Math 2.0

statement theorem

First Sylow theorem

Statement

Let GG be a finite group, let pp be a prime number, and let k≥0k \geq 0 with pk∣∣G∣p^k \mid |G|. Then GG has a subgroup of order pkp^k. In particular, GG has a Sylow p-subgroup.

Remarks

For k=1k = 1, this is Cauchy's theorem. Let pnp^n be the largest power of pp that divides ∣G∣|G|. A proof by Wielandt lets GG act by left multiplication on the subsets of GG of size pnp^n. For example, a group of order 1212 has a subgroup of order 44 and a subgroup of order 33.

History

Ludwig Sylow proved the Sylow theorems in 1872, and he published them in Mathematische Annalen.

Explanations

L1 human

The converse of Lagrange's theorem is false in general. But it is true for prime powers: each prime power that divides the order of the group is the order of a subgroup. The largest such subgroups are the Sylow subgroups.

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What this needs

To read this node, read these 33 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…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 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. Sylow p-subgroup definition
  26. Order of a group is the index times the order of the subgroup statement
  27. Normal core definition
  28. Action on left cosets example
  29. Orbit-stabiliser theorem statement
  30. Size of an orbit times the size of the stabiliser statement
  31. Fixed points of a p-group action statement
  32. Cauchy's theorem statement
  33. First Sylow theorem statement

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

Sources

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

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