These are my compiled notes from Modern Algebra I, reorganized into a narrative that builds the theory of groups from first principles. Abstract algebra strips away the specifics of numbers, matrices, and functions to study the structure of mathematical operations themselves. The payoff is enormous: a single theorem about groups can simultaneously tell you something about clock arithmetic, symmetries of molecules, Rubik’s cube solutions, and error-correcting codes.
This is Part 1 of a three-part series. Here we build the foundations — from operations on sets to the definition of a group, through the first major examples and subgroup theory.
What This Post Covers
- Operations on Sets — What it means to combine elements, and how to organize combinations in a table
- Properties of Operations — Commutativity, identity elements, and inverses
- Associativity and the Definition of a Group — The three axioms that make a group
- Elementary Group Properties — Uniqueness theorems, cancellation, and the “Sudoku property”
- Symmetry Groups — The dihedral group of the triangle, where algebra meets geometry
- Subgroups — Direct products, the subgroup criterion, cyclic subgroups, and generators
Operations on Sets
The starting point of abstract algebra is almost absurdly simple. Forget about addition, multiplication, and all the specific operations you know. Instead, ask: what does it mean to combine two things?
Definition. An operation on a set $A$ is a rule that assigns exactly one element of $A$ to every ordered pair of elements of $A$. Equivalently, it is a function $*: A \times A \to A$.
We write $a * b$ (or $a + b$, $a \cdot b$, etc.) instead of $\ast(a, b)$ because it’s more natural. The key word is closed — the output must land back in $A$. Addition is an operation on $\mathbb{Z}$, but division is not an operation on $\mathbb{Z}$ (since $1 \div 2 \notin \mathbb{Z}$).
Some familiar examples:
- $+$ on $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$
- $\div$ on $\mathbb{Q} - \lbrace 0\rbrace$, $\mathbb{R} - \lbrace 0\rbrace$, $\mathbb{C} - \lbrace 0\rbrace$
- $\cdot$ on $n \times n$ matrices
How Many Operations Exist?
Here’s a fun counting question: how many operations are there on the set $A = \lbrace 0, 1\rbrace$?
An operation on $A$ must assign an output to every ordered pair. The possible inputs are $(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$ — four pairs. For each pair, we can choose either $0$ or $1$ as the output. By the product principle, there are $2^4 = 16$ possible operations.
We can organize any operation on a finite set into a Cayley table (also called an operation table). Here are two of the sixteen operations on $\lbrace0, 1\rbrace$:
| * | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 1 |
| * | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 0 |
The left table is ordinary multiplication. The right is addition modulo 2. To read these: the entry in row $a$, column $b$ gives $a * b$.
In general, an $n$-element set has $n^{n^2}$ possible operations. This grows absurdly fast — a 3-element set has $3^9 = 19{,}683$ operations. Most of these are unstructured noise. The goal of algebra is to focus on the “nice” ones.
Properties of Operations
What makes an operation “nice”? Three properties stand out.
Commutativity
Definition. An operation $*$ on a set $A$ is commutative if $a * b = b * a$ for all $a, b \in A$.
Addition on $\mathbb{Z}$ is commutative. Matrix multiplication is not — in general, $AB \neq BA$. There’s a neat visual test: an operation is commutative if and only if its Cayley table is symmetric across the main diagonal. If $G$ is the matrix representing the Cayley table, the operation is commutative iff $G^T = G$.
Identity Elements
Definition. An operation $*$ on $A$ has an identity element if there is some $e \in A$ such that $a * e = e * a = a$ for every $a \in A$.
The identity for addition is $0$. The identity for multiplication is $1$. For matrix multiplication, it’s the identity matrix $I$. A critical point: the identity must be a fixed constant — it cannot depend on which element $a$ you’re combining it with.
Non-example. Exponentiation on $\lbrace0, 1\rbrace$: we’d need $x^e = x$ for all $x$, which gives $e = 1$. But we also need $e^x = x$ for all $x$, which gives $1^0 = 0$ — false. So exponentiation has no identity.
In a Cayley table, the identity element is easy to spot: its row and column reproduce the header row and column exactly.
| * | a | b | c |
|---|---|---|---|
| a | b | b | a |
| b | a | c | b |
| c | a | b | c |
Here $c$ is the identity — the highlighted row and column just reproduce the labels.
Inverses
Definition. If an operation $*$ on a set $A$ has an identity element $e$, then an element $a \in A$ has an inverse $a^{-1} \in A$ if $a * a^{-1} = e = a^{-1} * a$.
