Use remainders, congruences, and arithmetic mod n as a first reliable playground for algebra. You will compute units, solve simple congruences, and see why closure and inverses matter.

Treat permutations as actions that rearrange objects. Work with cycle notation, composition, inverses, parity, and common mistakes caused by order of operations.

Trace how equations, geometry, crystallography, and physics pushed mathematicians toward the modern idea of a group. This chapter connects Galois, Cayley, Noether, Lie, and later computational work to the tools used today.

Turn examples into the formal definition of a group. Check the axioms, identify identities and inverses, and separate true groups from structures that almost work.

Use Cayley tables, generators, and relations to describe finite groups without listing every calculation by hand. Practice finding orders of elements and spotting structure from compact data.

Find smaller groups living inside larger ones. Work with generated subgroups, cyclic groups, element orders, and the special behavior of groups generated by one element.

Divide a group into equal-sized pieces using left and right cosets. Prove Lagrange’s theorem and use it to rule out impossible subgroup sizes and element orders.

Compare groups by maps that preserve the operation. Build homomorphisms, find kernels and images, and decide when two groups are the same in structure even if they look different.

Use normal subgroups to collapse a group into a smaller group while keeping the operation well defined. Build quotient groups and connect them to kernels of homomorphisms.

Apply the first, second, third, and correspondence isomorphism theorems as working tools rather than memorized statements. Use them to simplify quotient calculations and compare subgroup structures.

Combine groups in controlled ways with direct products and direct sums. Classify finite abelian groups and recognize how prime-power pieces determine the whole group.

Model the rotations and reflections of polygons with dihedral groups. Use them as a full case study in presentations, subgroups, conjugacy, quotients, and group actions.

Work deeply with symmetric and alternating groups. Use cycle types, transpositions, parity, conjugacy classes, and simplicity of A_n in the cases where it holds.

Let groups act on sets to turn abstract structure into movement. Use orbits, stabilizers, fixed points, and the orbit-stabilizer theorem to count and classify objects.

Use conjugation as a group acting on itself. Derive class equations, study centers and centralizers, and apply these ideas to p-groups and finite group structure.

Use the Sylow theorems to find and count large prime-power subgroups inside finite groups. Apply them to prove that many group orders allow only certain structures.

Follow a real classification workflow from a group’s order to a short list of possible structures. Combine Lagrange’s theorem, normal subgroups, actions, quotients, and Sylow arguments to make and justify decisions.

Break groups into layers using subnormal series, composition series, and the Jordan-Hölder theorem. Connect solvable groups to repeated abelian quotients and to the limits of solving equations by radicals.

Study groups with no nontrivial normal subgroups and why they act like building blocks. Meet cyclic prime groups, alternating groups, groups of Lie type, and sporadic groups through the story of the finite simple group classification.

Describe groups by named generators and rules they must satisfy. Move between presentations, Cayley graphs, quotient groups, and concrete examples built from relations.

Use free groups as the most flexible groups generated by a set. Reduce words, build homomorphisms by choosing generator images, and see how relations create quotients.

Build groups where one group controls another through an action. Use semidirect products to construct non-abelian examples and to explain many groups of small order.

Turn hand calculations into reproducible work with GAP and SageMath. Create groups, compute subgroups and quotients, test isomorphism, inspect character tables, and document results clearly.

Use the main ideas behind computational group theory. Work with permutation-group algorithms, stabilizer chains, matrix groups over finite fields, and practical limits of brute-force calculation.

Bring linear algebra into group theory through matrices that form groups under multiplication. Study GL(n), SL(n), orthogonal and unitary groups, eigenvalue clues, and actions on vector spaces.

Represent abstract group elements as matrices so their structure can be studied with linear algebra. Build representations, subrepresentations, irreducible representations, and regular representations.

Use traces of representation matrices to compress large amounts of group information. Build character tables, decompose representations, and apply orthogonality relations to finite groups.

Use groups to describe symmetries of graphs, polyhedra, surfaces, and tilings. Connect automorphism groups, Cayley graphs, fundamental groups, and covering spaces at a working level.

Handle groups whose elements vary continuously, such as rotations of a sphere. Use matrix Lie groups, tangent spaces, Lie algebras, exponentials, and one-parameter subgroups.

Connect group theory to polynomial equations through field automorphisms. Build the needed field and polynomial language, compute small Galois groups, and see why symmetry controls solvability.

Use group structures that appear in modular arithmetic, elliptic curves, and modern cryptography. Study cyclic groups, discrete logarithms, elliptic curve groups, and the algebra behind security assumptions.

Apply groups to real symmetry problems in science. Use point groups, space groups, representation ideas, and selection rules in chemistry, crystallography, particle physics, and quantum systems.

Study infinite groups through generators, relations, and the shapes of their Cayley graphs. Work with growth, amenability, hyperbolic groups, and examples that behave very differently from finite groups.

Add topology to groups so limits, continuity, and compactness matter. Work with topological groups, profinite groups, inverse limits, and why these ideas appear in number theory and Galois theory.

Measure when group extensions split and when hidden obstructions remain. Use low-dimensional group cohomology to organize extensions, actions, cocycles, and projective representations.

Use Lean to write machine-checkable group theory proofs. Formalize definitions, subgroup arguments, quotient constructions, and short theorem chains while seeing how modern proof libraries are organized.

Pull the course together through a substantial project such as classifying a family of groups, computing a character table, modeling a symmetry group, or formalizing a theorem. The focus is on choosing tools, checking results, and communicating a clean mathematical argument.

Map the paths that use group theory in research, teaching, physics, chemistry, cryptography, computation, and pure mathematics. Build proof-of-skill artifacts, find books and databases, follow seminars and preprints, and choose next topics with confidence.