Let $S$ be a non-empty set. A binary operation on $S$ is a function

that assigns to each ordered pair $(a, b)$ of elements of $S$ a unique element $a \star b$ of $S$.

The following are binary operations:

1. Addition on $\mathbb{Z}$: for any $a, b \in \mathbb{Z}$, we have $a + b \in \mathbb{Z}$
2. Multiplication on $\mathbb{R}$: for any $a, b \in \mathbb{R}$, we have $a \cdot b \in \mathbb{R}$
3. Composition of functions: if $f, g : X \to X$, then $f \circ g : X \to X$

The following fail to be binary operations:

1. Division on $\mathbb{R}$: $a / b$ is undefined when $b = 0$
2. Subtraction on $\mathbb{N}$: $2 - 5 = -3 \notin \mathbb{N}$ (closure fails)

Let $\star$ be a binary operation on a set $S$. We say that $\star$ is:

1. Associative if $(a \star b) \star c = a \star (b \star c)$ for all $a, b, c \in S$
2. Commutative if $a \star b = b \star a$ for all $a, b \in S$

Associativity allows us to write $a \star b \star c$ without ambiguity. Commutativity, while a pleasant property, is not required for most of our theory—and indeed, some of the most important groups are non-commutative.

A group is an ordered pair $(G, \star)$ where $G$ is a non-empty set and $\star$ is a binary operation on $G$ satisfying the following axioms:

(G1) Associativity. For all $a, b, c \in G$:

(G2) Identity. There exists an element $e \in G$ such that for all $a \in G$:

(G3) Inverses. For each $a \in G$, there exists an element $a^{-1} \in G$ such that:

Several observations are in order:

1. The axioms are listed in order of decreasing generality. Many algebraic structures satisfy (G1) alone (semigroups) or (G1) and (G2) (monoids). Groups require all three.

2. Closure is implicit in the statement that $\star$ is a binary operation on $G$.

3. The identity axiom requires $e$ to be a two-sided identity: it must work on both the left and the right. Similarly, the inverse axiom requires two-sided inverses.

A group $(G, \star)$ is called abelian (or commutative) if for all $a, b \in G$:

The term honors Niels Henrik Abel (1802–1829), who made fundamental contributions to the theory of equations.

The order of a group $G$, denoted $|G|$, is the cardinality of the underlying set. If $|G|$ is finite, we say $G$ is a finite group; otherwise, $G$ is an infinite group.

In any group $(G, \star)$, the identity element is unique.

In any group $(G, \star)$, each element has a unique inverse.

Let $(G, \star)$ be a group and let $a, b, c \in G$. Then:

1. Left cancellation: If $a \star b = a \star c$, then $b = c$
2. Right cancellation: If $b \star a = c \star a$, then $b = c$

Let $(G, \star)$ be a group and let $a, b \in G$. Then:

The formula $(ab)^{-1} = b^{-1}a^{-1}$ is sometimes called the "socks and shoes" rule: to undo putting on socks then shoes, you must first remove shoes, then socks. The order reverses.

For any element $a$ in a group $G$: $(a^{-1})^{-1} = a$.

The set $\mathbb{Z}$ of integers forms a group under addition, denoted $(\mathbb{Z}, +)$:

• Closure: The sum of two integers is an integer
• Associativity: $(a + b) + c = a + (b + c)$ for all $a, b, c \in \mathbb{Z}$
• Identity: The integer $0$ satisfies $0 + a = a + 0 = a$
• Inverses: For each $a \in \mathbb{Z}$, the integer $-a$ satisfies $a + (-a) = 0$

This group is abelian and infinite.

Similarly, $(\mathbb{Q}, +)$, $(\mathbb{R}, +)$, and $(\mathbb{C}, +)$ are all abelian groups under addition.

The set $\mathbb{Q}^* = \mathbb{Q} \setminus \{0\}$ forms a group under multiplication:

• Identity: The number $1$ satisfies $1 \cdot a = a \cdot 1 = a$
• Inverses: For each $a \in \mathbb{Q}^*$, the number $1/a$ satisfies $a \cdot (1/a) = 1$

Note that we must exclude $0$ because it has no multiplicative inverse.

The set $\mathbb{R}^+ = \{x \in \mathbb{R} : x > 0\}$ forms an abelian group under multiplication.

