Math 2.0

statement theorem

Cauchy's theorem

Statement

Let GG be a finite group and let pp be a prime number that divides ∣G∣|G|. Then GG has an element of order pp.

Remarks

Equivalently, GG has a subgroup of order pp, namely the cyclic subgroup generated by such an element. It is a partial converse of Lagrange's theorem: a prime divisor of ∣G∣|G| is always the order of a subgroup. One proof lets Z/pZ\mathbb{Z}/p\mathbb{Z} act by cyclic shifts on the set of tuples (g1,…,gp)(g_1, \dots, g_p) with g1g2⋯gp=eg_1 g_2 \cdots g_p = e, and uses the fixed point lemma. The first Sylow theorem generalises the result to prime powers.

History

Augustin-Louis Cauchy discovered the theorem in 1845.

Explanations

L1 human

Lagrange says that the order of an element divides the order of the group. Cauchy says that, for prime numbers, each divisor really appears as the order of an element. For example, a group of order 1515 has an element of order 33 and an element of order 55.

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To read this node, read these 31 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 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 actiondefinitionFixed points of an action, definition, no evidence levelFixed points of an ac…definitionOrder 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 theoremstatement
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. Intersection of subgroups is a subgroup statement
  22. Orbits partition the set statement
  23. Transitive action definition
  24. Fixed points of an action definition
  25. Order of a group is the index times the order of the subgroup statement
  26. Normal core definition
  27. Action on left cosets example
  28. Orbit-stabiliser theorem statement
  29. Size of an orbit times the size of the stabiliser statement
  30. Fixed points of a p-group action statement
  31. Cauchy's theorem statement

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

Sources

  1. Web https://en.wikipedia.org/wiki/Cauchy's_theorem_(group_theory), lead section
  2. Web https://en.wikipedia.org/wiki/Cauchy's_theorem_(group_theory), Statement and proof

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