The proof of Ore's conjecture [after Ellers-Gordeev and Liebeck-O'Brien-Shalev-Tiep]

Malle, Gunter

Séminaire Bourbaki volume 2012/2013 : exposés 1059-1073 - Avec table par noms d'auteurs de 1948/49 à 2012/13, Astérisque, no. 361 (2014), Exposé no. 1069, 24 p.

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MR Zbl

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     author = {Malle, Gunter},
     title = {The proof of {Ore's} conjecture [after {Ellers-Gordeev} and {Liebeck-O'Brien-Shalev-Tiep]}},
     booktitle = {S\'eminaire Bourbaki volume 2012/2013 : expos\'es 1059-1073 - Avec table par noms d'auteurs de 1948/49 \`a 2012/13},
     series = {Ast\'erisque},
     note = {talk:1069},
     pages = {325--348},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {361},
     year = {2014},
     mrnumber = {3289286},
     zbl = {1356.20008},
     language = {en},
     url = {http://archive.numdam.org/item/AST_2014__361__325_0/}
}

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Malle, Gunter. The proof of Ore's conjecture [after Ellers-Gordeev and Liebeck-O'Brien-Shalev-Tiep], dans Séminaire Bourbaki volume 2012/2013 : exposés 1059-1073 - Avec table par noms d'auteurs de 1948/49 à 2012/13, Astérisque, no. 361 (2014), Exposé no. 1069, 24 p. http://archive.numdam.org/item/AST_2014__361__325_0/

Bibliographie
Cité par

[AH85] Z. Arad & M. Herzog (eds.) - Products of conjugacy classes in groups. Lecture Notes in Math., vol. 1112. Springer-Verlag, Berlin, 1985. | MR | Zbl

[Ber72] E. Bertram - "Even permutations as a product of two conjugate cycles", J. Combinatorial Theory Ser. A 12 (1972), p. 368-380. | DOI | MR | Zbl

[Bl94] H. I. Blau - "A fixed-point theorem for central elements in quasisimple groups". Proc. Amer. Math. Soc. 122 (1994). no. 1, p. 79-84. | DOI | MR | Zbl

[Bo93] O. Bonten - Über Kommutatoren in endlichen einfachen Gruppen, Aach-ener Beitrage zur Math., vol. 7, Verlag der Augustinus-Buchhandlung, Aachen, 1993. | Zbl

[Bor83] A. Borel - "On free subgroups of semisimple groups", Enseign. Math. (2) 29 (1983), no. 1-2, p. 151-164. | MR | Zbl

[Br83] J. L. Brenner - "Covering theorems for finasigs. X. The group $G = P S L (n, q)$ had a class $C$ such that $C C = G$ ", Ars Combin. 16 (1983), p. 57-67. | MR | Zbl

[Ca85] R. W. Carter - Finite groups of Lie type, Pure Appl. Math. (New York), John Wiley & Sons Inc., New York, 1985. | MR | Zbl

[Hs65] X. Cheng-Hao - "The commutators of the alternating group", Sci. Sinica 14 (1965), p. 339-342. | MR | Zbl

[EG94] E. W. Ellers & N. Gordeev - "Gauss decomposition with prescribed semisimple part in classical Chevalley groups", Comm. Algebra 22 (1994), no. 14, p. 5935-5950. | DOI | MR | Zbl

[EG95] E. W. Ellers & N. Gordeev, "Gauss decomposition with prescribed semisimple part in Chevalley groups II. Exceptional cases", Comm. Algebra 23 (1995), no. 8, p. 3085-3098. | DOI | MR | Zbl

[EG96] E. W. Ellers & N. Gordeev, "Gauss decomposition with prescribed semisimple part in Chevalley groups. III. Finite twisted groups", Comm. Algebra 24 (1996), no. 14, p. 4447-4475. | DOI | MR | Zbl

[EG98] E. W. Ellers & N. Gordeev, - "On the conjectures of J. Thompson and O. Ore", Trans. Amer. Math. Soc. 350 (1998), no. 9, p. 3657-3671. | DOI | MR | Zbl

[EGH99] E. W. Ellers, N. Gordeev & M. Herzog - "Covering numbers for Chevalley groups", Israel J. Math. 111 (1999), p. 339-372. | DOI | MR | Zbl

[FA87] E. Fisman & Z. Arad - "A proof of Szep's conjecture on nonsimplicity of certain finite groups", J. Algebra 108 (1987), no. 2, p. 340-354. | DOI | MR | Zbl

