A solvable version of the Baer-Suzuki theorem

Author:
Simon Guest

Journal:
Trans. Amer. Math. Soc. **362** (2010), 5909-5946

MSC (2000):
Primary 20F14, 20D10

DOI:
https://doi.org/10.1090/S0002-9947-2010-04932-3

Published electronically:
June 2, 2010

MathSciNet review:
2661502

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that is a finite group and has prime order . Then is contained in the solvable radical of , , if (and only if) is solvable for all . If is an almost simple group and has prime order , then this implies that there exists such that is not solvable. In fact, this is also true when with very few exceptions, which are described explicitly.

**[ABN]**Rachel Abbott, John Bray, Simon Nickerson, Steve Linton, Simon Norton, Richard Parker, Ibrahim Suleiman, Jonathan Tripp, Peter Walsh, and Robert Wilson,*A www-atlas of finite group representations.***[AG84]**M. Aschbacher and R. Guralnick,*Some applications of the first cohomology group*, J. Algebra**90**(1984), no. 2, 446-460. MR**760022 (86m:20060)****[Bos]**Wieb Bosma, John Cannon, and Catherine Playoust.*The Magma algebra system*. I.*The user language*. J. Symbolic Comput.**24**(3-4): 235-265, 1997. MR**1484478****[Bur04]**Timothy C. Burness,*Fixed point spaces in actions of classical algebraic groups*, J. Group Theory**7**(2004), no. 3, 311-346. MR**2063000 (2005c:14054)****[CCN85]**J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson,*Atlas of finite groups*, Oxford University Press, Eynsham, 1985, Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray. MR**827219 (88g:20025)****[Enn62]**Veikko Ennola,*On the conjugacy classes of the finite unitary groups*, Ann. Acad. Sci. Fenn. Ser. A I No.**313**(1962), 13pp. MR**0139651 (25:3082)****[FGG]**Paul Flavell, Robert Guralnick, and Simon Guest,*Characterizations of the solvable radical*, Proc. Amer. Math. Soc.**138**(2010), no. 4, 1161-1170. MR**2578510**.**[GGKP08a]**Nikolai Gordeev, Fritz Grunewald, Boris Kunyavskii, and Eugene Plotkin,*A commutator description of the solvable radical of a finite group*, Groups Geom. Dyn.**2**(2008), no. 1, 85-120. MR**2367209 (2008j:20057)****[GGKP08b]**-,*A description of Baer-Suzuki type of the solvable radical of a finite group*, J. Pure and Applied Algebra**213**(2009), 250-258. MR**2467402 (2009i:20045)****[GHL96]**Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, and Götz Pfeiffer,*CHEVIE--a system for computing and processing generic character tables*, Appl. Algebra Engrg. Comm. Comput.**7**(1996), no. 3, 175-210, Computational methods in Lie theory (Essen, 1994). MR**1486215 (99m:20017)****[GK00]**Robert M. Guralnick and William M. Kantor,*Probabilistic generation of finite simple groups*, J. Algebra**234**(2000), no. 2, 743-792, Special issue in honor of Helmut Wielandt. MR**1800754 (2002f:20038)****[GL83]**Daniel Gorenstein and Richard Lyons,*The local structure of finite groups of characteristic type*, Mem. Amer. Math. Soc.**42**(1983), no. 276, vii+731pp. MR**690900 (84g:20025)****[GLS98]**Daniel Gorenstein, Richard Lyons, and Ronald Solomon,*The classification of the finite simple groups. Number 3*, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. MR**1490581 (98j:20011)****[GPPS99]**Robert Guralnick, Tim Penttila, Cheryl E. Praeger, and Jan Saxl,*Linear groups with orders having certain large prime divisors*, Proc. London Math. Soc. (3)**78**(1999), no. 1, 167-214. MR**1658168 (99m:20113)****[GPS07]**Robert Guralnick, Eugene Plotkin, and Aner Shalev,*Burnside-type problems related to solvability*, Internat. J. Algebra Comput.