Math 2.0

Group theory and generalizations Finite groups Theorem

Burnside's p^a q^b theorem

Builds on Solvable groupSimple group

Statement

Let pp and qq be prime numbers, let a,b≥0a, b \geq 0 be integers, and let GG be a finite group of order paqbp^a q^b. Then GG is solvable.

Remarks

A consequence is that the order of each finite simple group that is not abelian has at least three distinct prime divisors. For example, A5A_5 has order 60=22⋅3⋅560 = 2^2 \cdot 3 \cdot 5. For b=0b = 0, the result says that finite pp-groups are solvable, which follows from their nilpotency.

History

William Burnside proved the theorem in 1904, with the representation theory of finite groups. Burnside, Jordan and Frobenius had proved special cases before, in 1897, 1898 and 1902. John G. Thompson later showed that his work on the N-group theorem gives a proof without representation theory.

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To understand this page from the ground up, read these 16 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionSimple group, definition, no evidence levelSimple groupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionSolvable group, definition, no evidence levelSolvable groupdefinitionBurnside's p^a q^b theorem, statement, no evidence levelBurnside's p^a q^b th…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.
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  1. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Conjugate elements definition
  5. Subgroup definition
  6. Abelian group definition
  7. Conjugate subgroup definition
  8. Left coset definition
  9. Right coset definition
  10. Normal subgroup definition
  11. Simple group definition
  12. Normality via left and right cosets statement
  13. Quotient group definition
  14. Subnormal series definition
  15. Solvable group definition
  16. Burnside's p^a q^b theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Burnside's_theorem, lead section
  2. Web https://en.wikipedia.org/wiki/Burnside's_theorem, History
About this page
Tag0FMAQSB1SC
URIm2://0FMAQSB1SC
Aliasesburnsides-theorem
Kindstatement, theorem
WritingW1 stated
MSC20D10
StatusThis copy of the site has no build data. Run m2 status 0FMAQSB1SC.
UsesSolvable group, Simple group
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