Products of squares in finite simple groups
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- by Martin W. Liebeck, E. A. O’Brien, Aner Shalev and Pham Huu Tiep PDF
- Proc. Amer. Math. Soc. 140 (2012), 21-33 Request permission
Abstract:
The Ore conjecture, proved by the authors, states that every element of every finite non-abelian simple group is a commutator. In this paper we use similar methods to prove that every element of every finite simple group is a product of two squares. This can be viewed as a non-commutative analogue of Lagrange’s four squares theorem. Results for higher powers are also obtained.References
- Edward Bertram, Even permutations as a product of two conjugate cycles, J. Combinatorial Theory Ser. A 12 (1972), 368–380. MR 297853, DOI 10.1016/0097-3165(72)90102-1
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- Michel Broué and Jean Michel, Blocs et séries de Lusztig dans un groupe réductif fini, J. Reine Angew. Math. 395 (1989), 56–67 (French). MR 983059, DOI 10.1515/crll.1989.395.56
- François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841, DOI 10.1017/CBO9781139172417
- Erich W. Ellers and Nikolai Gordeev, On the conjectures of J. Thompson and O. Ore, Trans. Amer. Math. Soc. 350 (1998), no. 9, 3657–3671. MR 1422600, DOI 10.1090/S0002-9947-98-01953-9
- J. Fulman and R. M. Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups, Trans. Amer. Math. Soc. (to appear).
- The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.12, 2008, http://www.gap-system.org.
- Shelly Garion and Aner Shalev, Commutator maps, measure preservation, and $T$-systems, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4631–4651. MR 2506422, DOI 10.1090/S0002-9947-09-04575-9
- Robert M. Guralnick and Frank Lübeck, On $p$-singular elements in Chevalley groups in characteristic $p$, Groups and computation, III (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 169–182. MR 1829478
- R.M. Guralnick and G. Malle, Products of conjugacy classes and fixed point spaces, arXiv:1005.3756 (preprint).
- Robert M. Guralnick and Pham Huu Tiep, Cross characteristic representations of even characteristic symplectic groups, Trans. Amer. Math. Soc. 356 (2004), no. 12, 4969–5023. MR 2084408, DOI 10.1090/S0002-9947-04-03477-4
- Gerhard Hiss and Gunter Malle, Low-dimensional representations of special unitary groups, J. Algebra 236 (2001), no. 2, 745–767. MR 1813499, DOI 10.1006/jabr.2000.8513
- Dale H. Husemoller, Ramified coverings of Riemann surfaces, Duke Math. J. 29 (1962), 167–174. MR 136726
- Adalbert Kerber and Bernd Wagner, Gleichungen in endlichen Gruppen, Arch. Math. (Basel) 35 (1980), no. 3, 252–262 (German). MR 583596, DOI 10.1007/BF01235344
- Alexander S. Kleshchev and Pham Huu Tiep, Representations of finite special linear groups in non-defining characteristic, Adv. Math. 220 (2009), no. 2, 478–504. MR 2466423, DOI 10.1016/j.aim.2008.09.011
- Michael Larsen and Aner Shalev, Characters of symmetric groups: sharp bounds and applications, Invent. Math. 174 (2008), no. 3, 645–687. MR 2453603, DOI 10.1007/s00222-008-0145-7
- Michael Larsen and Aner Shalev, Word maps and Waring type problems, J. Amer. Math. Soc. 22 (2009), no. 2, 437–466. MR 2476780, DOI 10.1090/S0894-0347-08-00615-2
- M. Larsen, A. Shalev and P.H. Tiep, The Waring problem for finite simple groups, Annals of Math. (to appear).
- Martin W. Liebeck, E. A. O’Brien, Aner Shalev, and Pham Huu Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 4, 939–1008. MR 2654085, DOI 10.4171/JEMS/220
- M.W. Liebeck and G.M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, preprint.
- Frank Lübeck, Data for Finite Groups of Lie Type and Related Algebraic Groups. www.math.rwth-aachen.de/$\sim$Frank.Luebeck/chev.
- George Lusztig, Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR 742472, DOI 10.1515/9781400881772
- Dan Segal, Words: notes on verbal width in groups, London Mathematical Society Lecture Note Series, vol. 361, Cambridge University Press, Cambridge, 2009. MR 2547644, DOI 10.1017/CBO9781139107082
- Aner Shalev, Word maps, conjugacy classes, and a noncommutative Waring-type theorem, Ann. of Math. (2) 170 (2009), no. 3, 1383–1416. MR 2600876, DOI 10.4007/annals.2009.170.1383
- Pham Huu Tiep and Alexander E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra 24 (1996), no. 6, 2093–2167. MR 1386030, DOI 10.1080/00927879608825690
- Pham Huu Tiep and Alexander E. Zalesskii, Some characterizations of the Weil representations of the symplectic and unitary groups, J. Algebra 192 (1997), no. 1, 130–165. MR 1449955, DOI 10.1006/jabr.1996.6943
- W. R. Unger, Computing the character table of a finite group, J. Symbolic Comput. 41 (2006), no. 8, 847–862. MR 2246713, DOI 10.1016/j.jsc.2006.04.002
Additional Information
- Martin W. Liebeck
- Affiliation: Department of Mathematics, Imperial College, Queen’s Gate, London SW7 2BZ, United Kingdom
- MR Author ID: 113845
- ORCID: 0000-0002-3284-9899
- Email: m.liebeck@imperial.ac.uk
- E. A. O’Brien
- Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
- MR Author ID: 251889
- Email: e.obrien@auckland.ac.nz
- Aner Shalev
- Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
- MR Author ID: 228986
- ORCID: 0000-0001-9428-2958
- Email: shalev@math.huji.ac.il
- Pham Huu Tiep
- Affiliation: Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721
- MR Author ID: 230310
- Email: tiep@math.arizona.edu
- Received by editor(s): May 21, 2010
- Received by editor(s) in revised form: November 1, 2010
- Published electronically: May 6, 2011
- Additional Notes: The first author acknowledges the support of a Maclaurin Fellowship from the New Zealand Institute of Mathematics and its Applications
The second author acknowledges the support of the Marsden Fund of New Zealand (grant UOA 0721)
The third author acknowledges the support of ERC Advanced Grant 247034, an EPSRC Visiting Fellowship, an Israel Science Foundation Grant, and Bi-National Science Foundation grant United States-Israel 2008194.
The fourth author acknowledges the support of the NSF (grant DMS-0901241) - Communicated by: Jonathan I. Hall
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 21-33
- MSC (2010): Primary 20C33, 20D06
- DOI: https://doi.org/10.1090/S0002-9939-2011-10878-5
- MathSciNet review: 2833514