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Groups Subgroups And Generated StructuresGroup AxiomsInteger AdditionAdditive Inverses

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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Groups Subgroups And Generated StructuresNonassociativitySubtraction OperationAxiom Counterexamples

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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Groups Subgroups And Generated StructuresMultiplicative GroupNonzero RealsDomain Restriction

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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Groups Subgroups And Generated StructuresMonoid UnitsInverse ProductInherited Operation

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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Groups Subgroups And Generated StructuresIdentity UniquenessTwo Sided IdentityAxiom Unpacking

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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Groups Subgroups And Generated StructuresInverse UniquenessAssociativityGroup Identities

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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Groups Subgroups And Generated StructuresLeft CancellationRight CancellationInverse Multiplication

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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Groups Subgroups And Generated StructuresGroup EquationsFactor OrderSolution Uniqueness

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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Groups Subgroups And Generated StructuresTransported StructureBijectionGroup Construction

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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Groups Subgroups And Generated StructuresShifted OperationExplicit IsomorphismNonstandard Identity

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:011The three-part subgroup test

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Groups Subgroups And Generated StructuresSubgroup CriterionClosure PropertiesInherited Associativity

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:012The one-step subgroup test

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Groups Subgroups And Generated StructuresOne Step Subgroup TestNonempty HypothesisInverse Closure

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:013Arbitrary intersections of subgroups

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Groups Subgroups And Generated StructuresSubgroup IntersectionIndexed FamiliesEmpty Intersection

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:014When the union of two subgroups is a subgroup

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Groups Subgroups And Generated StructuresSubgroup UnionContainment DichotomyProduct Closure

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:015The center is a subgroup

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Groups Subgroups And Generated StructuresGroup CenterCommuting ElementsSubgroup Test

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:016Centralizers pass the subgroup test

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Groups Subgroups And Generated StructuresCentralizerVacuous ConditionCenter Identification

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:017Even integers and odd integers

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Groups Subgroups And Generated StructuresEven IntegersOdd IntegersAdditive Subgroups

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:018Positive reals inside the multiplicative group

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Groups Subgroups And Generated StructuresPositive RealsSign ClosureMultiplicative Subgroups

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:019Finite product-closed subsets

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Groups Subgroups And Generated StructuresFinite SubsetPower RepetitionProduct Closure

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:020A subgroup hidden in a congruence

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Groups Subgroups And Generated StructuresModular CongruenceInteger SubgroupsResidue Classes

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:021Generated subgroup as an intersection

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Groups Subgroups And Generated StructuresGenerated SubgroupMinimalitySubgroup Intersection

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:022Subgroups generated inside the integers

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Groups Subgroups And Generated StructuresBezout IdentityGcd Generated SubgroupInteger Linear Combinations

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:023All integer powers form a subgroup

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Groups Subgroups And Generated StructuresCyclic SubgroupInteger PowersSingle Generator

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:024Powers that generate the same finite cyclic subgroup

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Groups Subgroups And Generated StructuresPower GeneratorsCoprimalityBezout Exponents

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:025Two transpositions generate $S_3$

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Groups Subgroups And Generated StructuresSymmetric GroupTransposition GeneratorsWord Enumeration

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:026Adjacent transpositions generate every permutation

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Groups Subgroups And Generated StructuresAdjacent TranspositionsSymmetric Group GeneratorsInversion Reduction

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:027Normal forms in a dihedral generating set

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Groups Subgroups And Generated StructuresDihedral Normal FormGenerator RelationsWord Reduction

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:028A finite closure algorithm

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Groups Subgroups And Generated StructuresFinite Generation AlgorithmClosure IterationStabilization

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:029Joining two subgroups

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Groups Subgroups And Generated StructuresSubgroup JoinGenerated UnionStrict Containment

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:030Recognizing a generating set from relations

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Groups Subgroups And Generated StructuresGeneration CriterionNormal Form HypothesisUniversal Coverage

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:031An element and its inverse have the same order

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Groups Subgroups And Generated StructuresElement OrderInverse PowersInfinite Order

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:032Conjugation preserves element order

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Groups Subgroups And Generated StructuresConjugate ElementsOrder InvariancePower Conjugation

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:033Order of a power

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Groups Subgroups And Generated StructuresOrder OF A PowerGcd FormulaDivisibility

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:034Order of a product of commuting elements

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Groups Subgroups And Generated StructuresCommuting ElementsCoprime OrdersProduct Order

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:035The commutativity hypothesis really matters

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Groups Subgroups And Generated StructuresProduct Order CounterexampleNoncommuting TranspositionsHypothesis Audit

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:036Left cosets partition the group

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Groups Subgroups And Generated StructuresLeft CosetsPartitionCoset Overlap

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:037Integer cosets are congruence classes

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Groups Subgroups And Generated StructuresInteger CosetsModular CongruenceSubgroup Index

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:038Every coset has the size of its subgroup

