Math 2.0

statement theorem

Finite p-groups are nilpotent

Statement

Let pp be a prime number and let GG be a finite p-group. Then GG is nilpotent.

Remarks

Each quotient G/ZiG/Z_i of the upper central series is a pp-group, so it has a nontrivial centre when it is not trivial. So the upper central series grows strictly until it reaches GG, and GG is finite. For ∣G∣=pn|G| = p^n with n≥2n \geq 2, the nilpotency class of GG is at most n−1n - 1.

Explanations

L1 human

A nontrivial finite pp-group always has a nontrivial centre. Divide by it, and the quotient is again a pp-group with a nontrivial centre. Repeat this, and you climb the upper central series to the whole group.

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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 levelGroupconceptCentre of a group, definition, no evidence levelCentre of a groupdefinitionConjugate 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 groupdefinitionOrder of an element, definition, no evidence levelOrder of an elementdefinitionGroup action, definition, no evidence levelGroup actiondefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionCentraliser, definition, no evidence levelCentraliserdefinitionLeft 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 levelOrbitdefinitionAction by conjugation, example, no evidence levelAction by conjugationexampleStabiliser, definition, no evidence levelStabiliserdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionIntersection 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 actiondefinitionNormal core, definition, no evidence levelNormal coredefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementSubgroup generated by a subset, definition, no evidence levelSubgroup generated by…definitionConjugacy class, definition, no evidence levelConjugacy classdefinitionAction on left cosets, example, no evidence levelAction on left cosetsexampleQuotient group, definition, no evidence levelQuotient groupdefinitionLattice of subgroups, concept, no evidence levelLattice of subgroupsconceptNilpotent group, definition, no evidence levelNilpotent groupdefinitionOrbit-stabiliser theorem, statement, no evidence levelOrbit-stabiliser theo…statementCorrespondence theorem, statement, no evidence levelCorrespondence theoremstatementUpper central series, definition, no evidence levelUpper central seriesdefinitionSize of a conjugacy class is the index of the centraliser, statement, no evidence levelSize of a conjugacy c…statementClass equation, statement, no evidence levelClass equationstatementA nontrivial finite p-group has nontrivial centre, statement, no evidence levelA nontrivial finite p…statementFinite p-groups are nilpotent, statement, no evidence levelFinite p-groups are n…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. Centre of a group definition
  3. Conjugate elements definition
  4. Group homomorphism definition
  5. Laws of exponents in a group statement
  6. Subgroup definition
  7. Symmetric group definition
  8. Order of an element definition
  9. Group action definition
  10. Conjugate subgroup definition
  11. Centraliser definition
  12. Left coset definition
  13. Right coset definition
  14. Subgroup criterion statement
  15. P-group definition
  16. Orbit definition
  17. Action by conjugation example
  18. Stabiliser definition
  19. Normal subgroup definition
  20. Intersection of subgroups is a subgroup statement
  21. Orbits partition the set statement
  22. Transitive action definition
  23. Normal core definition
  24. Normality via left and right cosets statement
  25. Subgroup generated by a subset definition
  26. Conjugacy class definition
  27. Action on left cosets example
  28. Quotient group definition
  29. Lattice of subgroups concept
  30. Nilpotent group definition
  31. Orbit-stabiliser theorem statement
  32. Correspondence theorem statement
  33. Upper central series definition
  34. Size of a conjugacy class is the index of the centraliser statement
  35. Class equation statement
  36. A nontrivial finite p-group has nontrivial centre statement
  37. Finite p-groups are nilpotent statement

What depends on this

1 node uses this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Nilpotent_group, Examples
  2. Web https://en.wikipedia.org/wiki/P-group, Non-trivial center

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