The inverse of $3$ under addition is $-3$. The inverse of $3$ under multiplication is $\frac{1}{3}$ (but only if we’re in $\mathbb{Q}$ or $\mathbb{R}$, not $\mathbb{Z}$). Zero has no multiplicative inverse. You cannot have inverses without first having an identity element.
Finding inverses in a Cayley table amounts to looking for copies of $e$: to find $a^{-1}$, scan row $a$ for $e$, and the column header gives you $a^{-1}$.
An operation can be commutative and/or have an identity and/or have inverses for every element — these properties are independent.
Associativity and the Birth of Groups
Associativity
Definition. An operation $*$ on a set $A$ is associative if $(a * b) * c = a * (b * c)$ for all $a, b, c \in A$.
Addition, multiplication, and matrix multiplication are all associative. Subtraction is not: $8 - (4 - 2) = 6$, but $(8 - 4) - 2 = 2$. Unlike commutativity and identity elements, there is no simple visual test for associativity from a Cayley table — you just have to check.
Example. Consider the operation $a * b = -a - b + 1$ on $\mathbb{Z}$.
- Commutative? Yes: $-a - b + 1 = -b - a + 1$.
- Identity? We need $a * e = a$, i.e., $-a - e + 1 = a$, giving $e = 1 - 2a$. But $e$ depends on $a$ — so no identity exists.
- Associative? $(a * b) * c = (-a - b + 1) * c = -(-a-b+1) - c + 1 = a + b - c$. Meanwhile, $a * (b * c) = a * (-b - c + 1) = -a - (-b-c+1) + 1 = -a + b + c$. These aren’t equal, so not associative.
The Definition of a Group
Now we can state the central definition of the course.
Definition. A group $(G, )$ is a set $G$ together with an operation $$ such that:
- $*$ is associative: $(a * b) * c = a * (b * c)$ for all $a, b, c \in G$
- $*$ has an identity element $e \in G$
- Every element of $G$ has an inverse under $*$
That’s it. Three axioms. Everything in group theory flows from these.
A group is not necessarily commutative. Commutative groups are called abelian groups (after Niels Henrik Abel).
First Examples
- $(\mathbb{Z}, +)$, $(\mathbb{Q}, +)$, $(\mathbb{R}, +)$ are abelian groups.
- $(\mathbb{Q} - \lbrace 0\rbrace, \cdot)$, $(\mathbb{R} - \lbrace 0\rbrace, \cdot)$ are abelian groups.
- $(\mathbb{R}^{pos}, \cdot)$ and $([0, \infty), \cdot)$ are abelian groups… wait, is $[0, \infty)$ a group under multiplication? No — $0$ has no multiplicative inverse!
- Modular addition gives us finite groups: $\mathbb{Z}_n = \lbrace 0, 1, 2, \ldots, n-1\rbrace$ with addition modulo $n$.
Here are the Cayley tables for $\mathbb{Z}_2$ and $\mathbb{Z}_3$:
| + | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 0 |
| + | 0 | 1 | 2 |
|---|---|---|---|
| 0 | 0 | 1 | 2 |
| 1 | 1 | 2 | 0 |
| 2 | 2 | 0 | 1 |
$\mathbb{Z}_n$ is an abelian group for any $n$, and it’s our first example of a cyclic group — every element can be obtained by repeatedly adding $1$.
The set of invertible $n \times n$ real matrices under multiplication is a non-commutative group. This is a big deal: it means group theory captures structures where order matters.
Elementary Properties of Groups
What can we deduce from just the three axioms? Quite a lot.
Uniqueness of the Identity
Proposition. Every group has exactly one identity element.
Proof. Suppose $e_1$ and $e_2$ are both identity elements in $(G, *)$. Then:
\[e_1 = e_1 * e_2 = e_2\]The first equality holds because $e_2$ is an identity, and the second holds because $e_1$ is an identity. So $e_1 = e_2$. $\square$
Uniqueness of Inverses
Proposition. For any element $a$ in a group $(G, *)$:
- $a^{-1}$ is unique (no element has two different inverses)
- $(a^{-1})^{-1} = a$
- $(a * b)^{-1} = b^{-1} * a^{-1}$
Watch the order in (3)! The inverse of a product reverses the order. This is the same reason you take off your shoes before your socks. Only when the group is abelian do we get $(a * b)^{-1} = a^{-1} * b^{-1}$.
More generally, $(a_1 * a_2 * \cdots * a_n)^{-1} = a_n^{-1} * \cdots * a_2^{-1} * a_1^{-1}$.
