3.6 Groups

Groups are the building blocks of abstract algebra. They give you a precise framework for studying symmetry and structure across many areas of mathematics. A group is just a set paired with an operation that satisfies four specific rules. Once you internalize those rules, you can recognize group structure everywhere, from integer arithmetic to the symmetries of a square.

This topic covers the formal definition, key properties, important types of groups, and how groups relate to each other through subgroups, homomorphisms, and quotient groups.

Definition of groups

A group is a set $G$ together with a binary operation $*$ that satisfies four axioms. These axioms are what separate a group from just "a set with some operation." Every time you want to check whether something is a group, you run through these four requirements.

Group axioms

1. Closure: For any two elements $a, b \in G$, the result $a * b$ is also in $G$. The operation never kicks you out of the set.
2. Associativity: For all $a, b, c \in G$, $(a * b) * c = a * (b * c)$. You can regroup operations without changing the result.
3. Identity element: There exists a unique element $e \in G$ such that $a * e = e * a = a$ for every $a \in G$. This element does nothing when you combine it with anything else.
4. Inverse elements: For every $a \in G$, there exists some $b \in G$ such that $a * b = b * a = e$. Every element has a partner that "undoes" it.

All four must hold. If even one fails, you don't have a group.

Examples of groups

• Integer addition $(\\mathbb{Z}, +)$: The set of all integers with addition. The identity is 0, and the inverse of any integer $n$ is $-n$.
• Non-zero reals under multiplication $(\\mathbb{R}^*, \\times)$: All real numbers except zero, with multiplication. The identity is 1, and the inverse of $a$ is $\\frac{1}{a}$. Zero is excluded precisely because it has no multiplicative inverse.
• Symmetry groups: The symmetries of a square (4 rotations and 4 reflections) form a group called the dihedral group $D_4$, with 8 elements.
• Matrix groups: The general linear group $GL(n, \\mathbb{R})$ consists of all invertible $n \\times n$ matrices under matrix multiplication.

Non-examples of groups

These are just as important for building intuition. In each case, identify which axiom fails:

• Natural numbers under addition $(\\mathbb{N}, +)$: No inverse elements. There's no natural number you can add to 5 to get 0.
• Even integers under multiplication: Closure fails. $2 \\times 4 = 8$ is even, but there's no identity element (1 is odd, so it's not in the set).
• Rationals under division: Multiple failures. Division isn't associative, there's no consistent identity element, and division by zero is undefined, so it's not even a well-defined binary operation on all of $\\mathbb{Q}$.
• All 2×2 matrices under multiplication: Not every matrix has an inverse (singular matrices with determinant 0 don't), so the inverse axiom fails.

Group properties

Closure property

Closure means the operation never produces a result outside the set. Formally: $\\forall \\, a, b \\in G, \\; a * b \\in G$. This keeps the group self-contained. Without closure, you couldn't reliably combine elements and stay within the same structure.

Associativity property

Associativity says $(a * b) * c = a * (b * c)$ for all $a, b, c \\in G$. This lets you write $a * b * c$ without ambiguity. Note that associativity is not the same as commutativity. The order in which you group operations doesn't matter, but the order of the elements themselves might.

Identity element

The identity $e$ is the unique element satisfying $a * e = e * a = a$ for all $a$. It's unique because if you had two identity elements $e$ and $e'$, then $e = e * e' = e'$, so they'd have to be the same. Common examples: 0 for addition, 1 for multiplication, the identity matrix for matrix multiplication.

Inverse elements

For each $a \\in G$, there's a unique element (written $a^{-1}$ in multiplicative notation or $-a$ in additive notation) such that $a * a^{-1} = a^{-1} * a = e$. Inverses are what let you "solve equations" inside a group. If $a * x = b$, you can multiply both sides by $a^{-1}$ to get $x = a^{-1} * b$.

Types of groups

Finite vs infinite groups

A finite group has a finite number of elements. The number of elements is called the order of the group, written $|G|$. For example, the symmetry group of a square has order 8.

An infinite group has infinitely many elements. These can be countably infinite (like $\\mathbb{Z}$ under addition) or uncountably infinite (like $\\mathbb{R}^*$ under multiplication). Finite and infinite groups often require different proof techniques.

Abelian vs non-abelian groups

An abelian group (named after Niels Henrik Abel) satisfies commutativity: $a * b = b * a$ for all elements. Integer addition and real number multiplication are both abelian. Many theorems become simpler in the abelian case.

A non-abelian group has at least one pair of elements where $a * b \\neq b * a$. Matrix multiplication groups are a classic example: in general, $AB \\neq BA$. Symmetry groups of most geometric objects (like $D_4$) are also non-abelian.

