Closure

Associativity

Identity element

Inverses

Multiplicative: ab, identity e

Additive (common in abelian groups): a+b, identity 0, inverse -a

A symmetry is a transformation that preserves some object or structure

Examples of preserved structure

Doing symmetry A then B equals composed symmetry B ∘ A

Identity symmetry: do nothing

Inverse symmetry: undo

The set of all symmetries of an object forms a group under composition

Finite: limited number of symmetries (e.g., polygon rotations)

Infinite: unbounded symmetries (e.g., integer translations)

Abelian: ab=ba (e.g., Z under addition)

Non-abelian: order matters (e.g., many rotation groups in 3D)

Discrete: separated elements (e.g., permutations)

Continuous: smoothly varying symmetries (e.g., rotations by any angle)

Identity: 0

Inverse: -n

Models translation symmetry along a line in integer steps

Clock arithmetic

Models rotational symmetry by n equal steps

Scaling symmetries (dilations) on a line

Elements are permutations of n objects

Operation: composition of permutations

Captures symmetries of labeled configurations and combinatorial structures

Symmetries: rotations of a regular n-gon

Structure: generated by one rotation

Symmetries: rotations + reflections of a regular n-gon

Non-abelian for n ≥ 3

Presentation idea

SO(2): all planar rotations (continuous)

SO(3): all 3D rotations (continuous, non-abelian)

Translations, rotations, reflections, glide reflections

Wallpaper groups: 17 types of planar repeating pattern symmetries

A group G acts on a set X by transformations of X

Connects abstract group elements to concrete symmetries

Orbit of x: all points reachable by applying group elements

Stabilizer of x: symmetries that keep x fixed

Orbit–stabilizer principle (finite case)

Two groups are isomorphic if they are structurally identical

Means two different-looking symmetry systems can be the same group

Many groups described by a small set of basic symmetries (generators)

Relations encode constraints (e.g., rotate n times gives identity)

Smaller symmetry sets within a symmetry system

Example: rotations form a subgroup of dihedral symmetries

Normal subgroup: compatible with forming symmetry modulo symmetry

Quotient group: compresses a group by identifying elements differing by a normal subgroup

Useful for simplifying complex symmetry systems

Structure-preserving maps between groups

Kernel measures symmetries that become invisible under a map

Elements related by g a g^{-1} represent the same type of symmetry in different positions

Conjugacy classes often correspond to symmetry types (e.g., reflections)

The order of a subgroup divides the order of the group

Strong constraint on possible symmetry counts

Finite abelian groups decompose into products of cyclic groups

Simple groups: no nontrivial normal subgroups (building blocks)

Classification of finite simple groups (major achievement)

Conservation laws from symmetries (via group/Lie symmetry ideas)

Particle physics uses groups like SU(2), SU(3)

Crystallography uses discrete symmetry groups

Molecular symmetry groups predict spectra and bonding patterns

Permutation groups in algorithms and complexity

Error-correcting codes and cryptography use algebraic group structures

Rotation/rigid motion groups SO(3), SE(3) for orientation and motion

Number theory (Galois groups), geometry, topology, combinatorics

Must satisfy the four axioms

Includes invariance of equations, operations, and abstract structures

Many important symmetry systems are non-abelian (order matters)