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    • ▾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

    Basic group theory proofs

    In this section you will get comfortable with the definition of a group by proving some basic properties. You will also witness some of the consequences and effects of a group being abelian.

    Lesson

    The identity element is unique.
    1973Proof: Identity Element of a Group is Unique | Abstract Algebra
    Wrath of Math
    46352Proof that the identity element of a group is unique
    The Math Sorcerer
    70205Abstract Algebra | General properties of groups.
    Michael Penn
    70624Abstract Algebra 2.3: Properties of Groups
    Patrick Jones
    775Identity of Group is Unique
    ProofWiki
    The inverse of an element is unique.
    70447Proof: Group Elements Have Unique Inverses | Group Theory, Abstract Algebra
    Wrath of Math
    44145Proof that inverses in a group are unique
    The Math Sorcerer
    70205Abstract Algebra | General properties of groups.
    Michael Penn
    776Inverse in Group is Unique
    ProofWiki
    Shoes and socks \((ab)^{-1} = b^{-1}a^{-1}\)
    70543Inverse of a Product of Group Elements (Socks-Shoes Property) | Group Theory
    Wrath of Math
    44927Group Theory: Proof of the Formula for the Inverse of a Product
    The Math Sorcerer
    70205Abstract Algebra | General properties of groups.
    Michael Penn
    779Inverse of Group Product
    ProofWiki
    Left cancellation law \(ab = ac \longrightarrow b = c\)
    45095Proof of the Cancellation Laws in a Group
    The Math Sorcerer
    70199Cancellation Laws hold in a group proof (Abstract Algebra)
    BriTheMathGuy
    70205Abstract Algebra | General properties of groups.
    Michael Penn
    49234Prove The Left Cancellation Law for Groups
    Ms Shaws Math Class
    Right cancellation law \(ba = ca \longrightarrow b = c\)
    45095Proof of the Cancellation Laws in a Group
    The Math Sorcerer
    70199Cancellation Laws hold in a group proof (Abstract Algebra)
    BriTheMathGuy
    778Cancellation Laws
    ProofWiki

    Practice

    Let \(G\) be a finite group of even order. Show that \(G\) has an element \(a \ne e\) such that \(a^2 = e.\)
    70203(Abstract Algebra 1) Basic Group Proof 3
    learnifyable
    Let \(G\) be a group and fix \(g_0 \in G\). Prove that \(f : G \longrightarrow G\) given by \(f(x) = xg_0\) is a bijection.
    45678Proof that f(x) = xg_0 is a Bijection
    The Math Sorcerer