Math 2.0

Group theory and generalizations Theorem

A nontrivial finite p-group has nontrivial centre

Builds on P-groupClass equationCentre of a group

Statement

Let pp be a prime number and let GG be a finite p-group with G≠{e}G \neq \{e\}. Then the centre Z(G)Z(G) is not {e}\{e\}.

Remarks

In the class equation ∣G∣=∣Z(G)∣+∑[G:CG(ai)]|G| = |Z(G)| + \sum [G : C_G(a_i)], the left side and each term of the sum are divisible by pp. So pp divides ∣Z(G)∣|Z(G)|, and ∣Z(G)∣≥p|Z(G)| \geq p. Equivalently, apply the fixed point lemma to the action of GG on itself by conjugation. A consequence is that each finite pp-group is nilpotent.

Reading path

To understand this page from the ground up, read these 28 pages in order. The foundations come first.

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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 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 coredefinitionConjugacy class, definition, no evidence levelConjugacy classdefinitionAction on left cosets, example, no evidence levelAction on left cosetsexampleOrbit-stabiliser theorem, statement, no evidence levelOrbit-stabiliser theo…statementSize 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…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.
  • 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. Subgroup criterion statement
  14. P-group definition
  15. Orbit definition
  16. Action by conjugation example
  17. Stabiliser definition
  18. Normal subgroup definition
  19. Intersection of subgroups is a subgroup statement
  20. Orbits partition the set statement
  21. Transitive action definition
  22. Normal core definition
  23. Conjugacy class definition
  24. Action on left cosets example
  25. Orbit-stabiliser theorem statement
  26. Size of a conjugacy class is the index of the centraliser statement
  27. Class equation statement
  28. A nontrivial finite p-group has nontrivial centre statement

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/P-group, Non-trivial center
About this page
Tag0RZCTT30H5
URIm2://0RZCTT30H5
Kindstatement, theorem
WritingW0 stub
MSC20D15
StatusThis copy of the site has no build data. Run m2 status 0RZCTT30H5.
UsesP-group, Class equation, Centre of a group

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