Groups – Symmetry as the Language of Mathematics¶

Summary

What do rotations of a square, addition of integers, and the symmetries of equations have in common? They are all groups – the most fundamental algebraic structure of modern mathematics.

Prerequisites¶

None – this article is the entry point to abstract algebra.

1. Symmetry in Everyday Life¶

A square can be rotated (by 90°, 180°, 270°) and reflected (across four axes) – that makes a total of eight symmetries. Each individual symmetry maps the square onto itself, and two symmetries performed in succession yield another symmetry.

This observation can be generalised: wherever there are symmetries – in geometry, in physics, in number theory – there is a mathematical structure behind them that always obeys the same rules. This structure is called a group.

Some examples that at first glance seem to have nothing in common:

The rotations and reflections of a regular \(n\)-gon
Addition of integers: \(3 + 5 = 8\), \(7 + (-7) = 0\)
The permutations of a set: rearrangements of \(\{1, 2, 3\}\)
The symmetries of the roots of a polynomial (→ Galois theory)

All these examples satisfy the same four rules.

2. The Group Axioms¶

A group is a pair \((G, \cdot)\) consisting of a set \(G\) and an operation \(\cdot : G \times G \to G\) satisfying four axioms:

(G1) Closure. For all \(a, b \in G\), \(a \cdot b \in G\).

(G2) Associativity. For all \(a, b, c \in G\), \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).

(G3) Identity element. There exists an element \(e \in G\) with \(e \cdot a = a \cdot e = a\) for all \(a \in G\).

(G4) Inverses. For every \(a \in G\) there exists an element \(a^{-1} \in G\) with \(a \cdot a^{-1} = a^{-1} \cdot a = e\).

If additionally \(a \cdot b = b \cdot a\) for all \(a, b \in G\), the group is called abelian (or commutative) – named after Niels Henrik Abel.

Notation

For abelian groups, the operation is often written as \(+\) instead of \(\cdot\) and the identity element as \(0\) instead of \(e\). The inverse of \(a\) is then \(-a\).

3. First Examples¶

The integers \((\mathbb{Z}, +)\)¶

The simplest infinite group: the integers under addition.

Closed: \(a + b \in \mathbb{Z}\) for all \(a, b \in \mathbb{Z}\) ✓
Associative: \((a + b) + c = a + (b + c)\) ✓
Identity element: \(0\) (since \(a + 0 = a\)) ✓
Inverses: \(-a\) (since \(a + (-a) = 0\)) ✓
Abelian: \(a + b = b + a\) ✓

Residue classes \((\mathbb{Z}/n\mathbb{Z}, +)\)¶

For \(n \geq 1\), the residue classes modulo \(n\) form a finite abelian group. For example, \(\mathbb{Z}/4\mathbb{Z} = \{0, 1, 2, 3\}\) with addition modulo \(4\):

\(+\)	\(0\)	\(1\)	\(2\)	\(3\)
\(0\)	\(0\)	\(1\)	\(2\)	\(3\)
\(1\)	\(1\)	\(2\)	\(3\)	\(0\)
\(2\)	\(2\)	\(3\)	\(0\)	\(1\)
\(3\)	\(3\)	\(0\)	\(1\)	\(2\)

This group has order \(4\) (four elements). It is cyclic: every element can be written as a multiple of \(1\).

The symmetric group \(S_n\)¶

The symmetric group \(S_n\) consists of all permutations (rearrangements) of the set \(\{1, 2, \ldots, n\}\), with composition as the operation.

\(S_3\) has \(3! = 6\) elements:

\[ \text{id}, \quad (12), \quad (13), \quad (23), \quad (123), \quad (132) \]

Here \((12)\) means "swap \(1\) and \(2\)", and \((123)\) means "send \(1 \to 2 \to 3 \to 1\)".

Caution: \(S_3\) is not abelian! We have \((12) \circ (13) = (132)\), but \((13) \circ (12) = (123)\).

The dihedral group \(D_n\)¶

The symmetry group of a regular \(n\)-gon is called the dihedral group \(D_n\). It has \(2n\) elements: \(n\) rotations and \(n\) reflections. For \(n = 4\) (square), \(|D_4| = 8\).

4. Subgroups and Order¶

A subset \(H \subseteq G\) is called a subgroup of \(G\) if \(H\) itself forms a group under the restricted operation. One writes \(H \leq G\).

