Math 2.0

Group theory and generalizations Theorem

Order of a group is the index times the order of the subgroup

Builds on Index of a subgroupCosets have the same cardinality as the subgroupOrder of a group

Statement

Let GG be a finite group and let HH be a subgroup of GG. Then ∣G∣=[G:H]⋅∣H∣|G| = [G : H] \cdot |H|, where [G:H][G : H] is the index of HH in GG.

Remarks

The left cosets of HH partition GG, and [[each coset has ∣H∣|H| elements|0DKMBWTD7P]]. This statement gives Lagrange's theorem: the order of HH divides the order of GG. The formula ∣G∣=[G:H]⋅∣H∣|G| = [G : H] \cdot |H| is also true for infinite groups as an equation of cardinal numbers.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptOrder of a group, definition, no evidence levelOrder of a groupdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionIndex of a subgroup, definition, no evidence levelIndex of a subgroupdefinitionCosets have the same cardinality as the subgroup, statement, no evidence levelCosets have the same …statementOrder of a group is the index times the order of the subgroup, statement, no evidence levelOrder of a group is t…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. Order of a group definition
  3. Subgroup definition
  4. Left coset definition
  5. Right coset definition
  6. Index of a subgroup definition
  7. Cosets have the same cardinality as the subgroup statement
  8. Order of a group is the index times the order of the subgroup statement

Where this leads

18 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory), lead section
  2. Web https://en.wikipedia.org/wiki/Index_of_a_subgroup, lead section
About this page
Tag04DC0SRN2K
URIm2://04DC0SRN2K
Kindstatement, theorem
WritingW0 stub
MSC20D60
StatusThis copy of the site has no build data. Run m2 status 04DC0SRN2K.
UsesIndex of a subgroup, Cosets have the same cardinality as the subgroup, Order of a group

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