--- title: "Abstract Algebra" description: "15 topics in Mathematics, in the order they build on each other." canonical: https://lightmysky.com/learn/mathematics/areas/abstract-algebra source: https://lightmysky.com/learn/mathematics/areas/abstract-algebra.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Abstract Algebra 15 topics in Mathematics, in the order they build on each other. Page: https://lightmysky.com/learn/mathematics/areas/abstract-algebra - [Groups, Subgroups and Symmetry](https://lightmysky.com/learn/mathematics/groups-subgroups-and-symmetry-mt_HFAhcaGo2L): One operation, associative, with an identity and inverses. Symmetries of an object, permutations and modular arithmetic all satisfy the same four axioms, so anything proved once holds for all of them. - [Homomorphisms, Cosets and Lagrange's Theorem](https://lightmysky.com/learn/mathematics/homomorphisms-cosets-and-lagranges-theorem-mt_yKiZ8sg7yN): Maps that preserve the operation, the classes a subgroup carves out, and the counting result that a subgroup's order divides the group's. Kernels are what make quotient groups possible. - [Rings, Fields and Their First Properties](https://lightmysky.com/learn/mathematics/rings-fields-and-their-first-properties-mt_BmhIook13o): Two operations instead of one. Rings cover the integers and polynomials, fields are the rings where division works, and the difference explains why some equations are solvable and others are not. - [Normal Subgroups and Quotient Groups](https://lightmysky.com/learn/mathematics/normal-subgroups-and-quotient-groups-mt_RU7A-t7wQ5): Cosets can be multiplied consistently exactly when the subgroup is normal. The quotient group that results turns the kernel of a homomorphism into a complete description of the map. - [Group Actions, Orbits and the Class Equation](https://lightmysky.com/learn/mathematics/group-actions-orbits-and-the-class-equation-mt_urut0TFTOR): A group acting on a set splits it into orbits, and each orbit has the size of an index of a stabiliser. Counting one set two ways with that relation proves results no direct argument reaches. - [The Sylow Theorems and Groups of Small Order](https://lightmysky.com/learn/mathematics/the-sylow-theorems-and-groups-of-small-order-mt_vg2VYkLuz8): Subgroups of prime-power order exist, are all conjugate, and their number is pinned down by two arithmetic conditions. That is enough to rule out simple groups of many orders and to finish the classification of small ones. - [Ideals and Quotient Rings](https://lightmysky.com/learn/mathematics/ideals-and-quotient-rings-mt_uQBA9E1JIP): The ring counterpart of a normal subgroup is an ideal, and quotienting by it declares its elements to be zero. Whether an ideal is prime or maximal is read directly off the ring the quotient produces. - [Polynomial Rings, Irreducibility and Unique Factorisation](https://lightmysky.com/learn/mathematics/polynomial-rings-irreducibility-and-unique-factorisation-mt_m2G9PIHj6y): Polynomials over a field divide with remainder, and that single fact forces unique factorisation into irreducibles. Deciding whether a given polynomial is irreducible over the rationals needs tests rather than inspection. - [Modules: Linear Algebra Over a Ring](https://lightmysky.com/learn/mathematics/modules-linear-algebra-over-a-ring-mt_0GMVp-s8wD): A module is a vector space whose scalars form a ring rather than a field. Bases may fail to exist, and what survives is a structure theorem that delivers the Jordan form and the classification of finite abelian groups at the same time. - [Field Extensions and Their Degrees](https://lightmysky.com/learn/mathematics/field-extensions-and-their-degrees-mt_CjOMNt5_i0): Adjoining a root to a field produces a larger field, which is a vector space over the smaller one. Its dimension is called the degree, and degrees multiply along a tower of extensions. - [Group Representations and the Group Algebra](https://lightmysky.com/learn/mathematics/group-representations-and-the-group-algebra-mt_y5JlvQGKYW): Realise an abstract group as matrices acting on a vector space, and recognise a representation as a module over the group algebra. - [Maschke's Theorem and Complete Reducibility](https://lightmysky.com/learn/mathematics/maschkes-theorem-and-complete-reducibility-mt_FVNCwUqeTx): Prove that over a field of the right characteristic every representation splits into irreducible pieces, and see exactly where the proof needs that hypothesis. - [The Galois Correspondence](https://lightmysky.com/learn/mathematics/the-galois-correspondence-mt_30kyliQR_L): The symmetries of a field extension form a group whose subgroups match the intermediate fields, in reverse order. Whether that group can be built from abelian pieces is what decides if the roots can be written with radicals. - [Characters and the Orthogonality Relations](https://lightmysky.com/learn/mathematics/characters-and-the-orthogonality-relations-mt_RleR7YGXjX): Reduce a representation to the trace of each group element, and prove the resulting functions are orthogonal for an inner product on class functions. - [Decomposing a Representation from Its Character](https://lightmysky.com/learn/mathematics/decomposing-a-representation-from-its-character-mt_ccEcWW9xAa): Build the character table of a finite group and use it to read off how any representation breaks into irreducibles.