Group Axioms and First Examples

A group is the mathematical distillation of symmetry: a set of transformations that can be composed and undone. This page fixes the axioms, the basic vocabulary (order, abelian, generators), and a standard menagerie of examples that recur throughout the folder and the physics tree. It is the root of the Group Theory section; everything else builds on it.

The group axioms

A group $(G, \cdot)$ is a set $G$ together with a binary operation $\cdot : G \times G \to G$ satisfying:

(G1) Closure: $g_{1}, g_{2} \in G \Rightarrow g_{1} \cdot g_{2} \in G$ .
(G2) Associativity: $(g_{1} \cdot g_{2}) \cdot g_{3} = g_{1} \cdot (g_{2} \cdot g_{3})$ for all $g_{1}, g_{2}, g_{3} \in G$ .
(G3) Identity: there exists $e \in G$ such that $e \cdot g = g \cdot e = g$ for all $g \in G$ .
(G4) Inverses: for every $g \in G$ there exists $g^{- 1} \in G$ such that $g \cdot g^{- 1} = g^{- 1} \cdot g = e$ .

Closure is often folded into the statement " $\cdot$ is a binary operation on $G$ ". We write $g h$ for $g \cdot h$ and $g^{n}$ for the $n$ -fold product when no confusion arises.

If additionally $g_{1} g_{2} = g_{2} g_{1}$ for all elements, $G$ is abelian (commutative); the operation is then often written additively, $+$ , with identity $0$ and inverse $- g$ .

Immediate consequences (each provable from the axioms alone):

Uniqueness of the identity. If $e, e^{'}$ both satisfy (G3), then $e = e e^{'} = e^{'}$ .
Uniqueness of inverses. If $h, h^{'}$ both invert $g$ , then $h = h (g h^{'}) = (h g) h^{'} = h^{'}$ .
Cancellation. $g a = g b \Rightarrow a = b$ (multiply by $g^{- 1}$ ); likewise on the right.
Socks–shoes. $(g h)^{- 1} = h^{- 1} g^{- 1}$ — the inverse of a product reverses the order.
Left-multiplication is a bijection. For fixed $g$ , the map $x \mapsto gx$ permutes $G$ . (This is the seed of Cayley's theorem in group-actions.md.)

Order

The order of the group $∣ G ∣$ is its cardinality (number of elements). A group is finite or infinite accordingly.
The order of an element $g$ , written $ord (g)$ or $∣ g ∣$ , is the least $n \geq 1$ with $g^{n} = e$ (or $\infty$ if no such $n$ exists). It equals the size of the cyclic subgroup $⟨ g ⟩ = {e, g, g^{2}, \dots}$ that $g$ generates.

In a finite group every element has finite order, and (by Lagrange's theorem) $ord (g)$ divides $∣ G ∣$ .

Generators

A subset $S \subseteq G$ generates $G$ if every element is a finite product of elements of $S$ and their inverses; we write $G = ⟨ S ⟩$ . A group is cyclic if it is generated by a single element, and finitely generated if $S$ can be chosen finite. The systematic study of "generators and relations" is presentations.md.

A standard menagerie

These examples are referenced throughout the folder. The right-hand column notes where each reappears.

Group	Description	Order	Abelian?	Appears in
$Z$	integers under $+$	$\infty$	yes	cyclic groups
$Z_{n}$	integers mod $n$ under $+$	$n$	yes	cyclic-abelian
$Z_{n}^{\times}$	units mod $n$ under $\times$	$φ (n)$	yes	number theory
$S_{n}$	permutations of $n$ symbols	$n!$	no ( $n \geq 3$ )	permutation-groups
$A_{n}$	even permutations	$n! /2$	no ( $n \geq 4$ )	simple-groups
$D_{n}$	symmetries of a regular $n$ -gon	$2 n$	no ( $n \geq 3$ )	products
$Q_{8}$	quaternion group ${\pm 1, \pm i, \pm j, \pm k}$	$8$	no	character-theory
$V_{4}$	Klein four-group $Z_{2} \times Z_{2}$	$4$	yes	products
$G L (n, F)$	invertible $n \times n$ matrices	$\infty$	no ( $n \geq 2$ )	matrix-groups
$S U (2)$	$2 \times 2$ unitary, $det = 1$	$\infty$	no	examples-classical

Notes on the small cases.

$Z_{n} ≅ ⟨ r ∣ r^{n} = e ⟩$ is the prototypical cyclic group; the $n$ -th roots of unity under multiplication give the same group.
$S_{3}$ (order $6$ ) is the smallest non-abelian group; it is also the dihedral group $D_{3}$ (symmetries of an equilateral triangle).
$Q_{8}$ and $D_{4}$ are the two non-abelian groups of order $8$ ; they have the same order but are not isomorphic (their character tables are the same, but their element orders differ — $Q_{8}$ has a single element of order $2$ , $D_{4}$ has several).
$V_{4}$ and $Z_{4}$ are the two groups of order $4$ , both abelian; $V_{4}$ has every non-identity element of order $2$ , $Z_{4}$ has an element of order $4$ . This distinction reappears in the split/non-split extension discussion in products.md.

Group tables (Cayley tables)

For a small finite group the operation is fully specified by its Cayley table, an $∣ G ∣ \times ∣ G ∣$ grid with entry $g h$ in row $g$ , column $h$ . The cancellation law forces every row and every column to be a permutation of $G$ — a Latin square. Not every Latin square is a group table (associativity is an extra constraint), but the Latin-square property is a quick sanity check.

Example — $Z_{3} = {0, 1, 2}$ under addition mod $3$ :

$+$	0	1	2
0	0	1	2
1	1	2	0
2	2	0	1

Where this goes next

The axioms are inert until we ask about substructure and maps:

Substructure — subgroups, cosets, and the divisibility constraint of Lagrange's theorem.
Maps — homomorphisms and the isomorphism theorems, which say that quotients and images are two views of the same data.
Symmetry in action — group actions, where a group finally does what it was invented for: permute a set.

References

Dummit & Foote, Abstract Algebra, Ch. 1 — the standard first exposure.
Artin, Algebra, Ch. 2 — a more geometric route in.
Herstein, Topics in Algebra — terse and classic.

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