Abstract Algebra I
Skills
Every skill in the course, and how you are doing on each.
Groups, subgroups, and generated structures
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Nonempty hypothesis
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Subgroup intersection
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Product closure
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Subgroup union
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Commuting elements
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Group center
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Subgroup test
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Centralizer
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Modular congruence
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Bezout identity
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Cyclic subgroup
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Bezout exponents
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Coprime orders
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Hypothesis audit
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Subgroup index
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Coset bijection
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Finite cardinality
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Coset partition
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Index two
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Abelian criterion
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Subgroup product
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Closure counterexample
117 more topics with one problem
- Additive inverses
- Group axioms
- Integer addition
- Axiom counterexamples
- Nonassociativity
- Subtraction operation
- Domain restriction
- Multiplicative group
- Nonzero reals
- Inherited operation
- Inverse product
- Monoid units
- Axiom unpacking
- Identity uniqueness
- Two sided identity
- Associativity
- Group identities
- Inverse uniqueness
- Inverse multiplication
- Left cancellation
- Right cancellation
- Factor order
- Group equations
- Solution uniqueness
- Bijection
- Group construction
- Transported structure
- Explicit isomorphism
- Nonstandard identity
- Shifted operation
- Closure properties
- Inherited associativity
- Subgroup criterion
- Inverse closure
- One step subgroup test
- Empty intersection
- Indexed families
- Containment dichotomy
- Center identification
- Vacuous condition
- Additive subgroups
- Even integers
- Odd integers
- Multiplicative subgroups
- Positive reals
- Sign closure
- Finite subset
- Power repetition
- Integer subgroups
- Residue classes
- Generated subgroup
- Minimality
- Gcd generated subgroup
- Integer linear combinations
- Integer powers
- Single generator
- Coprimality
- Power generators
- Transposition generators
- Word enumeration
- Adjacent transpositions
- Inversion reduction
- Symmetric group generators
- Dihedral normal form
- Generator relations
- Word reduction
- Closure iteration
- Finite generation algorithm
- Stabilization
- Generated union
- Strict containment
- Subgroup join
- Generation criterion
- Normal form hypothesis
- Universal coverage
- Element order
- Infinite order
- Inverse powers
- Conjugate elements
- Order invariance
- Power conjugation
- Divisibility
- Gcd formula
- Order of a power
- Product order
- Noncommuting transpositions
- Product order counterexample
- Coset overlap
- Left cosets
- Partition
- Integer cosets
- Left translation
- Left right cosets
- Nonnormal subgroup
- Permutation computation
- Coordinatewise operation
- Product inverse
- Opposite group
- Reversed multiplication
- Permutable subgroups
- Setwise equality
- Subgroup cover
- Union counterexample
- Infinite dihedral group
- Torsion elements
- Abelian torsion
- Finite order elements
- Lcm exponent
- Cyclicity
- Lagrange consequence
- Prime order group
- Exponent two
- Self inverse elements
- Noncyclic counterexample
- Unique involution
- Element order analysis
- Order four classification
Cyclic and permutation groups
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Homomorphism image
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Least common multiple
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Well definedness
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Cyclic automorphisms
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Order twenty four
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Explicit homomorphism
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Commuting permutations
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Gcd cycle count
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Sign homomorphism
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Dihedral center
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Dihedral quotient
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Lagrange theorem
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Group exponent
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Fermat little theorem
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Coset representatives
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Coset enumeration
120 more topics with one problem
- Abelian group
- Integer exponents
