Math 2.0

statement corollary

Order of an element divides the order of the group

Statement

Let GG be a finite group and let g∈Gg \in G. Then the order of gg divides the order ∣G∣|G| of GG.

Remarks

The order of gg is the order of the cyclic subgroup ⟨g⟩\langle g \rangle, by this fact, and Lagrange's theorem applies to ⟨g⟩\langle g \rangle. For example, in a group of order 1010, each element has order 1,2,51, 2, 5 or 1010.

Explanations

L1 human

An element generates a subgroup with as many elements as its order. The size of a subgroup always divides the size of the group. So the order of an element divides the order of the group.

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Group, concept, no evidence levelGroupconceptLagrange'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 levelSubgroupdefinitionOrder of an element, definition, no evidence levelOrder of an elementdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitiong to the n equals 1 iff the order divides n, statement, no evidence levelg to the n equals 1 i…statementOrder of a cyclic subgroup equals the order of its generator, statement, no evidence levelOrder of a cyclic sub…statementOrder of an element divides the order of the group, statement, no evidence levelOrder of an element d…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. Lagrange's theorem statement
  3. Laws of exponents in a group statement
  4. Order of a group definition
  5. Subgroup definition
  6. Order of an element definition
  7. Cyclic subgroup generated by an element definition
  8. g to the n equals 1 iff the order divides n statement
  9. Order of a cyclic subgroup equals the order of its generator statement
  10. Order of an element divides the order of the group statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Lagrange's_theorem_(group_theory), Applications

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