Cancellation Laws
Proposition. In any group $(G, *)$ with elements $a, b, c$:
- If $a * b = a * c$, then $b = c$ (left cancellation)
- If $b * a = c * a$, then $b = c$ (right cancellation)
Proof of (1). Multiply both sides on the left by $a^{-1}$:
\[a^{-1} * (a * b) = a^{-1} * (a * c)\] \[(a^{-1} * a) * b = (a^{-1} * a) * c \quad \text{(associativity)}\] \[e * b = e * c \quad \text{(inverses)}\] \[b = c \quad \text{(identity)} \quad \square\]WARNING. In a non-abelian group, $a * b = c * a$ does not imply $b = c$. Similarly, $b * a = a * c$ does not imply $b = c$. You can only cancel from the same side.
The “Sudoku Property”
The cancellation laws have a beautiful consequence: in the Cayley table of any group, every element appears exactly once in every row and exactly once in every column. This is sometimes called the Sudoku property.
Why? If some element $d$ appeared twice in row $a$ — say in columns $b$ and $c$ with $b \neq c$ — then $a * b = d = a * c$, which by left cancellation gives $b = c$, a contradiction.
This property is surprisingly powerful. Suppose you know $(\lbrace e, a\rbrace, *)$ is a group. The table must start with:
| * | e | a |
|---|---|---|
| e | e | a |
| a | a | e |
The identity row and column are forced, and then the Sudoku property forces $a * a = e$. So up to renaming elements, there is exactly one group of size 2.
Similarly, there is exactly one group of size 3, but there are two groups of size 4 (up to isomorphism — a concept we’ll develop in Part 2).
From here on, we adopt multiplicative notation: we write $ab$ instead of $a * b$, and $a^n$ for the product of $n$ copies of $a$.
Symmetry: The Dihedral Group
Groups become vivid when we connect them to geometry. Consider an equilateral triangle with vertices labeled $A$, $B$, $C$. What are all the ways we can pick up this triangle and set it back down so that it occupies the same space?
There are exactly six symmetries:
- $e$: do nothing (identity)
- $r$: rotate 120° clockwise
- $s$: rotate 240° clockwise (equivalently, 120° counter-clockwise)
- $a$: reflect across the line through vertex $A$
- $b$: reflect across the line through vertex $B$
- $c$: reflect across the line through vertex $C$
The set is $D_3 = \lbrace e, r, s, a, b, c\rbrace$ and the operation is composition, applied right to left. To compute $r * a$ (“do $a$ first, then $r$”), we track where each vertex goes.
This group is called the dihedral group $D_3$. Its full Cayley table is:
| * | e | r | s | a | b | c |
|---|---|---|---|---|---|---|
| e | e | r | s | a | b | c |
| r | r | s | e | c | a | b |
| s | s | e | r | b | c | a |
| a | a | b | c | e | r | s |
| b | b | c | a | s | e | r |
| c | c | a | b | r | s | e |
Notice that this table is not symmetric across the diagonal — $D_3$ is non-abelian. For instance, $r * a = c$ but $a * r = b$. The rotations commute with each other, and each reflection is its own inverse, but mixing rotations and reflections does not commute.
The dihedral group $D_n$ generalizes to the symmetries of a regular $n$-gon. $D_4$ (the square) has 8 elements, $D_5$ (the pentagon) has 10, and in general $\lvert D_n \rvert = 2n$.
Subgroups
Direct Products
Definition. Given groups $(G_1, *_1)$ and $(G_2, *_2)$, the direct product $G_1 \times G_2$ consists of all ordered pairs $\lbrace(a_1, a_2) : a_1 \in G_1, a_2 \in G_2\rbrace$ with componentwise operation:
\[(a_1, a_2) * (b_1, b_2) = (a_1 *_1 b_1, \; a_2 *_2 b_2)\]For example, $\mathbb{Z}_2 \times \mathbb{Z}_2$ has elements $\lbrace(0,0), (0,1), (1,0), (1,1)\rbrace$ with modular addition in each coordinate. Its Cayley table:
| + | (0,0) | (0,1) | (1,0) | (1,1) |
|---|---|---|---|---|
| (0,0) | (0,0) | (0,1) | (1,0) | (1,1) |
| (0,1) | (0,1) | (0,0) | (1,1) | (1,0) |
| (1,0) | (1,0) | (1,1) | (0,0) | (0,1) |
| (1,1) | (1,1) | (1,0) | (0,1) | (0,0) |
The Subgroup Criterion
Definition. If $(G, *)$ is a group and $H \subseteq G$, then $H$ is a subgroup of $G$ if $(H, *)$ is itself a group.