- Cyclic image
- Generator mapping
- Cyclic subgroup membership
- Linear congruences
- Cyclic order twelve
- Euler totient
- Generator count
- Additive order
- Gcd computation
- Integers mod n
- Cyclic subgroups
- Division algorithm
- Least positive exponent
- Cyclic subgroup lattice
- Divisor classification
- Unique subgroup
- Cyclic intersection
- Exponent divisibility
- Cyclic join
- Greatest common divisor
- Lcm order
- Product of cyclic groups
- Homomorphism classification
- Image of one
- Integer universal property
- Finite cyclic homomorphism
- Generator relation
- Annihilated elements
- Cyclic hom count
- Modular equation
- Composition law
- Units modulo n
- Image size
- Kernel size
- Multiplication endomorphism
- Quotient generator
- Cyclic lattice
- Subgroup containment
- Cyclic injectivity
- Generator order
- Kernel test
- Cyclic surjection
- Order divisibility
- Cyclic endomorphism ring
- Scalar maps
- Unit automorphisms
- Noncommutativity
- Permutation composition
- Right to left convention
- Cycle reversal
- Disjoint cycles
- Permutation inverse
- Disjoint support
- Pointwise proof
- Cycle lengths
- Permutation order
- Cycle powers
- Orbit step size
- Cycle conjugation
- Permutation action
- Relabeling
- Cycle expansion
- Transposition decomposition
- Cycle type counting
- Overcount control
- Symmetric group enumeration
- Cycle centralizer
- Image determination
- Cycle type audit
- Permutation equation
- Square root
- Permutation parity
- Cycle parity
- Cycle sign
- Transposition count
- Sign kernel
- Conjugacy parity
- Sign invariance
- Sign inverse
- Adjacent swap length
- Inversion number
- Minimal factorization
- Alternating generators
- Paired transpositions
- Three cycles
- Dihedral computation
- Normal form reduction
- Square symmetries
- Dihedral reflections
- Reflection product
- Rotation order
- Generator commutation
- Odd even dichotomy
- Reflection parity
- Rotation subgroup
- Finite counting
- Element order divisibility
- Lagrange application
- Unit group mod prime
- Index multiplicativity
- Nested subgroups
- Coset pair map
- Finite index bound
- Intersection index
- Index computation
- Cosets
- Subgroups
- Converse lagrange
- Index two contradiction
- Even group order
- Inverse pairing
- Involution existence
- Normal three subgroup
- Order six groups
- Semidirect normal form
- Fiber counting
- Intersection overcount
- Subgroup product cardinality
Homomorphisms, kernels, and normal subgroups
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Klein four group
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Quaternion group
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Alternating group
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Kernel computation
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Matrix groups
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Special linear group
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Abelian groups
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Center
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Conjugation
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Inner automorphisms
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Automorphism classification
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Element orders
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General linear group
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Dihedral group
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Faithful action
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Automorphism invariance
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Characteristic subgroup
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Center quotient
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First isomorphism theorem
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Faithful permutation action
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Coset equality
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Normalizer
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Commutator subgroup
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Abelianization
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Normal subgroups
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Trivial kernel
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Quotient map
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Coset multiplication
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Normal image
85 more topics with one problem
- Permutations
- Cyclic homomorphisms
- Gcd
- Determinant map
- Product projection
- Surjective maps
- Power map
- Finite cyclic groups
- Image order
- Counterexample
- Trace
- Normal kernel
- Symmetric three
- Finite abelian groups
- Isomorphism invariants
- Basis images
- Elementary abelian group
- Automorphism computation
- Units modulo eight
- Involutions
- Inner automorphism group
- Automorphism group
- Free abelian groups
- Unimodular matrices
- Conjugate generators
- Minimal normal subgroup
- Normal closure
- Normality criteria
- Conjugation invariance
- Normal intersection
- Subgroup families
- Intersection of conjugates
- Maximal normal subgroup
- Subgroup core
- Conjugation stabilizer
- Maximality property
- Characteristic subgroups
- Universal quotient