[Fi02] W. B. Fite - "On metabelian groups", Trans. Amer. Math. Soc. 3 (1902), no. 3, p. 331-353. | DOI | JFM | MR

[FG12] J. Fulman & R. Guralnick - "Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements", Trans. Amer. Math. Soc. 364 (2012), no. 6, p. 3023-3070. | DOI | MR | Zbl

[Ger77] P. Gérardin - "Weil representations associated to finite fields", J. Algebra 46 (1977), no. 1, p. 54-101. | DOI | MR | Zbl

[Go49] M. Gotô - "A theorem on compact semi-simple groups", J. Math. Soc. Japan 1 (1949), p. 270-272. | DOI | MR | Zbl

[Gow88] R. Gow - "Commutators in the symplectic group", Arch. Math. (Basel) 50 (1988), no. 3, p. 204-209. | DOI | MR | Zbl

[Gow00] R. Gow - "Commutators in finite simple groups of Lie type", Bull. London Math. Soc. 32 (2000), no. 3, p. 311-315. | DOI | MR | Zbl

[Gu80] R. Guralnick - "Expressing group elements as commutators", Rocky Mountain J. Math. 10 (1980), no. 3. p. 651-654. | DOI | MR | Zbl

[Gu10] R. Guralnick, "Commutators and wreath products", in Character theory of finite groups, Contemp. Math., vol. 524. Amer. Math. Soc., Providence, 2010, p. 79-82. | DOI | MR | Zbl

[GMST02] R. Guralnick, K. Magaard, J. Saxl & P. H. Tiep - "Cross characteristic representations of symplectic and unitary groups", J. Algebra 257 (2002), no. 2, p. 291-347. | DOI | MR | Zbl

[GM13] R. Guralnick & G. Malle - "Products and commutators of classes in algebraic groups", submitted, arXiv: 1302.0182. | DOI | MR | Zbl

[GM12] R. Guralnick & G. Malle, "Products of conjugacy classes and fixed point spaces", J. Amer. Math. Soc. 25 (2012), no. 1, p. 77-121. | DOI | MR | Zbl

[GMT13] R. M. Guralnick, G. Malle & P. H. Tiep - "Products of conjugacy classes in finite and algebraic simple groups", Adv. Math. 234 (2013), p. 618-652. | MR | Zbl

[GT13] R. M. Guralnick & P. H. Tiep - "The Waring problem for finite quasisimple groups II", preprint, arXiv: 1302.0333.

[Is77] I. M. Isaacs - "Commutators and the commutator subgroup", Amer. Math. Monthly 84 (1977), no. 9, p. 720-722. | DOI | MR | Zbl

[Ito51] N. Ito - "A theorem on the alternating group $𝔄_{n} (n \geq 5)$ ", Math. Japonicae 2 (1951), p. 59-60. | MR | Zbl

[JL013] S. Jambor, M. W. Liebeck & E. A. O'Brien - "Some word maps that are non-surjective on infinitely many finite simple groups", Bull. Lond. Math. Soc. 45 (2013), no. 5, p. 907-910. | DOI | MR | Zbl

[La04] M. Larsen - "Word maps have large image", Israel J. Math. 139 (2004), p. 149-156. | DOI | MR | Zbl

[LS09] M. Larsen & A. Shalev - "Word maps and Waring type problems", J. Amer. Math. Soc. 22 (2009), no. 2, p. 437-466. | DOI | MR | Zbl

[LST11] M. Larsen, A. Shalev & P. H. Tiep - "The Waring problem for finite simple groups", Ann. of Math. (2) 174 (2011), no. 3, p. 1885-1950. | DOI | MR | Zbl

[LL98] R. Lawther & M. W. Liebeck - "On the diameter of a Cayley graph of a simple group of Lie type based on a conjugacy class", J. Combin. Theory Ser. A 83 (1998), no. 1, p. 118-137. | DOI | MR | Zbl

[Lev94] A. Lev - "Products of cyclic similarity classes in the groups $G L_{n} (F)$ ", Linear Algebra Appl. 202 (1994), p. 235-266. | DOI | MR | Zbl

[LOST] M. W. Liebeck, E. A. O'Brien, A. Shalev & P. H. Tiep - "The Ore conjecture", J. Eur. Math, Soc. (JEMS) 12 (2010), no. 4, p. 939-1008. | DOI | EuDML | MR | Zbl