For example, $(\mathbb{Z}, +)$ is a subgroup of $(\mathbb{Q}, +)$, which is a subgroup of $(\mathbb{R}, +)$.
You don’t need to re-verify all three axioms from scratch. Since the operation is inherited from $G$, associativity is automatic. You only need to check:
- Closed under $*$: for all $a, b \in H$, $a * b \in H$
- Closed under inverses: for all $a \in H$, $a^{-1} \in H$
(These two conditions together guarantee the identity is in $H$ — can you see why?)
Example. What are the subgroups of $\mathbb{Z}_4 = \lbrace 0, 1, 2, 3\rbrace$?
- $\lbrace0\rbrace$ — the trivial subgroup (always exists)
- $\lbrace0, 2\rbrace$ — closed under addition mod 4
- $\lbrace0, 1, 2, 3\rbrace$ — the whole group (always a subgroup of itself)
Note that any divisor $d$ of $n$ yields a subgroup $\lbrace0, d, 2d, \ldots\rbrace$ of $\mathbb{Z}_n$ of size $\frac{n}{d}$. So the number of subgroups of $\mathbb{Z}_n$ equals the number of divisors of $n$.
Cyclic Subgroups and Generators
Definition. Given an element $a$ in a group $(G, *)$, the subgroup generated by $a$ is:
\[\langle a \rangle = \{e, a, a^2, a^3, \ldots, a^{-1}, a^{-2}, \ldots\}\]This is always a subgroup (the smallest one containing $a$).
Definition. A group $G$ is cyclic if there exists some element $a$ such that $\langle a \rangle = G$ — that is, every element of $G$ can be obtained by repeatedly applying $a$ or $a^{-1}$.
Examples:
- $(\mathbb{Z}, +) = \langle 1 \rangle = \langle -1 \rangle$, so $\mathbb{Z}$ is cyclic.
- $\mathbb{Z}_n = \langle 1 \rangle$, so every $\mathbb{Z}_n$ is cyclic.
- $\mathbb{Z}_2 \times \mathbb{Z}_2$ is not cyclic — no single element generates the whole group. Each non-identity element generates a subgroup of size 2: $\langle(0,1)\rangle = \lbrace(0,0), (0,1)\rbrace$, $\langle(1,0)\rangle = \lbrace(0,0), (1,0)\rbrace$, and $\langle(1,1)\rangle = \lbrace(0,0), (1,1)\rbrace$.
Groups like $\mathbb{Z}_m \times \mathbb{Z}_n$ may or may not be cyclic, depending on $m$ and $n$. We’ll see the precise criterion in Part 2.
Generators and Relations
We can describe a group by specifying a set of generators and relations between them. For instance:
- $(\mathbb{Z}, +)$ can be presented as $\langle 1 \rangle$ — one generator, no relations (other than the group axioms).
- $\mathbb{Z}_n$ can be presented as $\langle 1 \mid n \cdot 1 = 0 \rangle$ (meaning $n$ copies of the generator sum to the identity).
- The dihedral group $D_3$ can be presented as $\langle r, s \mid rrr, ss, rsrs \rangle$ — two generators (a rotation and a reflection) with relations encoding that $r^3 = e$, $s^2 = e$, and $rsrs = e$.
The free group $\langle A \rangle$ on an alphabet $A$ consists of all “words” in the letters of $A$ (and their inverses), with concatenation as the operation and cancellation ($aa^{-1} = e$) as the only relation. Adding more relations restricts which words are considered equal, carving out a quotient of the free group — a concept we’ll formalize in Part 3.
Intersections of Subgroups
Proposition. If $H$ and $K$ are both subgroups of $(G, *)$, then $H \cap K$ is also a subgroup of $(G, *)$.
Proof. We need $H \cap K$ to be closed under $*$ and inverses. If $a, b \in H \cap K$, then $a * b \in H$ (because $H$ is closed) and $a * b \in K$ (because $K$ is closed), so $a * b \in H \cap K$. Similarly for inverses. $\square$
Caution. Unions of subgroups are generally not subgroups. For instance, in $(\mathbb{Z}, +)$, the subgroups $2\mathbb{Z}$ and $3\mathbb{Z}$ have union containing both 2 and 3, but $2 + 3 = 5 \notin 2\mathbb{Z} \cup 3\mathbb{Z}$.
Looking Ahead
We’ve built the foundations: operations, the group axioms, basic properties, symmetry groups, and subgroups. In Part 2, we’ll develop the machinery of permutation groups and cycle notation, then use isomorphisms to understand when two groups are “really the same” despite looking different. The punchline — Cayley’s theorem — will show that every group is secretly a permutation group.