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Groups Subgroups And Generated StructuresCoset BijectionLeft TranslationFinite Cardinality

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:039Left and right cosets can disagree

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Groups Subgroups And Generated StructuresLeft Right CosetsNonnormal SubgroupPermutation Computation

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:040Every subgroup of index two is normal

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Groups Subgroups And Generated StructuresIndex TwoNormal SubgroupCoset Partition

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:041Direct products are groups

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Groups Subgroups And Generated StructuresDirect ProductCoordinatewise OperationProduct Inverse

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:042The opposite group

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Groups Subgroups And Generated StructuresOpposite GroupReversed MultiplicationAbelian Criterion

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:043When a product of subgroups is a subgroup

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Groups Subgroups And Generated StructuresSubgroup ProductPermutable SubgroupsSetwise Equality

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:044Three proper subgroups can cover a group

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Groups Subgroups And Generated StructuresKlein Four GroupSubgroup CoverUnion Counterexample

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:045Torsion elements need not form a subgroup

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Groups Subgroups And Generated StructuresTorsion ElementsInfinite Dihedral GroupClosure Counterexample

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:046Torsion elements in an abelian group

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Groups Subgroups And Generated StructuresAbelian TorsionFinite Order ElementsLcm Exponent

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:047Groups of prime order are cyclic

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Groups Subgroups And Generated StructuresPrime Order GroupCyclicityLagrange Consequence

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:048Exponent two forces commutativity

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Groups Subgroups And Generated StructuresExponent TwoAbelian CriterionSelf Inverse Elements

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:049A unique involution does not force cyclicity

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Groups Subgroups And Generated StructuresQuaternion GroupUnique InvolutionNoncyclic Counterexample

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:050Classifying groups of order four

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Groups Subgroups And Generated StructuresOrder Four ClassificationKlein Four GroupElement Order Analysis

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Groups, subgroups, and generated structuresAbstract Algebra I · Mixed review

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Problem library

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Search by course or topic.

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IndependentSolved with helpIncorrect / unresolvedDraft in progressNot attemptedReview due

Problem results

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Groups Subgroups And Generated StructuresGroup AxiomsInteger AdditionAdditive Inverses

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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Groups Subgroups And Generated StructuresNonassociativitySubtraction OperationAxiom Counterexamples

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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Groups Subgroups And Generated StructuresMultiplicative GroupNonzero RealsDomain Restriction

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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Groups Subgroups And Generated StructuresMonoid UnitsInverse ProductInherited Operation

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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Groups Subgroups And Generated StructuresIdentity UniquenessTwo Sided IdentityAxiom Unpacking

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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Groups Subgroups And Generated StructuresInverse UniquenessAssociativityGroup Identities

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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Groups Subgroups And Generated StructuresLeft CancellationRight CancellationInverse Multiplication

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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Groups Subgroups And Generated StructuresGroup EquationsFactor OrderSolution Uniqueness

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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Groups Subgroups And Generated StructuresTransported StructureBijectionGroup Construction

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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Groups Subgroups And Generated StructuresShifted OperationExplicit IsomorphismNonstandard Identity

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:011The three-part subgroup test

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Groups Subgroups And Generated StructuresSubgroup CriterionClosure PropertiesInherited Associativity

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:012The one-step subgroup test

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Groups Subgroups And Generated StructuresOne Step Subgroup TestNonempty HypothesisInverse Closure

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:013Arbitrary intersections of subgroups

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Groups Subgroups And Generated StructuresSubgroup IntersectionIndexed FamiliesEmpty Intersection

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:014When the union of two subgroups is a subgroup

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Groups Subgroups And Generated StructuresSubgroup UnionContainment DichotomyProduct Closure

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:015The center is a subgroup

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Groups Subgroups And Generated StructuresGroup CenterCommuting ElementsSubgroup Test

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:016Centralizers pass the subgroup test

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Groups Subgroups And Generated StructuresCentralizerVacuous ConditionCenter Identification

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:017Even integers and odd integers

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Groups Subgroups And Generated StructuresEven IntegersOdd IntegersAdditive Subgroups

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:018Positive reals inside the multiplicative group

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Groups Subgroups And Generated StructuresPositive RealsSign ClosureMultiplicative Subgroups

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:019Finite product-closed subsets

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Groups Subgroups And Generated StructuresFinite SubsetPower RepetitionProduct Closure

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:020A subgroup hidden in a congruence

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Groups Subgroups And Generated StructuresModular CongruenceInteger SubgroupsResidue Classes

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:021Generated subgroup as an intersection

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Groups Subgroups And Generated StructuresGenerated SubgroupMinimalitySubgroup Intersection

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:022Subgroups generated inside the integers

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Groups Subgroups And Generated StructuresBezout IdentityGcd Generated SubgroupInteger Linear Combinations

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:023All integer powers form a subgroup

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Groups Subgroups And Generated StructuresCyclic SubgroupInteger PowersSingle Generator