- Converse counterexample
- Conjugate subgroups
- Nonnormal sylow
- Quotient to image
- Reduction map
- Determinant quotient
- Alternating quotient
- Sign map
- Preimage subgroup
- Subgroup correspondence
- Surjective image
- Correspondence theorem
- Normality transfer
- Quotient subgroups
- Restricted quotient map
- Second isomorphism theorem
- Induced map
- Iterated quotient
- Third isomorphism theorem
- Cyclic kernel
- Finite image
- First isomorphism application
- Cancellation
- Injectivity criterion
- Cyclic example
- Noninjective homomorphism
- Nontrivial image
- Infinite domain
- Surjective not injective
- Infinite group
- Injective not surjective
- Integer endomorphism
- Cyclic target
- Distinct homomorphisms
- Same kernel
- Equal images
- Projection kernels
- Map nonuniqueness
- Nonnormal quotient
- Well definedness failure
- Klein four subgroup
- Normality not transitive
- Subgroup inclusion
- Surjectivity hypothesis
- Conjugation transfer
- Normal preimage
- Surjective normal image
Quotient groups and direct products
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Direct product
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Cyclic quotient
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Integer quotient
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Normality criterion
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Quotient order
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Factorization
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Subgroup enumeration
112 more topics with one problem
- Quotient well definedness
- Addition table
- Cyclic generator
- Alternating cosets
- Parity
- Quotient table
- Presentation relations
- Coset orders
- Quaternion quotient
- Explicit quotient map
- Kernel identification
- Mixed cyclic product
- Coset cardinality
- Representative test
- Subgroup membership
- Abelian quotient
- Commutator containment
- Quotient criterion
- Kernel containment
- Quotient universal property
- Induced map uniqueness
- Quotient surjectivity
- Diagonal subgroup
- Normality characterization
- Product quotient
- Cyclic center quotient
- Normal correspondence
- Simple quotient
- Subgroup interval
- Generator image
- Composite kernel
- Quotient image
- Surjective composition
- Coordinate quotient
- Projection kernel
- Induced isomorphism
- Surjective quotient transfer
- Cyclic correspondence
- Quotient lattice
- Coordinatewise powers
- Product element order
- Finite direct product
- Coordinatewise commutation
- Product center
- Structural invariant
- Coordinatewise generation
- Derived subgroup
- Product commutator
- Index formula
- Product subgroup
- Graph subgroup
- Nonrectangular subgroup
- Klein four lattice
- Order two subgroups
- Element order count
- Finite abelian group
- Product computation
- Coordinate factors
- Trivial intersection
- Complementary normal subgroups
- Multiplication isomorphism
- Commuting images
- Homomorphism construction
- Product universal map
- Map into product
- Paired homomorphism
- Projection uniqueness
- Coordinate projections
- Homomorphism equality
- Product map extensionality
- Concrete factorization
- Induced cyclic map
- Induced map injectivity
- Kernel equality
- Quotient factorization
- Diagonal quotient map
- Injective induced map
- Intersection quotient
- Chinese remainder theorem
- Coprime moduli
- Cyclic product map
- Equalizer subgroup
- Fiber product
- Pullback construction
- Homomorphism fiber
- Kernel coset
- Solution set
- Canonical quotient map
- Nested quotients
- Conjugacy
- Normal subgroup classification
- Klein four normal
- Order three quotient
- Finite to infinite map
- Torsion free target
- Trivial homomorphism
- Center index
- Contradiction
- Prime order quotient
- Isomorphism transport
- Normality invariant
- Cyclic quotient classification
- Quotient isomorphism type
- Subgroup divisors
- Action on partitions
- Klein four kernel
- Symmetric four quotient
- Center computation
- Mixed direct product
- Unique order subgroup
- Coprime direct product
- Rectangular subgroups
Group actions, orbits, and stabilizers
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Symmetric group
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Normal subgroup
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Action kernel
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Free action
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Stabilizer intersection
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Coset action
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Fixed point congruence
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Orbit partition
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Centralizer index
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Class equation
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P group center
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Burnside count
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Cyclic burnside
113 more topics with one problem
- Dihedral action
- Orbit stabilizer
- Vertex stabilizer
- Action axioms
- Permutation representation
- Infinite orbit
- Translation action
- Point stabilizer
- Conjugation action
- Empty set edge case
- Singleton orbit
- Trivial action
- Orbit refinement
- Restricted action
