|
Products of squares in finite simple groups
Authors:
Martin W. Liebeck, E. A. O’Brien, Aner Shalev and Pham Huu Tiep
Journal:
Proc. Amer. Math. Soc. 140 (2012), 21-33
MSC (2010):
Primary 20C33, 20D06
Posted:
May 6, 2011
MathSciNet review:
2833514
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
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.
- 1.
Edward
Bertram, Even permutations as a product of two conjugate
cycles, J. Combinatorial Theory Ser. A 12 (1972),
368–380. MR 0297853
(45 #6905)
- 2.
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, http://dx.doi.org/10.1006/jsco.1996.0125
- 3.
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
(90b:20037), http://dx.doi.org/10.1515/crll.1989.395.56
- 4.
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
(92g:20063)
- 5.
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
(98k:20022), http://dx.doi.org/10.1090/S0002-9947-98-01953-9
- 6.
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).
- 7.
THE GAP GROUP, GAP - Groups, Algorithms, and Programming, Version 4.4.12, 2008, http://www.gap-system.org.
- 8.
Shelly
Garion and Aner
Shalev, Commutator maps, measure preservation,
and 𝑇-systems, Trans. Amer. Math.
Soc. 361 (2009), no. 9, 4631–4651. MR 2506422
(2010f:20019), http://dx.doi.org/10.1090/S0002-9947-09-04575-9
- 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
(2002d:20074)
- 10.
R.M. Guralnick and G. Malle, Products of conjugacy classes and fixed point spaces, arXiv:1005.3756 (preprint).
- 11.
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 (electronic). MR 2084408
(2005j:20012), http://dx.doi.org/10.1090/S0002-9947-04-03477-4
- 12.
Gerhard
Hiss and Gunter
Malle, Low-dimensional representations of special unitary
groups, J. Algebra 236 (2001), no. 2,
745–767. MR 1813499
(2001m:20019), http://dx.doi.org/10.1006/jabr.2000.8513
- 13.
Dale
H. Husemoller, Ramified coverings of Riemann surfaces, Duke
Math. J. 29 (1962), 167–174. MR 0136726
(25 #188)
- 14.
Adalbert
Kerber and Bernd
Wagner, Gleichungen in endlichen Gruppen, Arch. Math. (Basel)
35 (1980), no. 3, 252–262 (German). MR 583596
(81k:20015), http://dx.doi.org/10.1007/BF01235344
- 15.
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
(2009j:20022), http://dx.doi.org/10.1016/j.aim.2008.09.011
- 16.
Michael
Larsen and Aner
Shalev, Characters of symmetric groups: sharp bounds and
applications, Invent. Math. 174 (2008), no. 3,
645–687. MR 2453603
(2010g:20022), http://dx.doi.org/10.1007/s00222-008-0145-7
- 17.
Michael
Larsen and Aner
Shalev, Word maps and Waring type
problems, J. Amer. Math. Soc.
22 (2009), no. 2,
437–466. MR 2476780
(2010d:20019), http://dx.doi.org/10.1090/S0894-0347-08-00615-2
- 18.
M. Larsen, A. Shalev and P.H. Tiep, The Waring problem for finite simple groups, Annals of Math. (to appear).
- 19.
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
(2011e:20016), http://dx.doi.org/10.4171/JEMS/220
- 20.
M.W. Liebeck and G.M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, preprint.
- 21.
Frank Lübeck, Data for Finite Groups of Lie Type and Related Algebraic Groups. www.math.rwth-aachen.de/
 Frank.Luebeck/chev.
- 22.
George
Lusztig, Characters of reductive groups over a finite field,
Annals of Mathematics Studies, vol. 107, Princeton University Press,
Princeton, NJ, 1984. MR 742472
(86j:20038)
- 23.
Dan
Segal, Words: notes on verbal width in groups, London
Mathematical Society Lecture Note Series, vol. 361, Cambridge
University Press, Cambridge, 2009. MR 2547644
(2011a:20055)
- 24.
Aner
Shalev, Word maps, conjugacy classes, and a noncommutative
Waring-type theorem, Ann. of Math. (2) 170 (2009),
no. 3, 1383–1416. MR 2600876
(2011e:20017), http://dx.doi.org/10.4007/annals.2009.170.1383
- 25.
Pham
Huu Tiep and Alexander
E. Zalesskii, Minimal characters of the finite classical
groups, Comm. Algebra 24 (1996), no. 6,
2093–2167. MR 1386030
(97f:20018), http://dx.doi.org/10.1080/00927879608825690
- 26.
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
(99d:20074), http://dx.doi.org/10.1006/jabr.1996.6943
- 27.
W.
R. Unger, Computing the character table of a finite group, J.
Symbolic Comput. 41 (2006), no. 8, 847–862. MR 2246713
(2007b:20023), http://dx.doi.org/10.1016/j.jsc.2006.04.002
- 1.
- E. Bertram, Even permutations as a product of two conjugate cycles, J. Comb. Theory Ser. A 12
 , 368-380. MR 0297853 (45:6905)
- 2.
- W. Bosma, J. Cannon and C. Playoust, The MAGMA algebra system. I: The user language, J. Symbolic Comput. 24 (1997), 235-265. MR 1484478
- 3.
