Proof Index
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Abstract algebra
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Binary operations and relations
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Properties of binary operations
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What is a relation? (for undergraduates)
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What is an equivalence relation?
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Group theory
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Intro to groups
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Subgroups
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Group isomorphisms
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Group homomorphisms
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Kernel of a group homomorphism
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Intro to group homomorphisms
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Equivalence classes
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Cosets of a subgroup
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Cosets and Lagrange's theorem
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Basic group theory proofs
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Abelian groups
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Lesson
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Practice
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Alternating groups
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Tidbits on Burnside's lemma
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Converting between array notation and cycle notation
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Converting cycle notation to array notation
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Converting array notation to cycle notation
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Cyclic groups
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Dihedral group
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Direct product of finite cyclic groups
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Direct product of groups
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Group automorphisms
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Is the table a commutative semigroup?
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Kernel of a group homomorphism
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Klein four-group
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Multiplying permutations in array notation
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Multiplying permutations in cycle notation
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Order of an element in a group
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Proof: Centralizer of group element is subgroup
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Compositions of group morphisms
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Normal subgroups
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Proof: Finite subgroup test
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Proof: First isomorphism theorem for groups
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Proof: Fundamental homomorphism theorem
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Proof: Group abelian iff cross cancellation property
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Proof: If \(y\) is a left or right inverse for \(x\) in a group, then \(y\) is the inverse of \(x\)
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Proof: Inverse of generator of cyclic group is generator
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Proof: Inverse of group inverse
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Proof: One-step subgroup test
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Proof: Order of element divides order of finite group
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Proof: Order of group element equals order of inverse
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Proof: Second isomorphism theorem for groups
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Proof: Special linear group is subgroup of general linear group
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Proof: Symmetric difference on power set forms group
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Proof: The conjugation map is an automorphism
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Proof: Two-step subgroup test
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Proving group homomorphisms
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Quotient groups
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Simple groups
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Solvable groups
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Sufficient conditions for a group to be abelian
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Symmetric group
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Symmetries of an equilateral triangle
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Verifying equivalence relations on groups
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What is cycle notation?
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Group actions
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Orbit stabilizer theorem
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Center of a group
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Ring theory
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Is it a ring?
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Basic ring theory proofs
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Integral ideals
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Intro to integral ideals
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Proving multiples form an integral ideal
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More ring theory proofs
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What is a subring?
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Abstract algebra
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Group theory
Group automorphisms
Prove that \(\phi\) is an automorphism: $$\phi : G \rightarrow G\quad\quad\phi(x) = g^{-1}xg$$
70200
Group Theory - Problem 3
Unbounded
YouTube videos
70191
Visual Group Theory, Lecture 4.6: Automorphisms
Professor Macauley
70195
GT11. Group Automorphisms
MathDoctorBob
70196
Group automorphisms in abstract algebra
Juan Klopper