Math 2.0

statement theorem

The number of Sylow subgroups divides the index

Statement

Let GG be a finite group, let pp be a prime number, and let PP be a Sylow p-subgroup of GG. Then npn_p divides the index [G:P][G : P].

Remarks

By this formula, np=[G:NG(P)]n_p = [G : N_G(P)], and [G:P]=[G:NG(P)]⋅[NG(P):P][G : P] = [G : N_G(P)] \cdot [N_G(P) : P]. If ∣G∣=pnm|G| = p^n m with p∤mp \nmid m, then npn_p divides mm. For example, in a group of order 1515, n3n_3 divides 55 and n3≡1(mod3)n_3 \equiv 1 \pmod 3, so n3=1n_3 = 1.

Explanations

L1 human

The number of Sylow subgroups is the size of a conjugation orbit. Orbit sizes divide the order of the group, and here even the part of the order without pp. This gives a strong limit on npn_p.

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What this needs

To read this node, read these 37 nodes in this order. The foundations come first, and the list ends with this node.

Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionGroup homomorphism, definition, no evidence levelGroup homomorphismdefinitionLagrange's theorem, statement, no evidence levelLagrange's theoremstatementLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementOrder of a group, definition, no evidence levelOrder of a groupdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionSymmetric group, definition, no evidence levelSymmetric groupdefinitionOrder of an element, definition, no evidence levelOrder of an elementdefinitionGroup action, definition, no evidence levelGroup actiondefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionSubgroup criterion, statement, no evidence levelSubgroup criterionstatementP-group, definition, no evidence levelP-groupdefinitionOrbit, definition, no evidence levelOrbitdefinitionIndex of a subgroup, definition, no evidence levelIndex of a subgroupdefinitionStabiliser, definition, no evidence levelStabiliserdefinitionCosets have the same cardinality as the subgroup, statement, no evidence levelCosets have the same …statementNormal subgroup, definition, no evidence levelNormal subgroupdefinitionNormaliser, definition, no evidence levelNormaliserdefinitionIntersection of subgroups is a subgroup, statement, no evidence levelIntersection of subgr…statementOrbits partition the set, statement, no evidence levelOrbits partition the …statementTransitive action, definition, no evidence levelTransitive actiondefinitionFixed points of an action, definition, no evidence levelFixed points of an ac…definitionSylow p-subgroup, definition, no evidence levelSylow p-subgroupdefinitionOrder of a group is the index times the order of the subgroup, statement, no evidence levelOrder of a group is t…statementNormal core, definition, no evidence levelNormal coredefinitionAction on left cosets, example, no evidence levelAction on left cosetsexampleOrbit-stabiliser theorem, statement, no evidence levelOrbit-stabiliser theo…statementSize of an orbit times the size of the stabiliser, statement, no evidence levelSize of an orbit time…statementFixed points of a p-group action, statement, no evidence levelFixed points of a p-g…statementCauchy's theorem, statement, no evidence levelCauchy's theoremstatementFirst Sylow theorem, statement, no evidence levelFirst Sylow theoremstatementSecond Sylow theorem, statement, no evidence levelSecond Sylow theoremstatementThe number of Sylow subgroups is the index of the normaliser, statement, no evidence levelThe number of Sylow s…statementThe number of Sylow subgroups divides the index, statement, no evidence levelThe number of Sylow s…statement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Group concept
  2. Conjugate elements definition
  3. Group homomorphism definition
  4. Lagrange's theorem statement
  5. Laws of exponents in a group statement
  6. Order of a group definition
  7. Subgroup definition
  8. Symmetric group definition
  9. Order of an element definition
  10. Group action definition
  11. Conjugate subgroup definition
  12. Left coset definition
  13. Right coset definition
  14. Subgroup criterion statement
  15. P-group definition
  16. Orbit definition
  17. Index of a subgroup definition
  18. Stabiliser definition
  19. Cosets have the same cardinality as the subgroup statement
  20. Normal subgroup definition
  21. Normaliser definition
  22. Intersection of subgroups is a subgroup statement
  23. Orbits partition the set statement
  24. Transitive action definition
  25. Fixed points of an action definition
  26. Sylow p-subgroup definition
  27. Order of a group is the index times the order of the subgroup statement
  28. Normal core definition
  29. Action on left cosets example
  30. Orbit-stabiliser theorem statement
  31. Size of an orbit times the size of the stabiliser statement
  32. Fixed points of a p-group action statement
  33. Cauchy's theorem statement
  34. First Sylow theorem statement
  35. Second Sylow theorem statement
  36. The number of Sylow subgroups is the index of the normaliser statement
  37. The number of Sylow subgroups divides the index statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Sylow_theorems, Statement

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