Suppose you hold a square card and are asked: *in how many ways can you put it back so that it occupies exactly the same region of the table?* You can rotate it by $0°$, $90°$, $180°$, or $270°$; you can also flip it about any of its four axes of symmetry. These eight operations can be composed (do one, then another) and undone (every flip and rotation has a reverse). This simple observation — that the symmetries of an object form a system closed under composition and inversion — is the starting point of **group theory**.

The power of the subject lies in abstraction: the same algebraic structure that describes the symmetries of a square also describes clock arithmetic modulo $n$, the permutations of a deck of cards, the invertible matrices acting on a vector space, and the conformal automorphisms of the Riemann sphere. By isolating the common axioms — closure, associativity, identity, inverses — we can prove theorems once and apply them everywhere.

These notes develop the theory of [groups](/page/Group) from first principles, assuming only familiarity with basic set theory, [functions](/page/Function), and elementary linear algebra. The eight chapters trace a path from the axioms to substantial structural results: basic definitions and examples; the dihedral and symmetric groups; cosets and Lagrange's theorem; normal subgroups, quotient groups, and the First Isomorphism Theorem; direct products and the classification of small groups; group actions, including Cayley's theorem, Cauchy's theorem, and the simplicity of $A_5$; matrix groups and the $\mathrm{SO}_3$ rotation theorem; and Möbius groups on the extended complex plane.

Several ideas recur throughout and are worth flagging at the outset. The **homomorphism philosophy**: to understand a group $G$, study its homomorphisms — the kernel tells you what information is lost, the image tells you where $G$ can be "seen," and the [First Isomorphism Theorem](/theorems/791) makes this precise. **Lagrange's constraint**: the order of every subgroup (and every element) divides $|G|$, a simple divisibility condition that, combined with its partial converses (Cauchy, Sylow), drives much of finite group theory. **Actions reveal structure**: watching a group act on a set — its own elements, its cosets, or an external geometric object — exposes internal structure invisible from the axioms alone.

# Basic Definitions and Examples

Throughout mathematics, we encounter [sets](/page/Set) equipped with operations: integers under addition, matrices under multiplication, symmetries of a geometric figure under composition. A natural question arises: *what common structure do these systems share, and what can we deduce from that structure alone?* The answer is the concept of a **group** — an algebraic structure that captures the essence of "reversible combination." The axioms are minimal (closure, associativity, identity, inverses), yet the consequences are remarkably deep: from the solvability of polynomial equations to the classification of crystal symmetries.

This chapter builds the foundational language. We begin by isolating the axioms, then immediately confront two questions that the axioms raise but do not obviously answer: *Is the identity unique? Are inverses unique?* After resolving these, we introduce the key structural tools — subgroups, homomorphisms, and cyclic groups — each motivated by a concrete problem that the preceding theory cannot yet solve.

## Groups and Their Axioms

Before we can study groups, we need a precise notion of "combining two elements." The familiar operations $+$ on $\mathbb{Z}$ and $\times$ on $\mathbb{R}$ share an important feature: given any two inputs from the set, they produce a single, unambiguous output in the same set. We isolate this as a definition.

[definition:Binary Operation] A **binary operation** $*$ on a set $X$ is a function $* : X \times X \to X$, $(x, y) \mapsto x * y$. [/definition]

The requirement that $*$ maps into $X$ (not into some larger set) is the **closure** condition: the operation never "escapes" the set. Subtraction on $\mathbb{N} = \{1, 2, 3, \ldots\}$ fails this condition, since $1 - 2 = -1 \notin \mathbb{N}$.

[definition:Group] A **group** is a pair $(G, *)$ where $G$ is a set and $*$ is a binary operation on $G$ satisfying: 1. **Closure:** $x * y \in G$ for all $x, y \in G$. 2. **Identity:** There exists $e \in G$ with $x * e = x = e * x$ for all $x \in G$. 3. **Inverses:** For each $x \in G$, there exists $y \in G$ with $x * y = e = y * x$. 4. **Associativity:** $(a * b) * c = a * (b * c)$ for all $a, b, c \in G$. [/definition]

The axioms deliberately say nothing about commutativity — the equation $x * y = y * x$ may or may not hold. When it does, the group is called **abelian**.