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:024Powers that generate the same finite cyclic subgroup

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Groups Subgroups And Generated StructuresPower GeneratorsCoprimalityBezout Exponents

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:025Two transpositions generate $S_3$

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Groups Subgroups And Generated StructuresSymmetric GroupTransposition GeneratorsWord Enumeration

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:026Adjacent transpositions generate every permutation

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Groups Subgroups And Generated StructuresAdjacent TranspositionsSymmetric Group GeneratorsInversion Reduction

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:027Normal forms in a dihedral generating set

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Groups Subgroups And Generated StructuresDihedral Normal FormGenerator RelationsWord Reduction

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:028A finite closure algorithm

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Groups Subgroups And Generated StructuresFinite Generation AlgorithmClosure IterationStabilization

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:029Joining two subgroups

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Groups Subgroups And Generated StructuresSubgroup JoinGenerated UnionStrict Containment

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:030Recognizing a generating set from relations

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Groups Subgroups And Generated StructuresGeneration CriterionNormal Form HypothesisUniversal Coverage

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:031An element and its inverse have the same order

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Groups Subgroups And Generated StructuresElement OrderInverse PowersInfinite Order

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:032Conjugation preserves element order

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Groups Subgroups And Generated StructuresConjugate ElementsOrder InvariancePower Conjugation

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Groups, subgroups, and generated structuresAbstract Algebra I · Proof

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AA:033Order of a power

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Groups Subgroups And Generated StructuresOrder OF A PowerGcd FormulaDivisibility

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:034Order of a product of commuting elements

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Groups Subgroups And Generated StructuresCommuting ElementsCoprime OrdersProduct Order

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:035The commutativity hypothesis really matters

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Groups Subgroups And Generated StructuresProduct Order CounterexampleNoncommuting TranspositionsHypothesis Audit

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:036Left cosets partition the group

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Groups Subgroups And Generated StructuresLeft CosetsPartitionCoset Overlap

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:037Integer cosets are congruence classes

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Groups Subgroups And Generated StructuresInteger CosetsModular CongruenceSubgroup Index

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:038Every coset has the size of its subgroup

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Groups Subgroups And Generated StructuresCoset BijectionLeft TranslationFinite Cardinality

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:039Left and right cosets can disagree

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Groups Subgroups And Generated StructuresLeft Right CosetsNonnormal SubgroupPermutation Computation

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:040Every subgroup of index two is normal

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Groups Subgroups And Generated StructuresIndex TwoNormal SubgroupCoset Partition

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:041Direct products are groups

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Groups Subgroups And Generated StructuresDirect ProductCoordinatewise OperationProduct Inverse

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:042The opposite group

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Groups Subgroups And Generated StructuresOpposite GroupReversed MultiplicationAbelian Criterion

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Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

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AA:043When a product of subgroups is a subgroup

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Groups Subgroups And Generated StructuresSubgroup ProductPermutable SubgroupsSetwise Equality

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:044Three proper subgroups can cover a group

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Groups Subgroups And Generated StructuresKlein Four GroupSubgroup CoverUnion Counterexample

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:045Torsion elements need not form a subgroup

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Groups Subgroups And Generated StructuresTorsion ElementsInfinite Dihedral GroupClosure Counterexample

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:046Torsion elements in an abelian group

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Groups Subgroups And Generated StructuresAbelian TorsionFinite Order ElementsLcm Exponent

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:047Groups of prime order are cyclic

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Groups Subgroups And Generated StructuresPrime Order GroupCyclicityLagrange Consequence

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:048Exponent two forces commutativity

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Groups Subgroups And Generated StructuresExponent TwoAbelian CriterionSelf Inverse Elements

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Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:049A unique involution does not force cyclicity

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Groups Subgroups And Generated StructuresQuaternion GroupUnique InvolutionNoncyclic Counterexample

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Groups, subgroups, and generated structuresAbstract Algebra I · Concept & counterexample

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AA:050Classifying groups of order four

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Groups Subgroups And Generated StructuresOrder Four ClassificationKlein Four GroupElement Order Analysis

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Groups, subgroups, and generated structuresAbstract Algebra I · Mixed review

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Problem results

AA:001The integers under addition

AA:002Why subtraction is not a group law

AA:003Nonzero reals under multiplication

AA:004Units of a monoid form a group

AA:005Uniqueness of the identity

AA:006Uniqueness of inverses

AA:007Cancellation comes from inverses

AA:008Solving a two-sided group equation

AA:009Transporting a group law across a bijection

AA:010A shifted group operation on the integers

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Problem results

AA:001The integers under addition

AA:002Why subtraction is not a group law

AA:003Nonzero reals under multiplication

AA:004Units of a monoid form a group

AA:005Uniqueness of the identity

AA:006Uniqueness of inverses

AA:007Cancellation comes from inverses

AA:008Solving a two-sided group equation

AA:009Transporting a group law across a bijection

AA:010A shifted group operation on the integers