Studies algebraic structures (“groups”) that capture symmetry and reversible transformations

Unifies patterns across mathematics, physics, chemistry, computer science, and more

Focuses on how actions/operations combine and invert while preserving structure

Symmetry as “structure-preserving change” (rotations, reflections, translations, permutations, phase shifts)

Symmetry operations form a system (compose, invert, identity)

Group theory formalizes these symmetry systems abstractly

G is a set (elements as “transformations”)

“·” is a binary operation combining two elements

Closure: for all a, b in G, a · b is also in G

Associativity: (a · b) · c = a · (b · c)

Identity element: exists e in G with e · a = a · e = a

Inverses: for each a, exists a⁻¹ with a · a⁻¹ = a⁻¹ · a = e

Multiplicative: a · b, ab

Additive (often abelian): a + b, identity 0, inverse −a

Function composition: (f ∘ g)(x) = f(g(x))

Elements: 4 rotations + 4 reflections

Operation: do one symmetry then another

Key idea: order matters (non-commutative)

3 rotations + 3 reflections

Models rigid motions mapping the triangle onto itself

Elements: {0,1,…,n−1}

Operation: addition mod n

Symmetry viewpoint: repeated rotation by 360°/n

Elements: all reorderings of n objects

Operation: composition of permutations

Models symmetries of labeled sets and combinatorial systems

a · b = b · a

Examples: (Z, +), Z/nZ, vector spaces under addition

a · b ≠ b · a in general

Examples: Dn (n ≥ 3), Sn (n ≥ 3), GL(n)

Finite: D4, Z/nZ, Sn

Infinite: (Z, +), (R, +), continuous symmetry groups

Generated by one element g

Every element is g^k (or k·g)

Elements: matrices under multiplication

GL(n, R): invertible n×n real matrices

O(n): orthogonal matrices (length-preserving rotations/reflections)

SO(n): rotations (determinant 1)

H ⊆ G is a subgroup if it is itself a group under the same operation

Nonempty and closed under products and inverses

2Z is a subgroup of Z

Rotations inside D4 form a subgroup (cyclic of order 4)

⟨S⟩: smallest subgroup containing S

⟨g⟩: cyclic subgroup generated by one element

Left coset: gH = {gh : h in H}

Cosets partition G into equal-size “copies” of H

[G : H] = number of cosets

If G is finite and H ≤ G, then |H| divides |G|

Consequence: element orders divide |G|

Map φ: G → K such that φ(ab) = φ(a)φ(b)

Preserves operation structure

ker(φ): elements mapping to identity; always a normal subgroup

im(φ): outputs; a subgroup of K

Bijective homomorphism

Groups are structurally the same (different labels, same algebra)

Isomorphisms G → G capturing internal symmetries

gN = Ng for all g in G

Equivalent: gNg⁻¹ = N

Elements: cosets of N

Operation: (aN)(bN) = (ab)N

“Factor out” symmetries considered equivalent

Central for classification and decomposition

G acts on X if each g gives a function X → X

Axioms: e·x = x and (ab)·x = a·(b·x)

Orbit(x): points reachable from x

Stabilizer(x): elements fixing x

|Orbit(x)| = |G| / |Stabilizer(x)|

Actions describe how symmetries transform objects (often the most concrete viewpoint)

Translations, rotations, reflections in plane/space

Often infinite (continuous and discrete)

Smooth manifolds with group structure

Examples: SO(3) for 3D rotations, SU(2) in quantum mechanics

Space groups and wallpaper groups classify repeating patterns

Combine translations with rotations/reflections/glide reflections

Number theory: modular arithmetic, Galois theory

Algebra: classification, rings/fields links, representation theory

Geometry/topology: symmetries of shapes and spaces

Physics: conservation laws via symmetry (Noether-type connections)

Chemistry: molecular symmetry, spectroscopy selection rules

Computer science: permutations, coding theory, cryptography, algorithms

Any system of reversible operations with composition likely forms (or approximates) a group

Definitions + examples (Z/nZ, Dn, Sn)

Subgroups and cyclic groups

Cosets + Lagrange’s theorem

Homomorphisms + isomorphisms

Normal subgroups + quotient groups

Group actions (symmetry of sets/objects)

Sylow theorems (finite group structure)

Representation theory (groups as matrices/linear transformations)

Lie groups and Lie algebras (continuous symmetry systems)

Galois theory (symmetries of polynomial roots)