Math 2.0

statement proposition

The number of Sylow subgroups is the index of the normaliser

Statement

Let GG be a finite group, let pp be a prime number, and let PP be a Sylow p-subgroup of GG. Then np=[G:NG(P)]n_p = [G : N_G(P)], where NG(P)N_G(P) is the normaliser of PP.

Remarks

By the second Sylow theorem, GG acts transitively by conjugation on Syl⁡p(G)\operatorname{Syl}_p(G). The stabiliser of PP is NG(P)N_G(P), so the orbit-stabiliser theorem gives the formula. In particular, PP is normal in GG if and only if np=1n_p = 1.

Explanations

L1 human

All Sylow pp-subgroups are conjugates of one of them, PP. The number of conjugates of PP is the number of cosets of the elements that do not move PP. These elements form the normaliser.

What this node needs for W1
  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 36 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…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

What depends on this

2 nodes use this node, directly or through other nodes.

Sources

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

What links here

Used by