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AA:001The integers under addition

Groups Subgroups And Generated StructuresGroup AxiomsInteger AdditionAdditive Inverses

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:002Why subtraction is not a group law

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Groups Subgroups And Generated StructuresNonassociativitySubtraction OperationAxiom Counterexamples

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

AA:003Nonzero reals under multiplication

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Groups Subgroups And Generated StructuresMultiplicative GroupNonzero RealsDomain Restriction

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:004Units of a monoid form a group

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Groups Subgroups And Generated StructuresMonoid UnitsInverse ProductInherited Operation

Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

AA:005Uniqueness of the identity

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Groups Subgroups And Generated StructuresIdentity UniquenessTwo Sided IdentityAxiom Unpacking

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:006Uniqueness of inverses

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Groups Subgroups And Generated StructuresInverse UniquenessAssociativityGroup Identities

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:007Cancellation comes from inverses

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Groups Subgroups And Generated StructuresLeft CancellationRight CancellationInverse Multiplication

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:008Solving a two-sided group equation

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Groups Subgroups And Generated StructuresGroup EquationsFactor OrderSolution Uniqueness

Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

AA:009Transporting a group law across a bijection

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Groups Subgroups And Generated StructuresTransported StructureBijectionGroup Construction

Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

AA:010A shifted group operation on the integers

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Groups Subgroups And Generated StructuresShifted OperationExplicit IsomorphismNonstandard Identity

Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

AA:011The three-part subgroup test

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Groups Subgroups And Generated StructuresSubgroup CriterionClosure PropertiesInherited Associativity

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:012The one-step subgroup test

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Groups Subgroups And Generated StructuresOne Step Subgroup TestNonempty HypothesisInverse Closure

Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

AA:013Arbitrary intersections of subgroups

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Groups Subgroups And Generated StructuresSubgroup IntersectionIndexed FamiliesEmpty Intersection

Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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

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Groups Subgroups And Generated StructuresSubgroup UnionContainment DichotomyProduct Closure

Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

AA:015The center is a subgroup

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Groups Subgroups And Generated StructuresGroup CenterCommuting ElementsSubgroup Test

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:016Centralizers pass the subgroup test

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Groups Subgroups And Generated StructuresCentralizerVacuous ConditionCenter Identification

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:017Even integers and odd integers

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Groups Subgroups And Generated StructuresEven IntegersOdd IntegersAdditive Subgroups

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

AA:018Positive reals inside the multiplicative group

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Groups Subgroups And Generated StructuresPositive RealsSign ClosureMultiplicative Subgroups

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:019Finite product-closed subsets

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Groups Subgroups And Generated StructuresFinite SubsetPower RepetitionProduct Closure

Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

AA:020A subgroup hidden in a congruence

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Groups Subgroups And Generated StructuresModular CongruenceInteger SubgroupsResidue Classes

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:021Generated subgroup as an intersection

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Groups Subgroups And Generated StructuresGenerated SubgroupMinimalitySubgroup Intersection

Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

AA:022Subgroups generated inside the integers

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Groups Subgroups And Generated StructuresBezout IdentityGcd Generated SubgroupInteger Linear Combinations

Groups, subgroups, and generated structuresAbstract Algebra I · Calculation

AA:023All integer powers form a subgroup

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Groups Subgroups And Generated StructuresCyclic SubgroupInteger PowersSingle Generator

Groups, subgroups, and generated structuresAbstract Algebra I · Proof

AA:024Powers that generate the same finite cyclic subgroup

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Groups Subgroups And Generated StructuresPower GeneratorsCoprimalityBezout Exponents

Groups, subgroups, and generated structuresAbstract Algebra I · Theorem

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AA:001The integers under additionView problem statement

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AA:001The integers under additionView problem statement

AA:001The integers under addition

AA:001The integers under addition