radical”, J. Algebra, 323:10 (2010), 2888–2904 | DOI | MR | Zbl

[10] N. Yan, Ch. U, D. O. Revin, “O tochnoi teoreme Bera–Suzuki dlya $\pi$-radikala: sporadicheskie gruppy”, Sib. matem. zh., 63:2 (2022), 464–472

[11] Ch. Van, V. Go, D. O. Revin, “K tochnoi teoreme Bera–Suzuki dlya $\pi$-radikala: isklyuchitelnye gruppy malogo ranga”, Algebra i logika, 62:1 (2023), 3–32

[12] D. O. Revin, A. V. Zavarnitsine, “On generations by conjugate elements in almost simple groups with socle $ ^2F_4(q^2)'$”, J. Group Theory, 27:1 (2024), 119–140 | MR | Zbl

[13] R. M. Guralnick, J. Saxl, “Generation of finite almost simple groups by conjugates”, J. Algebra, 268:2 (2003), 519–571 | DOI | MR | Zbl

[14] R. W. Carter, Simple groups of Lie type, Pure Appl. Math., 28, John Wiley Sons, a Wiley Intersci. Publ., London etc., 1972 | MR

[15] P. B. Kleidman, M. Liebeck, The subgroups structure of finite classical groups, Lond. Math. Soc. Lect. Note Ser., 129, Cambridge Univ. Press, Cambridge etc., 1990 | MR

[16] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985 | MR | Zbl

[17] J. N. Bray, D. F. Holt, C. M. Roney-Dougal, The maximal subgroups of the low-dimensional finite classical groups, Lond. Math. Soc. Lect. Note Ser., 407, Cambridge Univ. Press, Cambridge, 2013 | MR | Zbl

[18] D. Gorenstein, Finite groups, 2nd ed., Chelsea Pub. Co., New York, 1980 | MR | Zbl

[19] A. S. Kondratev, “Podgruppy konechnykh grupp Shevalle”, UMN, 41:1(247) (1986), 57–96 | MR | Zbl

[20] R. A. Wilson, The finite simple groups, Grad. Texts Math., 251, Springer, London, 2009 | DOI | MR | Zbl

[21] S. Gonshaw, M. W. Liebeck, E. A. O'Brien, “Unipotent class representatives for finite classical groups”, J. Group Theory, 20:3 (2017), 505–525 | DOI | MR | Zbl

[22] M. Aschbacher, G. M. Seitz, “Involutions in Chevalley groups over fields of even order”, Nagoya Math. J., 63 (1976), 1–91 | DOI | MR | Zbl