Math 2.0

statement theorem

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

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.

Explanations

L1 human

Cut a finite group into the left cosets of a subgroup. Each piece has the same size as the subgroup, and the number of pieces is the index. So the size of the group is the number of pieces times the size of one piece.

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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 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.
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  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

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Sources

  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

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