Group Axioms and First Examples

Arithmetic on the integers, the symmetries of a triangle, and the invertible matrices over a field have elements of entirely different kinds, yet they share a common shape: one operation that combines two elements into a third, an element that changes nothing, and for every element something that undoes it. Group theory studies that shape in isolation. Reducing the axioms to those few facts lets one theorem about groups apply at once to number systems, geometric symmetry, and permutations.

Binary operations

A group's operation is a rule for combining two elements of a set into one.

Two conditions are hidden in the phrase function $G\times G\to G$. The operation must be defined for every ordered pair, and its output must land back inside $G$. The second condition is closure; many candidate operations fail it.¹

• Ordinary $+$ and $\times$ are commutative binary operations on $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$.
• Subtraction is a binary operation on $\mathbb{Z}$ (noncommutative), but not on ${\mathbb{Z}}^{+}$: for $a<b$ with $a,b\in {\mathbb{Z}}^{+}$, $a-b\notin {\mathbb{Z}}^{+}$, so $-$ fails to map ${\mathbb{Z}}^{+}\times {\mathbb{Z}}^{+}$ into ${\mathbb{Z}}^{+}$.
• The cross product on ${\mathbb{R}}^3$ is a binary operation that is neither associative nor commutative.

When $H\subseteq G$ and $a\ast b\in H$ for all $a,b\in H$, the subset $H$ is closed under $\ast$. Associativity and commutativity are inherited for free: if $\ast$ is associative on $G$ and restricts to a binary operation on $H$, it is automatically associative on $H$.

The group axioms

Axiom (G2) forces $G$ to be nonempty, since it asserts $e$ exists. From here we drop the operation symbol: for an abstract group we write $ab$ for $a\ast b$ and denote the identity by $1$. When the operation is addition we write $0$ for the identity and $-a$ for the inverse of $a$.

Everything that follows is a consequence of these three axioms. First, the words the identity and the inverse are justified: both are unique.

Cancellation

Order

Two different quantities carry the name order, and they are related but not the same.

The notation $x^n$ abbreviates the $n$-fold product $xx\cdots x$; by the generalized associative law no bracketing is needed. Set $x^0=1$ and $x^{-n}=(x^{-1}{)}^n$. In an additive group $x^n$ becomes $nx$. The clash of symbols with set cardinality is deliberate: the order of $x$ turns out to equal the number of distinct powers of $x$, so the two meanings of order agree once we reach cyclic groups.

• An element has order $1$ exactly when it is the identity.
• In $(\mathbb{Z},+)$ every nonzero element has infinite order.
• In $(\mathbb{R}\setminus \{0\},\times )$ the element $-1$ has order $2$; every other nonidentity element has infinite order.
• In the additive group $\mathbb{Z}/9\mathbb{Z}$ the class $\bar{3}$ has order $3$, since $\bar{3}+\bar{3}=\bar{6}\ne \bar{0}$ but $\bar{3}+\bar{3}+\bar{3}=\bar{9}=\bar{0}$.
• In the multiplicative group $(\mathbb{Z}/7\mathbb{Z}{)}^{\times }$ the class $\bar{2}$ has order $3$ (its powers are $\bar{2},\bar{4},\bar{8}=\bar{1}$), and $\bar{3}$ has order $6$.

The running examples

Three groups built from the integers recur as test cases throughout.⁴

• $(\mathbb{Z},+)$ and its cousins. $\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$ are groups under addition with $e=0$ and inverse $-a$. Under multiplication, $\mathbb{Q}\setminus \{0\},\mathbb{R}\setminus \{0\},\mathbb{C}\setminus \{0\}$ are groups with $e=1$. But $\mathbb{Z}\setminus \{0\}$ is not a group under $\times$: the element $2$ has no integer inverse.
• $\mathbb{Z}/n\mathbb{Z}$ under addition. The residue classes mod $n$ form an abelian group of order $n$ under addition of classes, with identity $\bar{0}$ and inverse $-\bar{a}$. The construction of these classes is the content of the integers and modular arithmetic.
• $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ under multiplication. The classes $\bar{a}$ that have a multiplicative inverse mod $n$ form an abelian group under multiplication, with identity $\bar{1}$. The invertible classes are those $\bar{a}$ with $\gcd (a,n)=1$; the group is the units mod $n$.

The superscript $\times$ always signals that the operation is multiplication, so there is no confusion between $\mathbb{Z}/n\mathbb{Z}$ (all $n$ classes, added) and $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$ (the units, multiplied), even though the latter is a subset of the former.

Direct products

New groups assemble from old ones componentwise.

Each axiom holds in $A\times B$ because it holds in each coordinate separately. The identity is $(1_A,1_B)$ and $(a,b{)}^{-1}=(a^{-1},b^{-1})$. Taking $A=B=\mathbb{R}$ recovers the Euclidean plane $\mathbb{R}\times \mathbb{R}$ under vector addition. The order of $(a,b)$ is the least common multiple of $|a|$ and $|b|$, a fact we take up again in the study of direct products.

Reading a group from its table

For a finite group the entire operation is a square array.

Cancellation has a visible consequence in the table: no element repeats within a row or within a column, so every row and every column is a permutation of $G$ — a Latin square. The identity is the element whose row and column reproduce the header, and $x^{-1}$ is found by locating $1$ in the row of $x$.

The table is complete information about the group, but a poor way to see structure: it grows as $|G{|}^2$ and reveals nothing at a glance. Generators, subgroups, and symmetries expose the structure a table only encodes.

Symmetry as a group

The examples so far are arithmetic. The other source of groups is geometry: the symmetries of a figure, meaning the rigid motions that carry the figure onto itself, form a group under composition.

For the equilateral triangle the symmetries are three rotations (by $0$, $120$, and $240$ degrees) and three reflections across the medians. Composition of two symmetries is another symmetry, the identity motion leaves everything fixed, and every motion reverses, so the six form a group.

This group of six is the smallest nonabelian group, and it reappears under three names: as the dihedral group$D_6$ of the triangle, as the symmetric group $S_3$ of permutations of three labels, and as a matrix group. That three descriptions coincide is a first illustration of the abstract viewpoint, and it is the content of isomorphism: groups that look different are the same group when their tables match after relabeling.

Role of each axiom

Each axiom plays a distinct role.

Dropping any one axiom leaves a weaker structure (a semigroup, a monoid) with a much coarser theory.