- M. Broué and J. Michel, Blocs et séries de Lusztig dans un groupe réductif fini, J. Reine Angew. Math. 395
 , 56-67. MR 983059 (90b:20037)
- 4.
- F. Digne and J. Michel, Representations of Finite Groups of Lie Type, London Mathematical Society Student Texts,
 , Cambridge University Press,  . MR 1118841 (92g:20063)
- 5.
- E.W. Ellers and N. Gordeev, On the conjectures of J. Thompson and O. Ore, Trans. Amer. Math. Soc. 350 (1998), 3657-3671. MR 1422600 (98k:20022)
- 6.
- 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).
- 7.
- THE GAP GROUP, GAP - Groups, Algorithms, and Programming, Version 4.4.12, 2008, http://www.gap-system.org.
- 8.
- S. Garion and A. Shalev, Commutator maps, measure preservation, and
 -systems, Trans. Amer. Math. Soc. 361 (2009), 4631-4651. MR 2506422 (2010f:20019)
- 9.
- R.M. Guralnick and F. Lübeck, On
 -singular elements in Chevalley groups in characteristic  , in: Groups and computation, III (Columbus, OH, 1999), pp. 169-182, Ohio State Univ. Math. Res. Inst. Publ., 8, de Gruyter, Berlin, 2001. MR 1829478 (2002d:20074)
- 10.
- R.M. Guralnick and G. Malle, Products of conjugacy classes and fixed point spaces, arXiv:1005.3756 (preprint).
- 11.
- R. Guralnick and P.H. Tiep, Cross characteristic representations of even characteristic symplectic groups, Trans. Amer. Math. Soc. 356 (2004), 4969-5023. MR 2084408 (2005j:20012)
- 12.
- G. Hiss and G. Malle, Low dimensional representations of special unitary groups, J. Algebra 236
 , 745-767. MR 1813499 (2001m:20019)
- 13.
- D. Husemoller, Ramified coverings of Riemann surfaces, Duke Math. J. 29
 , 167-174. MR 0136726 (25:188)
- 14.
- A. Kerber and B. Wagner, Gleichungen in endlichen Gruppen, Arch. Math. 35
 , 252-262. MR 583596 (81k:20015)
- 15.
- A.S. Kleshchev and P.H. Tiep, Representations of finite special linear groups in non-defining characteristic, Adv. Math. 220 (2009), 478-504. MR 2466423 (2009j:20022)
- 16.
- M. Larsen and A. Shalev, Characters of symmetric groups: Sharp bounds and applications, Invent. Math. 174 (2008), 645-687. MR 2453603 (2010g:20022)
- 17.
- M. Larsen and A. Shalev, Word maps and Waring type problems, J. Amer. Math. Soc. 22 (2009), 437-466. MR 2476780 (2010d:20019)
- 18.
- M. Larsen, A. Shalev and P.H. Tiep, The Waring problem for finite simple groups, Annals of Math. (to appear).
- 19.
- M.W. Liebeck, E.A. O'Brien, A. Shalev and P.H. Tiep, The Ore Conjecture, J. Europ. Math. Soc. 12 (2010), 939-1008. MR 2654085
- 20.
- M.W. Liebeck and G.M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, preprint.
- 21.
- Frank Lübeck, Data for Finite Groups of Lie Type and Related Algebraic Groups. www.math.rwth-aachen.de/Frank.Luebeck/chev.
- 22.
- G. Lusztig, Characters of Reductive Groups over a Finite Field, Annals of Math. Studies, 107, Princeton Univ. Press, Princeton, 1984. MR 742472 (86j:20038)
- 23.
- D. Segal, Words: Notes on verbal width in groups, London Math. Soc. Lecture Note Series, 361, Cambridge University Press, Cambridge, 2009. MR 2547644 (2011a:20055)
- 24.
- A. Shalev, Word maps, conjugacy classes, and a non-commutative Waring-type theorem, Annals of Math. (2) 170 (2009), 1383-1416. MR 2600876
- 25.
- P.H. Tiep and A. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra 24 (1996), 2093-2167. MR 1386030 (97f:20018)
- 26.
- P.H. Tiep and A.E. Zalesskii, Some characterizations of the Weil representations of the symplectic and unitary groups, J. Algebra 192 (1997), 130-165. MR 1449955 (99d:20074)
- 27.
- W.R. Unger, Computing the character table of a finite group, J. Symbolic Comput. 41 (2006), 847-862. MR 2246713 (2007b:20023)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2010):
20C33,
20D06
Retrieve articles in all journals
with MSC (2010):
20C33,
20D06
Additional Information
Martin W. Liebeck
Affiliation:
Department of Mathematics, Imperial College, Queen’s Gate, London SW7 2BZ, United Kingdom
Email:
m.liebeck@imperial.ac.uk
E. A. O’Brien
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
Email:
e.obrien@auckland.ac.nz
Aner Shalev
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
Email:
shalev@math.huji.ac.il
Pham Huu Tiep
Affiliation:
Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721
Email:
tiep@math.arizona.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-2011-10878-5
PII:
S 0002-9939(2011)10878-5
Received by editor(s):
May 21, 2010
Received by editor(s) in revised form:
November 1, 2010
Posted:
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
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
|
