Geodesic
Minimal supplements of maximal tori in their normalizers for the groups $F_4(q)$
Izvestiya. Mathematics , Tome 86 (2022) no. 1, pp. 126-149.

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

Let $G$ be a finite group of Lie type $F_4$ and $W$ the Weyl group of $G$. For every maximal torus $T$ of $G$, we find the minimal order of a supplement of $T$ in its algebraic normalizer $N(G,T)$. In particular, we find all the maximal tori that have a complement in $N(G,T)$. Let $T$ correspond to an element $w$ of $W$. We find the minimal orders of the lifts of the elements $w$ in $N(G,T)$.
Keywords: finite group of Lie type $F_4$, Weyl group, algebraic normalizer, minimal supplement.
Mots-clés : maximal torus 
@article{IM2_2022_86_1_a3,
     author = {A. A. Gal't and A. M. Staroletov},
     title = {Minimal supplements of maximal tori in their normalizers for the groups $F_4(q)$},
     journal = {Izvestiya. Mathematics },
     pages = {126--149},
     publisher = {mathdoc},
     volume = {86},
     number = {1},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2022_86_1_a3/}
}
TY  - JOUR
AU  - A. A. Gal't
AU  - A. M. Staroletov
TI  - Minimal supplements of maximal tori in their normalizers for the groups $F_4(q)$
JO  - Izvestiya. Mathematics 
PY  - 2022
SP  - 126
EP  - 149
VL  - 86
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2022_86_1_a3/
LA  - en
ID  - IM2_2022_86_1_a3
ER  - 
%0 Journal Article
%A A. A. Gal't
%A A. M. Staroletov
%T Minimal supplements of maximal tori in their normalizers for the groups $F_4(q)$
%J Izvestiya. Mathematics 
%D 2022
%P 126-149
%V 86
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2022_86_1_a3/
%G en
%F IM2_2022_86_1_a3
A. A. Gal't; A. M. Staroletov. Minimal supplements of maximal tori in their normalizers for the groups $F_4(q)$. Izvestiya. Mathematics , Tome 86 (2022) no. 1, pp. 126-149. http://geodesic.mathdoc.fr/item/IM2_2022_86_1_a3/

[1] J. Tits, “Normalisateurs de tores. I. Groupes de coxeter étendus”, J. Algebra, 4 (1966), 96–116 | DOI | MR | Zbl

[2] J. Adams, Xuhua He, “Lifting of elements of Weyl groups”, J. Algebra, 485 (2017), 142–165 | DOI | MR | Zbl

[3] A. A. Gal't, “On the splitting of the normalizer of a maximal torus in the exceptional linear algebraic groups”, Izv. Math., 81:2 (2017), 269–285 | DOI | DOI | MR | Zbl

[4] A. Galt, “On splitting of the normalizer of a maximal torus in orthogonal groups”, J. Algebra Appl., 16:9 (2017), 1750174, 23 pp. | DOI | MR | Zbl

[5] A. Galt, “On splitting of the normalizer of a maximal torus in linear groups”, J. Algebra Appl., 14:7 (2015), 1550114, 20 pp. | DOI | MR | Zbl

[6] A. A. Gal't, “On the splitting of the normalizer of a maximal torus in symplectic groups”, Izv. Math., 78:3 (2014), 443–458 | DOI | DOI | MR | Zbl

[7] M. Curtis, A. Wiederhold, B. Williams, “Normalizers of maximal tori”, Localization in group theory and homotopy theory, and related topics (Battelle Seattle Res. Center, Seattle, WA, 1974), Lecture Notes in Math., 418, Springer, Berlin, 1974, 31–47 | DOI | MR | Zbl

[8] A. Galt, A. Staroletov, “On splitting of the normalizer of a maximal torus in $E_6(q)$”, Algebra Colloq., 26:2 (2019), 329–350 | DOI | MR | Zbl

[9] A. A. Galt, A. M. Staroletov, “On splitting of the normalizer of a maximal torus in $E_7(q)$ and $E_8(q)$”, Siberian Adv. Math., 31:4 (2021), 229–267 | DOI | DOI

[10] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups, Clarendon Press, Oxford, 1985, xxxiv+252 pp. | MR | Zbl

[11] R. W. Carter, Simple groups of Lie type, Pure Appl. Math., 28, John Wiley Sons, London–New York–Sydney, 1972, viii+331 pp. | MR | Zbl

[12] R. W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Pure Appl. Math., John Wiley Sons, Inc., New York, 1985, xii+544 pp. | MR | Zbl

[13] A. A. Buturlakin, M. A. Grechkoseeva, “The cyclic structure of maximal tori of the finite classical groups”, Algebra and Logic, 46:2 (2007), 73–89 | DOI | MR | Zbl

[14] W. Bosma, J. Cannon, C. Playoust, “The Magma algebra system. I. The user language”, J. Symbolic Comput., 24:3-4 (1997), 235–265 | DOI | MR | Zbl

[15] GAP – Groups, Algorithms, Programming – a system for computational discrete algebra, Version 4.10.1, 2019 https://www.gap-system.org

[16] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Ch. IV: Groupes de Coxeter et systèmes de Tits. Ch. V: Groupes engendrés par des réflexions. Ch. VI: Systèmes de racines, Actualites Sci. Indust., 1337, Hermann, Paris, 1968, 288 pp. | MR | MR | Zbl | Zbl

[17] K. Shinoda, “The conjugacy classes of Chevalley groups of type $(F_4)$ over finite fields of characteristic $2$”, J. Fac. Sci. Univ. Tokyo Sect. I A Math., 21 (1974), 133–159 | MR | Zbl

[18] R. Lawther, “The action of $F_4(q)$ on cosets of $B_4(q)$”, J. Algebra, 212:1 (1999), 79–118 | DOI | MR | Zbl