Examples: - \(2\mathbb{Z} = \{\ldots, -4, -2, 0, 2, 4, \ldots\} \leq \mathbb{Z}\) (the even integers) - \(\{e, (123), (132)\} \leq S_3\) (the rotations of the triangle)

The order \(|G|\) of a group is the number of its elements. The order \(\text{ord}(a)\) of an element \(a\) is the smallest positive integer \(n\) with \(a^n = e\).

Lagrange's theorem. If \(H \leq G\) with \(|G| < \infty\), then \(|H|\) divides \(|G|\).

Consequence: in a group with \(12\) elements, a subgroup can only have \(1\), \(2\), \(3\), \(4\), \(6\), or \(12\) elements. The order of every element divides \(|G|\).

Lagrange in action

Let \(G\) be a group with \(|G| = p\) (prime). Then \(G\) has no proper subgroups other than \(\{e\}\) and \(G\) itself. Hence \(G\) is cyclic: \(G \cong \mathbb{Z}/p\mathbb{Z}\).

5. Homomorphisms¶

A group homomorphism is a map \(\varphi: G \to H\) between two groups that preserves the structure:

\[ \varphi(a \cdot b) = \varphi(a) \cdot \varphi(b) \quad \text{for all } a, b \in G \]

Homomorphisms transport algebraic relationships from one group to another.

Examples: - \(\varphi: \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}\), \(a \mapsto a \bmod n\) (reduction modulo \(n\)) - \(\det: \text{GL}_n(\mathbb{R}) \to \mathbb{R}^*\), \(A \mapsto \det(A)\) (determinant)

The kernel of a homomorphism \(\varphi: G \to H\) is:

\[ \ker(\varphi) = \{a \in G \mid \varphi(a) = e_H\} \]

The kernel is always a subgroup of \(G\) – and even a special kind of subgroup: a normal subgroup.

6. Normal Subgroups and Quotient Groups¶

A subgroup \(N \leq G\) is called a normal subgroup (written \(N \trianglelefteq G\)) if \(gNg^{-1} = N\) for all \(g \in G\), that is, if \(N\) is invariant under conjugation.

In abelian groups, every subgroup is a normal subgroup (since \(gng^{-1} = n\) for all \(g, n\)).

Normal subgroups are important because they allow the formation of quotient groups (factor groups):

\[ G/N = \{gN \mid g \in G\} \]

The elements of \(G/N\) are the cosets \(gN = \{gn \mid n \in N\}\), and the operation is \((gN)(hN) = (gh)N\).

Example: \(\mathbb{Z}/n\mathbb{Z}\) is precisely the quotient group of \(\mathbb{Z}\) by the normal subgroup \(n\mathbb{Z}\).

The isomorphism theorem. For every homomorphism \(\varphi: G \to H\):

\[ G / \ker(\varphi) \cong \text{Im}(\varphi) \]

That is: the quotient group modulo the kernel is isomorphic to the image. This theorem connects homomorphisms, normal subgroups, and quotient groups into a unified picture.

Simple Groups¶

A group \(G \neq \{e\}\) is called simple if it has no normal subgroups other than \(\{e\}\) and \(G\) itself. Simple groups are the "atoms" of group theory – every finite group can be built from simple groups (Jordan–Hölder theorem).

The classification of all finite simple groups is one of the most monumental results in mathematics: it consists of the cyclic groups \(\mathbb{Z}/p\mathbb{Z}\), the alternating groups \(A_n\) (\(n \geq 5\)), 16 families of "Lie type" groups, and 26 sporadic groups.

7. Why Groups Matter for FLT¶

For Fermat's Last Theorem, groups play a key role through Galois theory: the symmetries of the roots of a polynomial form a group – the Galois group. This group controls the algebraic structure of the associated field extension.

In Wiles' proof, groups appear in several guises:

The absolute Galois group \(G_{\mathbb{Q}} = \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) – the central symmetry object of algebraic number theory. It acts on the division points of elliptic curves.
Matrix groups \(\text{GL}_2(\mathbb{F}_p)\) and \(\text{GL}_2(\mathbb{Z}_p)\) – as target groups of the Galois representations that link elliptic curves with linear algebra.
Hecke algebras – symmetries in the space of modular forms that generate algebraic structures on the Fourier coefficients.

The concept of a group is the common language in which all these objects communicate. Without groups, Wiles' proof would not even be formulable.