Math 2.0

Group theory and generalizations Counterexample

Converse of Lagrange's theorem fails for the alternating group A4

Builds on Lagrange's theorem

Counterexample

The converse of Lagrange's theorem would say this. If GG is a finite group and dd is a positive divisor of ∣G∣|G|, then GG has a subgroup of order dd. This is false. Let A4A_4 be the group of the even permutations of {1,2,3,4}\{1, 2, 3, 4\}, under composition. It has order 1212, and 66 divides 1212, but A4A_4 has no subgroup of order 66.

Remarks

The elements of A4A_4 are the identity, the three products of two disjoint transpositions, and the eight 33-cycles. A subgroup HH of order 66 would have index 22, and thus it would contain the square of each element of A4A_4. The square of each 33-cycle is a 33-cycle, and each 33-cycle is such a square. So HH would contain all eight 33-cycles, which is too many. The group A4A_4 is the smallest group for which the converse fails.

Reading path

To understand this page from the ground up, read these 36 pages in order. The foundations come first. The path takes the proof Proof of Lagrange's theorem.

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Element, definition, no evidence levelElementdefinitionGroup, concept, no evidence levelGroupconceptLeft coset, concept, no evidence levelLeft cosetconceptProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionUnion, definition, no evidence levelUniondefinitionSet-builder notation, concept, no evidence levelSet-builder notationconceptExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionDisjoint sets, definition, no evidence levelDisjoint setsdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionCosets have the same cardinality as the subgroup, statement, no evidence levelCosets have the same …statementUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionBinary relation, definition, no evidence levelBinary relationdefinitionPartition of a set, definition, no evidence levelPartition of a setdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionSymmetric relation, definition, no evidence levelSymmetric relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionEquivalence relation, definition, no evidence levelEquivalence relationdefinitionEquivalence class, definition, no evidence levelEquivalence classdefinitionEquivalence classes form a partition, statement, no evidence levelEquivalence classes f…statementProof of Lagrange's theorem, proof, no evidence levelProof of Lagrange's t…proofLagrange's theorem, statement, no evidence levelLagrange's theoremstatementConverse of Lagrange's theorem fails for the alternating group A4, counterexample, no evidence levelConverse of Lagrange'…counterexample
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. Element definition
  2. Group concept
  3. Left coset concept
  4. Proposition concept
  5. Set definition
  6. Subgroup definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Empty set object
  12. Family of sets definition
  13. Ordered pair definition
  14. Intersection definition
  15. Left coset definition
  16. Right coset definition
  17. Union definition
  18. Set-builder notation concept
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Disjoint sets definition
  22. Cartesian product definition
  23. Subset definition
  24. Cosets have the same cardinality as the subgroup statement
  25. Union of a family of sets definition
  26. Binary relation definition
  27. Partition of a set definition
  28. Reflexive relation definition
  29. Symmetric relation definition
  30. Transitive relation definition
  31. Equivalence relation definition
  32. Equivalence class definition
  33. Equivalence classes form a partition statement
  34. Proof of Lagrange's theorem proof
  35. Lagrange's theorem statement
  36. Converse of Lagrange's theorem fails for the alternating group A4 counterexample

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory), Existence of subgroups of given order
  2. Web https://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory), Counterexample of the converse of Lagrange's theorem
About this page
Tag0PDBVPP233
URIm2://0PDBVPP233
Kindcounterexample
WritingW0 stub
MSC20D60
UsesLagrange's theorem

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