- Diagonal action
- Product set
- Core subgroup
- Coset well definedness
- Finiteness audit
- Orbit stabilizer bijection
- Setwise stabilizer
- Subset action
- Single orbit
- Transitive action
- Equivariant map
- Injectivity counterexample
- Transitive image
- Conjugate stabilizers
- Coset model
- Transitive g set
- Divisibility obstruction
- Orbit decomposition
- Left regular action
- Regular action
- Unique transporter
- Orbit size
- P group action
- Consistency check
- Divisor constraint
- Index eight example
- Orbit divisibility
- Class size
- Conjugacy class
- Conjugation fixed point
- Singleton class
- Cycle type
- Conjugacy
- Dihedral conjugacy
- Conjugation orbits
- Centralizer order
- Class size divisibility
- Impossibility check
- Conjugacy closure
- Union of classes
- Class size count
- Transposition centralizer
- Centralizer sum
- Commuting probability
- Conjugacy class count
- Classification
- Two conjugacy classes
- Burnside lemma
- Double counting
- Fixed point average
- Binary necklace
- Cyclic rotation
- Dihedral coloring
- Reflection cycle type
- Three colors
- Triangle coloring
- Burnside classification
- Cube rotations
- Face coloring
- Coloring action
- Fixed coloring formula
- Permutation cycles
- Bell number
- Diagonal tuple action
- Equality pattern
- Necklace formula
- Burnside verification
- Cyclic distance
- Fixed weight necklace
- Burnside diagnosis
- Fixed point table
- Orbit double count
- Equivalence relation
- Orbit relation
- Claim repair
- Multiple orbits
- Stabilizer conjugacy
- Faithful not free
- Natural action
- Property separation
- Prescribed kernel
- Quotient action
- Transitive not faithful
- Action
- Disjoint regular orbits
- Free not transitive
- Coset construction
- Disjoint union action
- Prescribed orbits
- Lagrange
- Orbit obstruction
- Stabilizer order
- Cauchy theorem
- Cyclic rotation proof
- Cayley theorem
- Group embedding
- Left regular representation
- Permutation image
- Prime index normality
Sylow theory and finite-group structure
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Cyclic group
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Internal direct product
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Sylow count
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Element count
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Order twelve
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Sylow three
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Sylow seven
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Order thirty
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Element ledger
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Sylow conjugacy
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Normal sylow factors
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Abelian classification
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Nonsimplicity
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Semidirect product
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Sylow action
108 more topics with one problem
- Normalizer index
- P group fixed point
- Normal sylow
- Order twenty
- Uniqueness normality
- Order twenty one
- Sylow arithmetic limit
- Order twenty eight
- Multiple sylow counts
- Normality audit
- Cyclicity counterexample
- Order forty five
- Order fifty six
- Sylow seven count
- Cyclicity criterion
- Normal sylow products
- Order pq
- Prime order intersections
- Sylow element count
- Unique sylow
- Nilpotent decomposition
- Normal sylow product
- Order fifteen
- Order thirty three
- Order fifty five
- Order thirty five
- Index two normality
- Normal order nine
- Order eighteen
- Order three elements
- Sylow three count
- Conjugation on sylow
- Order forty
- Prime power components
- Subgroup sylow decomposition
- Normal sylow p
- Odd prime
- Order two p
- Abelian p group
- Order p squared
- Two group types
- Abelian order eight
- Element exponent
- Order eight
- Order ten classification
- Automorphism action
- Order twenty one classification
- Elementary abelian p group
- One dimensional subspace
- Subgroup count
- Order nine
- Order twelve nonsimple
- Sylow case split
- Normal sylow three
- Order eighteen nonsimple
- Simplicity
- Simplicity
- Conjugacy class arithmetic
- Normal union classes
- Order thirty nonsimple
- Sylow count contradiction
- Normal sylow seven
- Order forty two
- Normal sylow five
- Order forty nonsimple
- Simplicity obstruction
- Small index
- Simple group embedding
- Faithful conjugation
- Inner automorphism
- Simple group center
- Factorial divisibility
- Permutation embedding
- Simple action
- Order twenty four nonsimple
- Embedding
- Order obstruction
- Order thirty six nonsimple
- Sylow conjugation action
- Transitive sylow set
- Frattini argument
- Normalizer factorization
- Coset fixed points
- P group normalizer condition
- Proper containment
- Group transfer
- Computation
- Transfer kernel
- Normal order p squared
- Order p cubed
- Quotient lifting
- Maximal p subgroup
- Normalizer condition
- Prime index
- Fixed word count
- Word rotation
- Burnside transfer
- Normal p complement
- Sylow normalizer
- A4
- C12
- Sylow constraints limit
- Conjugate containment
- P subgroup containment
- Sylow extension
- Normal order fifteen
- Order thirty capstone
- Sylow product