Math 2.0

Group theory and generalizations Classification of finite simple groups Theorem

Classification of finite simple groups

Builds on Simple groupAlternating groupCyclic group

Statement

Each finite simple group is isomorphic to one of the following groups:

  1. a cyclic group ℤ/pℤ\mathbb{Z}/p\mathbb{Z} of prime order pp;
  2. an alternating group AnA_n with n≥5n \geq 5;
  3. a group of Lie type;
  4. one of 2626 sporadic groups.

Remarks

The groups of Lie type form infinite families, for example the groups PSLn(𝔽q)\mathrm{PSL}_n(\mathbb{F}_q), with a few exceptions for small nn and qq. Some authors count the Tits group as a 2727th sporadic group, because it is not strictly of Lie type. The largest sporadic group is the Monster group. The proof consists of tens of thousands of pages in several hundred journal articles.

History

Daniel Gorenstein announced in 1983 that the classification was complete, but this was premature. The gap was the case of the quasithin groups. Michael Aschbacher announced the completed proof in 2004, after Aschbacher and Stephen Smith published a proof of 12211221 pages for this case.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
Cycle notation, concept, no evidence levelCycle notationconceptGroup, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionKernel of a homomorphism, definition, no evidence levelKernel of a homomorph…definitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionDisjoint cycles commute, statement, no evidence levelDisjoint cycles commu…statementTransposition, definition, no evidence levelTranspositiondefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitionDecomposition of a permutation into disjoint cycles, statement, no evidence levelDecomposition of a pe…statementNormal subgroup, definition, no evidence levelNormal subgroupdefinitionCyclic group, definition, no evidence levelCyclic groupdefinitionSimple group, definition, no evidence levelSimple groupdefinitionDecomposition of a permutation into transpositions, statement, no evidence levelDecomposition of a pe…statementSign of a permutation, definition, no evidence levelSign of a permutationdefinitionEven and odd permutations, definition, no evidence levelEven and odd permutat…definitionSign is a homomorphism, statement, no evidence levelSign is a homomorphismstatementAlternating group, definition, no evidence levelAlternating groupdefinitionClassification of finite simple groups, statement, no evidence levelClassification of fin…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cycle notation concept
  2. Group concept
  3. Conjugate elements definition
  4. Group homomorphism definition
  5. Laws of exponents in a group statement
  6. Subgroup definition
  7. Symmetric group definition
  8. Kernel of a homomorphism definition
  9. Conjugate subgroup definition
  10. Disjoint cycles commute statement
  11. Transposition definition
  12. Cyclic subgroup generated by an element definition
  13. Decomposition of a permutation into disjoint cycles statement
  14. Normal subgroup definition
  15. Cyclic group definition
  16. Simple group definition
  17. Decomposition of a permutation into transpositions statement
  18. Sign of a permutation definition
  19. Even and odd permutations definition
  20. Sign is a homomorphism statement
  21. Alternating group definition
  22. Classification of finite simple groups statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups, lead section
  2. Web https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups, Statement of the classification theorem
About this page
Tag06J6P03MG7
URIm2://06J6P03MG7
Aliasesenormous-theorem
Kindstatement, theorem
WritingW1 stated
MSC20D05
StatusThis copy of the site has no build data. Run m2 status 06J6P03MG7.
UsesSimple group, Alternating group, Cyclic group
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