[LOST2] M. W. Liebeck, E. A. O'Brien, A. Shalev & P. H. Tiep, "Commutators in finite quasisimple groups", Bull. Lond. Math. Soc. 43 (2011), no. 6, p. 1079-1092. | DOI | MR | Zbl

[LOST3] M. W. Liebeck, E. A. O'Brien, A. Shalev & P. H. Tiep, "Products of squares in finite simple groups", Proc. Amer. Math. Soc. 140 (2012). no. 1, p. 21-33. | DOI | MR | Zbl

[LS01] M. W. Liebeck & A. Shalev - "Diameters of finite simple groups: sharp bounds and applications", Ann. of Math. (2) 154 (2001), no. 2, p. 383-406. | DOI | MR | Zbl

[LM99] F. Lübeck & G. Malle - " $(2, 3)$ -generation of exceptional groups", J. London Math. Soc. (2) 59 (1999), no. 1, p. 109-122. | DOI | MR | Zbl

[Lu84] G. Lusztig - Characters of reductive groups over a finite field, Ann. of Math. Stud., vol. 107, Princeton Univ. Press, Princeton, 1984. | MR | Zbl

[M88] G. Malle - "Exceptional groups of Lie type as Galois groups", J. reine angew. Math. 392 (1988), p. 70-109. | EuDML | MR | Zbl

[MSW94] G. Malle, J. Saxl & T. Weigel - "Generation of classical groups", Geom. Dedicata 49 (1994), no. 1, p. 85-116. | DOI | MR | Zbl

[MZ96] C. Martinez & E. Zelmanov - "Products of powers in finite simple groups", Israel J. Math. 96 (1996), no. part B, p. 469-479. | DOI | MR | Zbl

[My77] J. Mycielski - "Research Problems: Can One Solve Equations in Group?", Amer. Math. Monthly 84 (1977), no. 9, p. 723-726. | MR | Zbl

[NPC84] J. Neubüser, H. Pahlings & E. Cleuvers - "Each sporadic finasig $G$ has a class $C$ such that $C C = G$ ", Abstr. Amer. Math. Soc. 34 (1984), no. 6.

[NP11] N. Nikolov & L. Pyber - "Product decompositions of quasirandom groups and a Jordan type theorem", J. Eur. Math. Soc. (JEMS) 13 (2011), no. 4, p. 1063-1077. | DOI | EuDML | MR | Zbl

[NS07] N. Nikolov & D. Segal - "On finitely generated profinite groups. I. Strong completeness and uniform bounds", Ann. of Math. (2) 165 (2007), no. 1, p. 171-238. | DOI | MR | Zbl

[Ore] O. Ore -"Some remarks on commutators", Proc. Amer. Math. Soc. 2 (1951), p. 307-314. | DOI | MR | Zbl

[PW62] S. Pasiencier & H.-C. Wang "Commutators in a semi-simple Lie group", Proc. Amer. Math. Soc. 13 (1962), p. 907-913. | DOI | MR | Zbl

[Ree64] R. Ree - "Commutators in semi-simple algebraic groups", Proc. Arner. Math. Soc. 15 (1964), p. 457-460. | DOI | MR | Zbl

[SW97] J. Saxl & J. S. Wilson - "A note on powers in simple groups", Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 1, p. 91-94. | DOI | MR | Zbl

[Sh09] A. Shalev - "Word maps, conjugacy classes, and a noncommutative Waring-type theorem", Ann. of Math. (2) 170 (2009), no. 3, p. 1383-1416. | DOI | MR | Zbl

[So86] A. R. Sourour - "A factorization theorem for matrices", Linear and Multilinear Algebra 19 (1986), no. 2, p. 141-147. | DOI | MR | Zbl

[Th61] R. C. Thompson "Commutators in the special and general linear groups", Trans. Amer. Math. Soc. 101 (1961), p. 16-33. | DOI | MR | Zbl

[Th62] R. C. Thompson, "On matrix commutators", Portugal. Math. 21 (1962), p. 143-153. | EuDML | MR | Zbl

[Th62a] R. C. Thompson, "Commutators of matrices with coefficients from the field of two elements", Duke Math. J. 29 (1962), p. 367-373. | MR | Zbl

[TZ96] P. H. Tiep & A. E. Zalesskii "Minimal characters of the finite classical groups", Comm. Algebra 24 (1996), no. 6, p. 2093-2167. | DOI | MR | Zbl