Cyclic groups

A cyclic group is generated by a single element $g$, meaning every element in the group can be written as $g^n$ (or $ng$ in additive notation) for some integer $n$. The group of integers modulo $n$ under addition, written $\\mathbb{Z}_n$, is a finite cyclic group of order $n$. The integers $\\mathbb{Z}$ under addition form an infinite cyclic group generated by 1 (or by $-1$).

Cyclic groups are always abelian, and they're the simplest type of group to work with.

Symmetric groups

The symmetric group $S_n$ is the group of all permutations of $n$ elements. It has $n!$ elements. For instance, $S_3$ has $3! = 6$ elements. Symmetric groups are non-abelian for $n > 2$ and play a central role in the study of permutations and combinatorics.

Group operations

Binary operations

A binary operation on a set $S$ is a function that takes two elements of $S$ and returns one element of $S$. It must be well-defined, meaning each pair of inputs gives exactly one output. Addition, multiplication, and function composition are all binary operations. The group operation is always a binary operation on the underlying set.

Group tables

A group table (also called a Cayley table) displays the result of the group operation for every pair of elements. For a group with $n$ elements, it's an $n \\times n$ grid. The element in row $a$, column $b$ is $a * b$.

Useful things to spot in a Cayley table:

• If the table is symmetric across the main diagonal, the group is abelian.
• Every row and every column contains each group element exactly once (this is called the Latin square property).

Composition of elements

Applying the group operation repeatedly to the same element gives you powers of that element. In multiplicative notation: $a^1 = a$, $a^2 = a * a$, $a^3 = a * a * a$, and so on. In additive notation, you'd write $2a, 3a$, etc.

The order of an element $a$ is the smallest positive integer $n$ such that $a^n = e$. If no such $n$ exists, the element has infinite order. The set of all powers of $a$ forms a cyclic subgroup.

Subgroups

Definition of subgroups

A subgroup $H$ of a group $G$ is a subset of $G$ that is itself a group under the same operation. To verify that $H$ is a subgroup, you need to check:

1. $H$ is non-empty (it contains the identity element).
2. $H$ is closed under the group operation.
3. $H$ is closed under taking inverses.

Associativity is inherited from $G$, so you don't need to re-check it.

Proper vs improper subgroups

Every group $G$ has at least two subgroups: the trivial subgroup $\\{e\\}$ (containing only the identity) and $G$ itself. These are called improper subgroups. Any other subgroup is a proper subgroup.

For example, the even integers $2\\mathbb{Z}$ form a proper subgroup of $(\\mathbb{Z}, +)$.

Cyclic subgroups

The cyclic subgroup generated by $a$, written $\\langle a \\rangle$, is the set of all powers of $a$: $\\{\\ldots, a^{-2}, a^{-1}, e, a, a^2, \\ldots\\}$. If $a$ has finite order $n$, then $\\langle a \\rangle = \\{e, a, a^2, \\ldots, a^{n-1}\\}$ and $|\\langle a \\rangle| = n$.

Order of subgroups

The order of a subgroup is simply the number of elements it contains. Lagrange's Theorem (covered in more detail below) tells you that the order of any subgroup must divide the order of the group. So if $|G| = 12$, the only possible subgroup orders are 1, 2, 3, 4, 6, and 12.

Group homomorphisms

Definition of homomorphisms

A homomorphism is a function $f: G \\to H$ between two groups that preserves the group operation:

$f(a * b) = f(a) * f(b)$

for all $a, b \\in G$. The operation on the left side is in $G$; the operation on the right is in $H$. Homomorphisms let you relate different groups by showing how their structures correspond.

Kernel and image

The kernel of a homomorphism $f: G \\to H$ is the set of elements in $G$ that map to the identity in $H$:

$\\ker(f) = \\{g \\in G : f(g) = e_H\\}$

The kernel tells you how much information the homomorphism "loses." If $\\ker(f) = \\{e_G\\}$, then $f$ is injective (one-to-one). The kernel is always a normal subgroup of $G$.

The image of $f$ is the set of elements in $H$ that actually get mapped to:

$\\text{im}(f) = \\{f(g) : g \\in G\\}$

The image is always a subgroup of $H$. The First Isomorphism Theorem states that $G / \\ker(f) \\cong \\text{im}(f)$.

Isomorphisms

An isomorphism is a homomorphism that is also a bijection (one-to-one and onto). If an isomorphism exists between two groups, they are isomorphic, written $G \\cong H$. Isomorphic groups have identical algebraic structure; they're the "same group" wearing different labels.