For any positive integer $n$, the set

forms a group under addition modulo $n$:

• Identity: $0$
• Inverse of $a$: $n - a$ (when $a \neq 0$)

This is a finite abelian group of order $n$.

The set

forms a group under multiplication modulo $n$. The order of this group is $\phi(n)$, Euler's totient function.

The general linear group $\text{GL}_n(\mathbb{R})$ consists of all invertible $n \times n$ matrices with real entries, under matrix multiplication:

This is a non-abelian group for $n \geq 2$.

The special linear group

is a subgroup of $\text{GL}_n(\mathbb{R})$.

The symmetric group $S_n$ consists of all bijections from $\{1, 2, \ldots, n\}$ to itself, under composition of functions. These bijections are called permutations.

The order of $S_n$ is $n!$ (factorial). For $n \geq 3$, the group $S_n$ is non-abelian.

The dihedral group $D_n$ is the group of symmetries of a regular $n$-gon, including both rotations and reflections. It has order $2n$ and is non-abelian for $n \geq 3$.

Let $(G, \star)$ be a group. A subset $H \subseteq G$ is a subgroup of $G$, written $H \leq G$, if $(H, \star)$ is itself a group under the same operation.

Let $G$ be a group and let $H$ be a non-empty subset of $G$. Then $H$ is a subgroup of $G$ if and only if:

1. For all $a, b \in H$, we have $ab \in H$ (closure)
2. For all $a \in H$, we have $a^{-1} \in H$ (closure under inverses)

Let $G$ be a group and let $H$ be a non-empty subset of $G$. Then $H \leq G$ if and only if for all $a, b \in H$, we have $ab^{-1} \in H$.

Every group $G$ has at least two subgroups:

1. The trivial subgroup $\{e\}$
2. The improper subgroup $G$ itself

A subgroup $H$ with $\{e\} \subsetneq H \subsetneq G$ is called a proper non-trivial subgroup.

The subgroups of $(\mathbb{Z}, +)$ are precisely the sets $n\mathbb{Z} = \{nk : k \in \mathbb{Z}\}$ for $n \geq 0$. Thus:

Let $(G, \star)$ and $(H, \cdot)$ be groups. A function $\phi : G \to H$ is a homomorphism if for all $a, b \in G$:

Let $\phi : G \to H$ be a group homomorphism. Then:

1. $\phi(e_G) = e_H$
2. $\phi(a^{-1}) = \phi(a)^{-1}$ for all $a \in G$
3. $\phi(a^n) = \phi(a)^n$ for all $a \in G$ and $n \in \mathbb{Z}$

Let $\phi : G \to H$ be a homomorphism.

The kernel of $\phi$ is:

The image of $\phi$ is:

Let $\phi : G \to H$ be a homomorphism. Then:

1. $\ker(\phi)$ is a subgroup of $G$
2. $\text{im}(\phi)$ is a subgroup of $H$

A homomorphism $\phi : G \to H$ is called:

• A monomorphism (or injection) if $\phi$ is injective
• An epimorphism (or surjection) if $\phi$ is surjective
• An isomorphism if $\phi$ is bijective
• An automorphism if $\phi : G \to G$ is an isomorphism

A homomorphism $\phi : G \to H$ is injective if and only if $\ker(\phi) = \{e_G\}$.

Two groups $G$ and $H$ are isomorphic, written $G \cong H$, if there exists an isomorphism $\phi : G \to H$.

A group $G$ is cyclic if there exists an element $g \in G$ such that every element of $G$ can be written as a power of $g$:

The element $g$ is called a generator of $G$.

The group $(\mathbb{Z}, +)$ is cyclic with generator $1$ (or $-1$). Every integer is a "power" of $1$ under addition: $n = 1 + 1 + \cdots + 1$ ($n$ times).

The group $\mathbb{Z}/n\mathbb{Z}$ is cyclic with generator $1$. It is the unique cyclic group of order $n$, up to isomorphism.

Every cyclic group is isomorphic to either $\mathbb{Z}$ (if infinite) or $\mathbb{Z}/n\mathbb{Z}$ for some positive integer $n$ (if finite of order $n$).

Every subgroup of a cyclic group is cyclic.

Let $G = \langle g \rangle$ be a cyclic group of order $n$. Then:

1. For each divisor $d$ of $n$, there is exactly one subgroup of order $d$
2. This subgroup is $\langle g^{n/d} \rangle$