Examples

This gallery walks through every public feature of mathematicskit.abstract_algebra: cyclic and permutation groups, subgroups and cosets, finite fields, and polynomial ring arithmetic.

Each script in this gallery is self-contained and can be run directly with python examples/abstract_algebra/<section>/<script>.py.

Sections

• groups – cyclic and permutation groups, Cayley tables, and group-property checks.

• subgroups – subgroup and coset enumeration, and Lagrange’s theorem.

• finite_fields – GF(p) and GF(p^n) arithmetic.

• polynomial_ring – polynomial addition, multiplication, division, and gcd.

Group actions

Orbits of a group acting on a finite set, and Burnside’s orbit-counting lemma.

Burnside’s lemma: counting necklaces

Burnside's lemma: counting necklaces

Error-correcting codes

Reed-Solomon codes: polynomial evaluation over a finite field, and recovery from erasures.

Reed-Solomon codes: recovering from erasures

Reed-Solomon codes: recovering from erasures

Finite fields

GF(p) and GF(p^n) arithmetic.

GF(8): finding an irreducible polynomial and building the field

GF(8): finding an irreducible polynomial and building the field

Groups

Permutation groups of polynomial roots, dihedral and quaternion groups, Cayley tables, and group-property checks.

Galois’s permutation groups: S_3 permuting the roots of x^3 - 2

Galois's permutation groups: S_3 permuting the roots of x^3 - 2

Klein’s Erlangen program: the symmetry group of a square

Klein's Erlangen program: the symmetry group of a square

Hamilton’s quaternions: the group Q_8

Hamilton's quaternions: the group Q_8

Cayley’s abstract group and its multiplication table

Cayley's abstract group and its multiplication table

Homomorphisms

Structure-preserving maps between groups, their kernels and images, and the first isomorphism theorem.

Noether’s first isomorphism theorem

Noether's first isomorphism theorem

Polynomial ring arithmetic

Polynomial addition, multiplication, division, and gcd.

Polynomial long division and the Euclidean algorithm

Polynomial long division and the Euclidean algorithm

Group structure

Element orders, normal subgroups and quotients, solvability, Sylow subgroups, and composition series.

Cauchy’s theorem: elements of prime order

Cauchy's theorem: elements of prime order

Abel-Ruffini and solvable groups: why S_5 blocks the quintic

Abel-Ruffini and solvable groups: why S_5 blocks the quintic

Sylow’s theorems in S_4

Sylow's theorems in S_4

Hölder’s quotient groups: S_4 / V_4 is S_3

Hölder's quotient groups: S_4 / V_4 is S_3

Jordan-Hölder: composition series and their factors

Jordan-Hölder: composition series and their factors

Subgroups and cosets

Subgroup and coset enumeration, and Lagrange’s theorem.

Lagrange’s theorem: subgroup order divides group order

Lagrange's theorem: subgroup order divides group order