**17**(2007), no. 5-6, 1033-1048. MR**2355682 (2008j:20110)****[GS03]**Robert M. Guralnick and Jan Saxl,*Generation of finite almost simple groups by conjugates*, J. Algebra**268**(2003), no. 2, 519-571. MR**2009321 (2005f:20057)****[Gur98]**Robert M. Guralnick,*Some applications of subgroup structure to probabilistic generation and covers of curves*, Algebraic groups and their representations (Cambridge, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 301-320. MR**1670777 (2000d:20062)****[KL90]**Peter Kleidman and Martin Liebeck,*The subgroup structure of the finite classical groups*, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR**1057341 (91g:20001)****[Kle88a]**Peter B. Kleidman,*The maximal subgroups of the Chevalley groups with odd, the Ree groups , and their automorphism groups*, J. Algebra**117**(1988), no. 1, 30-71. MR**955589 (89j:20055)****[Kle88b]**-,*The maximal subgroups of the Steinberg triality groups and of their automorphism groups*, J. Algebra**115**(1988), no. 1, 182-199. MR**937609 (89f:20024)****[Law95]**R. Lawther,*Jordan block sizes of unipotent elements in exceptional algebraic groups*, Comm. Algebra**23**(1995), no. 11, 4125-4156. MR**1351124 (96h:20084)****[LS03]**Martin W. Liebeck and Gary M. Seitz,*A survey of maximal subgroups of exceptional groups of Lie type*, Groups, combinatorics & geometry (Durham, 2001), World Sci. Publ., River Edge, NJ, 2003, pp. 139-146. MR**1994964 (2004f:20089)****[LSS92]**Martin W. Liebeck, Jan Saxl, and Gary M. Seitz,*Subgroups of maximal rank in finite exceptional groups of Lie type*, Proc. London Math. Soc. (3)**65**(1992), no. 2, 297-325. MR**1168190 (93e:20026)****[LSS96]**-,*Factorizations of simple algebraic groups*, Trans. Amer. Math. Soc.**348**(1996), no. 2, 799-822. MR**1316858 (96g:20064)****[Mal91]**Gunter Malle,*The maximal subgroups of*, J. Algebra**139**(1991), no. 1, 52-69. MR**1106340 (92d:20068)****[Miz77]**Kenzo Mizuno,*The conjugate classes of Chevalley groups of type*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**24**(1977), no. 3, 525-563. MR**0486170 (58:5951)****[Miz80]**-,*The conjugate classes of unipotent elements of the Chevalley groups and*, Tokyo J. Math.**3**(1980), no. 2, 391-461. MR**605099 (82m:20046)****[NW02]**Simon P. Norton and Robert A. Wilson,*Anatomy of the Monster. II*, Proc. London Math. Soc. (3)**84**(2002), no. 3, 581-598. MR**1888424 (2003b:20023)****[Sei83]**Gary M. Seitz,*The root subgroups for maximal tori in finite groups of Lie type*, Pacific J. Math.**106**(1983), no. 1, 153-244. MR**694680 (84g:20085)****[Sho74]**Toshiaki Shoji,*The conjugacy classes of Chevalley groups of type over finite fields of characteristic*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**21**(1974), 1-17. MR**0357641 (50:10109)****[SS97]**Jan Saxl and Gary M. Seitz,*Subgroups of algebraic groups containing regular unipotent elements*, J. London Math. Soc. (2)**55**(1997), no. 2, 370-386. MR**1438641 (98m:20057)****[Suz62]**Michio Suzuki,*On a class of doubly transitive groups*, Ann. of Math. (2)**75**(1962), 105-145. MR**0136646 (25:112)****[Tho68]**John G. Thompson,*Nonsolvable finite groups all of whose local subgroups are solvable*, Bull. Amer. Math. Soc.**74**(1968), 383-437. MR**0230809 (37:6367)****[Wal63]**G. E. Wall,*On the conjugacy classes of classical groups*, J. Austral. Math. Soc.**3**(1963), 1-62. MR**0150210 (27:212)****[War66]**Harold N. Ward,*On Ree's series of simple groups*, Trans. Amer. Math. Soc.**121**(1966), 62-89. MR**0197587 (33:5752)****[Wie64]**Helmut Wielandt,*Finite permutation groups*, Translated from the German by R. Bercov, Academic Press, New York, 1964. MR**0183775 (32:1252)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
20F14,
20D10

Retrieve articles in all journals with MSC (2000): 20F14, 20D10

Additional Information

**Simon Guest**

Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, California 90089–2532

Address at time of publication:
Department of Mathematics, Baylor University, One Bear Place, #97328, Waco, Texas 76798

Email:
sguest@usc.edu

DOI:
https://doi.org/10.1090/S0002-9947-2010-04932-3

Received by editor(s):
January 25, 2008

Received by editor(s) in revised form:
September 14, 2008

Published electronically:
June 2, 2010

Additional Notes:
The author was partially supported by the NSF grant DMS 0653873

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.