Classification of finite simple groups

Groups of 2-rank 0, in other words groups of odd order, which are all by the Feit–Thompson theorem.

solvable

Groups of 2-rank 1. The Sylow 2-subgroups are either cyclic, which is easy to handle using the transfer map, or generalized , which are handled with the Brauer–Suzuki theorem: in particular there are no simple groups of 2-rank 1 except for the cyclic group of order two.

quaternion

Groups of 2-rank 2. Alperin showed that the Sylow subgroup must be dihedral, quasidihedral, wreathed, or a Sylow 2-subgroup of U₃(4). The first case was done by the which showed that the only simple groups are isomorphic to L₂(q) for q odd or A₇, the second and third cases were done by the Alperin–Brauer–Gorenstein theorem which implies that the only simple groups are isomorphic to L₃(q) or U₃(q) for q odd or M₁₁, and the last case was done by Lyons who showed that U₃(4) is the only simple possibility.

Gorenstein–Walter theorem

Groups of sectional 2-rank at most 4, classified by the .

Gorenstein–Harada theorem

The most important thing is that the correct, final statement of the theorem is now known. Simpler techniques can be applied that are known to be adequate for the types of groups we know to be finite simple. In contrast, those who worked on the first generation proof did not know how many sporadic groups there were, and in fact some of the sporadic groups (e.g., the ) were discovered while proving other cases of the classification theorem. As a result, many of the pieces of the theorem were proved using techniques that were overly general.

Janko groups

Because the conclusion was unknown, the first generation proof consists of many stand-alone theorems, dealing with important special cases. Much of the work of proving these theorems was devoted to the analysis of numerous special cases. Given a larger, orchestrated proof, dealing with many of these special cases can be postponed until the most powerful assumptions can be applied. The price paid under this revised strategy is that these first generation theorems no longer have comparatively short proofs, but instead rely on the complete classification.

Many first generation theorems overlap, and so divide the possible cases in inefficient ways. As a result, families and subfamilies of finite simple groups were identified multiple times. The revised proof eliminates these redundancies by relying on a different subdivision of cases.

Finite group theorists have more experience at this sort of exercise, and have new techniques at their disposal.

The

Schreier conjecture

The

Signalizer functor theorem

The

B conjecture

The for all groups (though this only uses the Feit–Thompson theorem).

Schur–Zassenhaus theorem

A transitive permutation group on a finite set with more than 1 element has a fixed-point-free element of prime power order.

The classification of .

2-transitive permutation groups

The classification of .

rank 3 permutation groups

The ^[2]

Sims conjecture

on the number of solutions of xⁿ = 1.

Frobenius's conjecture

O'Nan–Scott theorem

(2004). "The Status of the Classification of the Finite Simple Groups" (PDF). Notices of the American Mathematical Society. Vol. 51, no. 7. pp. 736–740.

Aschbacher, Michael

; Lyons, Richard; Smith, Stephen D.; Solomon, Ronald (2011), The Classification of Finite Simple Groups: Groups of Characteristic 2 Type, Mathematical Surveys and Monographs, vol. 172, ISBN 978-0-8218-5336-8

Aschbacher, Michael

; Curtis, Robert Turner; Norton, Simon Phillips; Parker, Richard A; Wilson, Robert Arnott (1985), Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Oxford University Press, ISBN 978-0-19-853199-9

Conway, John Horton

(1979), "The classification of finite simple groups. I. Simple groups and local analysis", Bulletin of the American Mathematical Society, New Series, 1 (1): 43–199, doi:10.1090/S0273-0979-1979-14551-8, ISSN 0002-9904, MR 0513750

Gorenstein, D.

(1982), Finite simple groups, University Series in Mathematics, New York: Plenum Publishing Corp., ISBN 978-0-306-40779-6, MR 0698782

Gorenstein, D.

(1983), The classification of finite simple groups. Vol. 1. Groups of noncharacteristic 2 type, The University Series in Mathematics, Plenum Press, ISBN 978-0-306-41305-6, MR 0746470

Gorenstein, D.

(1985), "The Enormous Theorem", Scientific American, December 1, 1985, vol. 253, no. 6, pp. 104–115.

Daniel Gorenstein

(1986), "Classifying the finite simple groups", Bulletin of the American Mathematical Society, New Series, 14 (1): 1–98, doi:10.1090/S0273-0979-1986-15392-9, ISSN 0002-9904, MR 0818060

Gorenstein, D.

Gorenstein, D.

Symmetry and the Monster, ISBN 978-0-19-280723-6, Oxford University Press, 2006. (Concise introduction for lay reader)

Mark Ronan

Finding Moonshine, Fourth Estate, 2008, ISBN 978-0-00-721461-7 (another introduction for the lay reader. American edition published in 2009 as Symmetry: A Journey into the Patterns of Nature)

Marcus du Sautoy

(1995) "On Finite Simple Groups and their Classification," Notices of the American Mathematical Society. (Not too technical and good on history. American version published in 2009 as Symmetry: A Journey into the Patterns of Nature)

Ron Solomon

Solomon, Ronald (2001), (PDF), Bulletin of the American Mathematical Society, New Series, 38 (3): 315–352, doi:10.1090/S0273-0979-01-00909-0, ISSN 0002-9904, MR 1824893, archived (PDF) from the original on 2001-06-15 – article won Levi L. Conant prize for exposition

"A brief history of the classification of the finite simple groups"

(1984), "Finite nonsolvable groups", in Gruenberg, K. W.; Roseblade, J. E. (eds.), Group theory. Essays for Philip Hall, Boston, MA: Academic Press, pp. 1–12, ISBN 978-0-12-304880-6, MR 0780566

Thompson, John G.

(2009), The finite simple groups, Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012

Wilson, Robert A.

Searchable database of representations and other data for many finite simple groups.

ATLAS of Finite Group Representations.

Elwes, Richard, "," Plus Magazine, Issue 41, December 2006. For laypeople.

An enormous theorem: the classification of finite simple groups

Madore, David (2003) Archived 2005-04-04 at the Wayback Machine Includes a list of all nonabelian simple groups up to order 10¹⁰.

Orders of nonabelian simple groups.

In what sense is the classification of all finite groups “impossible”?

Ornes, Stephen (2015). . Scientific American. 313 (1): 68–75. doi:10.1038/scientificamerican0715-68. PMID 26204718.

"Researchers Race to Rescue the Enormous Theorem before Its Giant Proof Vanishes"

. MathOverflow. (Last updated in February 2024)

"Where are the second- (and third-)generation proofs of the classification of finite simple groups up to?"