• TOC
  • Courses
  • Blog
  • Twitch
  • Shop
  • Search
    • Courses
    • Blog
    • Subreddit
    • Discord
    • Log in
    • Sign up
    • ▾Abstract algebra
      • ▸Binary operations and relations
        • •Properties of binary operations
        • •What is a relation? (for undergraduates)
        • •What is an equivalence relation?
      • ▾Group theory
        • •Intro to groups
        • •Subgroups
        • •Group isomorphisms
        • ▸Group homomorphisms
          • •Kernel of a group homomorphism
          • •Intro to group homomorphisms
        • •Equivalence classes
        • •Cosets of a subgroup
        • •Cosets and Lagrange's theorem
        • •Basic group theory proofs
        • ▸Abelian groups
          • •Lesson
          • •Practice
        • •Alternating groups
        • •Tidbits on Burnside's lemma
        • ▸Converting between array notation and cycle notation
          • •Converting cycle notation to array notation
          • •Converting array notation to cycle notation
        • •Cyclic groups
        • •Dihedral group
        • •Direct product of finite cyclic groups
        • •Direct product of groups
        • •Group automorphisms
        • •Is the table a commutative semigroup?
        • •Kernel of a group homomorphism
        • •Klein four-group
        • •Multiplying permutations in array notation
        • •Multiplying permutations in cycle notation
        • •Order of an element in a group
        • •Proof: Centralizer of group element is subgroup
        • •Compositions of group morphisms
        • •Normal subgroups
        • •Proof: Finite subgroup test
        • •Proof: First isomorphism theorem for groups
        • •Proof: Fundamental homomorphism theorem
        • •Proof: Group abelian iff cross cancellation property
        • •Proof: If \(y\) is a left or right inverse for \(x\) in a group, then \(y\) is the inverse of \(x\)
        • •Proof: Inverse of generator of cyclic group is generator
        • •Proof: Inverse of group inverse
        • •One-step subgroup test
        • •Proof: Order of element divides order of finite group
        • •Proof: Order of group element equals order of inverse
        • •Proof: Second isomorphism theorem for groups
        • •Proof: Special linear group is subgroup of general linear group
        • •Proof: Symmetric difference on power set forms group
        • •Proof: The conjugation map is an automorphism
        • •Proof: Two-step subgroup test
        • •Proving group homomorphisms
        • •Quotient groups
        • •Simple groups
        • •Solvable groups
        • •Sufficient conditions for a group to be abelian
        • •Symmetric group
        • •Symmetries of an equilateral triangle
        • •Verifying equivalence relations on groups
        • •What is cycle notation?
        • •Group actions
        • •Orbit stabilizer theorem
        • •Center of a group
      • ▸Ring theory
        • •Is it a ring?
        • •Basic ring theory proofs
        • ▸Integral ideals
          • •Intro to integral ideals
          • •Proving multiples form an integral ideal
        • •More ring theory proofs
        • •What is a subring?
     › Abstract algebra › Group theory

    Order of an element in a group

    Students will learn the definition for the order of an element in a group. They will also learn what it means for an element to have finite or infinite order. Then they'll see examples of groups and asked to determine the order of various elements. Here's a video on all that. Finally, students will see a proof of the following theorem: Let \(G\) be a group, \(g \in G,\) \(n \in \mathbb{N}.\) Prove that if \(g^n = e,\) then \(o(g)\) divides \(n.\) A proof can be seen here. Understanding the proof requires students to know the division algorithm.

    TODO: Is the converse true?