[definition:Abelian Group] A group $G$ is **abelian** (or commutative) if $xy = yx$ for all $x, y \in G$. [/definition]

[example:Groups And Non Groups] We verify the axioms in several settings, highlighting what goes wrong when they fail. 1. $(\mathbb{Z}, +)$ is a group: $e = 0$, and $x^{-1} = -x$. It is abelian since $a + b = b + a$. 2. $(\mathbb{Q}, +)$ is a group, as is $(\mathbb{R}, +)$ and $(\mathbb{C}, +)$. 3. $(\mathbb{Z}, -)$ is **not** a group: associativity fails, since $(1 - 2) - 3 = -4$ but $1 - (2 - 3) = 2$. 4. $(\mathbb{Z}, \times)$ is **not** a group: the integer $2$ has no multiplicative inverse in $\mathbb{Z}$ (there is no $n \in \mathbb{Z}$ with $2n = 1$). The set $\mathbb{Q} \setminus \{0\}$ under multiplication *is* a group, but $\mathbb{Z}$ is too small. 5. $(\{+1, -1\}, \times)$ is a group: $e = 1$, and each element is its own inverse. The Cayley table is: | $\times$ | $+1$ | $-1$ | |---|---|---| | $+1$ | $+1$ | $-1$ | | $-1$ | $-1$ | $+1$ | This is the smallest non-trivial group, isomorphic to $C_2$. [/example]

[example:Modular Arithmetic Group] The set $\{0, 1, 2\}$ under addition modulo $3$ is a group with $e = 0$. The Cayley table is: | $+_3$ | $0$ | $1$ | $2$ | |---|---|---|---| | $0$ | $0$ | $1$ | $2$ | | $1$ | $1$ | $2$ | $0$ | | $2$ | $2$ | $0$ | $1$ | Every row and column contains each element exactly once — this is a general feature of group Cayley tables (it follows from the cancellation law: if $ga = gb$ then $a = b$, so the map $x \mapsto gx$ is a bijection). This group is cyclic: $1 + 1 = 2$ and $1 + 1 + 1 = 0$, so $\langle 1 \rangle = \{0, 1, 2\}$. [/example]

[example:Symmetries of the Equilateral Triangle] Label the vertices of an equilateral triangle as $1, 2, 3$. The symmetries are: $e$ (identity), $r$ (rotation by $120°$, $1 \to 2 \to 3 \to 1$), $r^2$ (rotation by $240°$, $1 \to 3 \to 2 \to 1$), $s_1$ (reflection fixing $1$, swapping $2 \leftrightarrow 3$), $s_2$ (reflection fixing $2$, swapping $1 \leftrightarrow 3$), $s_3$ (reflection fixing $3$, swapping $1 \leftrightarrow 2$). To compute $r \circ s_1$: first $s_1$ sends $1 \to 1, 2 \to 3, 3 \to 2$; then $r$ sends $1 \to 2, 3 \to 1, 2 \to 3$. Composing: $1 \to 2$, $2 \to 1$, $3 \to 3$. This fixes $3$ and swaps $1 \leftrightarrow 2$, so $r \circ s_1 = s_3$. The full Cayley table is: | $\circ$ | $e$ | $r$ | $r^2$ | $s_1$ | $s_2$ | $s_3$ | |---|---|---|---|---|---|---| | $e$ | $e$ | $r$ | $r^2$ | $s_1$ | $s_2$ | $s_3$ | | $r$ | $r$ | $r^2$ | $e$ | $s_3$ | $s_1$ | $s_2$ | | $r^2$ | $r^2$ | $e$ | $r$ | $s_2$ | $s_3$ | $s_1$ | | $s_1$ | $s_1$ | $s_2$ | $s_3$ | $e$ | $r$ | $r^2$ | | $s_2$ | $s_2$ | $s_3$ | $s_1$ | $r^2$ | $e$ | $r$ | | $s_3$ | $s_3$ | $s_1$ | $s_2$ | $r$ | $r^2$ | $e$ | This group is $D_6 \cong S_3$, the smallest non-abelian group (note $r \circ s_1 = s_3 \neq s_2 = s_1 \circ r$). [/example]

The axioms demand that an identity exists and that each element has an inverse, but they do *not* assert uniqueness of either. Could a group have two different identity elements? If so, the notation $e$ and $g^{-1}$ would be ambiguous.

[quotetheorem:764]

The proof is short but structurally important: the uniqueness of the identity uses both the left- and right-identity properties simultaneously, while the uniqueness of inverses relies on associativity to "sandwich" one candidate against another. Without associativity, the proof breaks down — in non-associative structures (like quasigroups), inverses can fail to be unique.