Order of element divides order of group

Dependencies:

  1. Group
  2. Order of cyclic subgroup is order of generator
  3. Lagrange's Theorem

Let $G$ be a finite group and $g \in G$. Then $\operatorname{order}(g) \mid |G|$.

Proof

$$ g \in G \Rightarrow \langle g \rangle \subseteq G \Rightarrow |\langle g \rangle| \mid |G| \Rightarrow \operatorname{order}(g) \mid |G| $$

Dependency for:

  1. Group element to the power group size equals identity
  2. Group of prime order is cyclic

Info:

Transitive dependencies:

  1. Group
  2. Coset
  3. Size of coset equals size of subset
  4. Identity of a group is unique
  5. Order of element in finite group is finite
  6. Subgroup
  7. Inverse of a group element is unique
  8. gH = H iff g in H
  9. Two cosets are either identical or disjoint
  10. Lagrange's Theorem
  11. Conditions for a subset to be a subgroup
  12. Cyclic Group
  13. Integer Division Theorem
  14. Order of cyclic subgroup is order of generator