Combine two elements into another.
The operation can be addition, composition, rotation or another rule.
Side 135
Abstract Algebra
Algebra stripped to its reusable structure: operations, symmetries and mappings that reveal when apparently different mathematical systems behave in the same way.
operation→axioms→structure→mapping→symmetry
04lenses
16working concepts
V0content
SS-1.0standard
Groups formalize reversible composition and symmetry.
A group consists of elements with an operation satisfying closure, associativity, identity and inverses.
Leave every element unchanged.
Each group has a unique identity relative to its operation.
Undo an element under the operation.
Invertibility makes group structure especially natural for transformations and symmetries.
Find a smaller group inside a group.
Subgroups expose internal structure and recurring symmetries.
Structure-preserving maps reveal when systems are algebraically alike.
Homomorphisms keep the operation intact while translating between algebraic objects.
Preserve operations across systems.
A homomorphism can collapse detail while retaining algebraic relations.
Identify elements mapped to the identity.
The kernel measures what information the map loses.
Track reachable outputs.
The image is itself a structured subset of the target.
Preserve structure bijectively.
Isomorphic systems differ in representation but not in abstract algebraic form.
Adding a second operation creates richer algebraic structure.
Rings model addition and multiplication together, while fields make division possible away from zero.
Combine additive group structure with multiplication.
Integers and polynomial systems are canonical examples.
Generalize divisibility and compatible substructure.
Ideals allow quotient constructions and organize ring structure.
Allow addition, subtraction, multiplication and nonzero division.
Rational, real and complex numbers are fields with different properties.
Treat symbolic expressions as algebraic objects.
Factorization and roots connect ring structure to equations.
Algebra turns symmetry into something calculable.
Transformation groups encode what can change while preserving selected structure.
Rearrange objects while preserving membership.
Permutation groups provide a universal language for finite symmetry.
Track where an element can move under the group.
Orbits partition a set according to reachable symmetry states.
Track transformations that leave an element fixed.
Stabilizers connect local invariance to global group structure.
Collapse equivalent elements into classes.
Quotient structures reveal simpler systems hidden inside larger ones.
Abstraction earns its value through reuse. The point is not to remove meaning but to isolate the structural features that survive across numbers, permutations, matrices and geometric symmetries.