For example, $\\mathbb{Z}_2$ and $\\{1, -1\\}$ under multiplication are isomorphic. Both have two elements, and their Cayley tables match up perfectly.

Cosets and quotient groups

Left and right cosets

Given a subgroup $H$ of $G$ and an element $a \\in G$:

• Left coset: $aH = \\{a * h : h \\in H\\}$
• Right coset: $Ha = \\{h * a : h \\in H\\}$

Cosets partition $G$ into non-overlapping pieces, each the same size as $H$. Every element of $G$ belongs to exactly one left coset (and exactly one right coset). In an abelian group, left and right cosets are always the same.

Lagrange's theorem

Lagrange's Theorem states: if $H$ is a subgroup of a finite group $G$, then $|H|$ divides $|G|$. More precisely:

$|G| = |H| \\cdot [G:H]$

where $[G:H]$ is the index of $H$ in $G$ (the number of distinct cosets). This is one of the most important results in finite group theory. A direct consequence: the order of any element divides the order of the group.

Normal subgroups

A subgroup $N$ of $G$ is normal if $aN = Na$ for every $a \\in G$. Equivalently, $gng^{-1} \\in N$ for all $g \\in G$ and $n \\in N$ ($N$ is closed under conjugation). Every subgroup of an abelian group is automatically normal.

Normal subgroups matter because they're exactly the subgroups you can use to build quotient groups.

Quotient groups

If $N$ is a normal subgroup of $G$, the quotient group $G/N$ is the set of all cosets of $N$, with the operation $(aN)(bN) = (ab)N$. Normality of $N$ is what makes this operation well-defined.

You can think of a quotient group as "collapsing" everything in $N$ down to the identity. The result is a simpler group that captures the structure of $G$ "modulo" $N$. For example, $\\mathbb{Z}/n\\mathbb{Z} \\cong \\mathbb{Z}_n$, the integers mod $n$.

Group actions

Definition of group actions

A group action of $G$ on a set $X$ is a function $\\phi: G \\times X \\to X$ satisfying two properties:

1. Identity: $e \\cdot x = x$ for all $x \\in X$
2. Compatibility: $(gh) \\cdot x = g \\cdot (h \\cdot x)$ for all $g, h \\in G$ and $x \\in X$

Group actions formalize the idea of a group "acting as symmetries" of some set. For instance, the rotation group of a square acts on the set of vertices.

Orbits and stabilizers

The orbit of an element $x \\in X$ is the set of all elements reachable from $x$ by applying group elements:

$\\text{Orb}(x) = \\{g \\cdot x : g \\in G\\}$

Orbits partition $X$ into equivalence classes.

The stabilizer of $x$ is the set of group elements that fix $x$:

$\\text{Stab}(x) = \\{g \\in G : g \\cdot x = x\\}$

The stabilizer is always a subgroup of $G$. The Orbit-Stabilizer Theorem connects these: $|\\text{Orb}(x)| \\cdot |\\text{Stab}(x)| = |G|$ for finite groups.

Burnside's lemma

Burnside's lemma (also called the Cauchy-Frobenius lemma) counts the number of distinct orbits under a group action:

$|X/G| = \\frac{1}{|G|} \\sum_{g \\in G} |X^g|$

where $X^g = \\{x \\in X : g \\cdot x = x\\}$ is the set of elements fixed by $g$. In plain terms: the number of distinct orbits equals the average number of fixed points across all group elements.

This is especially useful in combinatorics. For example, if you want to count the number of distinct ways to color the faces of a cube (where rotations of the same coloring count as one), Burnside's lemma is the tool you'd reach for.

Applications of group theory

Symmetry in mathematics

Group theory is the mathematical language of symmetry. It classifies the symmetries of geometric objects, from regular polygons to higher-dimensional polytopes. In crystallography, the 230 space groups describe all possible symmetry patterns of three-dimensional crystals. In physics, symmetry groups underlie conservation laws through Noether's theorem.

Cryptography

Several modern cryptographic systems rely on group-theoretic problems that are easy to set up but hard to reverse. Elliptic curve cryptography operates on groups defined by points on elliptic curves. The security of systems like Diffie-Hellman key exchange depends on the difficulty of the discrete logarithm problem in certain groups.

Molecular structure

In chemistry, point groups classify molecules by their symmetry elements (rotations, reflections, inversions). Knowing a molecule's point group lets you predict its spectroscopic behavior, determine which molecular orbitals are allowed, and simplify quantum mechanical calculations. For example, water ($H_2O$) belongs to the point group $C_{2v}$, which has four symmetry operations.