Algebra, at its core, is the study of structure: the hidden patterns that make collections of objects behave in regular, predictable ways. Where analysis asks *how fast does this change?*, algebra asks *what is preserved?* Three layers of structure are the subject of these notes. [Groups](/page/Group) capture the pure essence of symmetry — the transformations of an object that leave it unchanged. Rings generalize the integers, equipping a set with two compatible operations and asking when the arithmetic of $\mathbb{Z}$ — divisibility, primality, factorization — extends to this broader setting. Modules generalize vector spaces by allowing scalars to come from a ring rather than a field; the reward is a unified theory that simultaneously classifies finite abelian groups and explains why every complex matrix is similar to a block-diagonal one.

The common thread through all three is the same methodology: identify the correct notion of *subobject*, form the *quotient* by identifying elements related by that subobject, and use the *isomorphism theorems* to read off the relationship between original, subobject, and quotient. Learning this methodology for groups — where it first appears in its cleanest form — is what makes the transition to rings and modules almost automatic.

These notes follow the Cambridge Part IB course in Groups, Rings, and Modules. Each chapter is self-contained, but the progression is deliberate: groups provide the language, rings provide the arithmetic, and modules provide the classification machinery that ties everything together.

# Groups

A group is the mathematical distillation of the concept of symmetry. This chapter begins with the basic definitions — groups, subgroups, cosets — and the first deep result, Lagrange's theorem. It then builds the structural theory: the notion of a normal subgroup and quotient group, the three isomorphism theorems, group actions and the orbit-stabilizer theorem, conjugacy classes and the class equation, Sylow's theorems, and finally the classification of finite abelian groups and the simplicity of the alternating groups. Each of these topics answers a progressively sharper version of the same question: how much can we determine about a group's structure from limited data?

## Symmetry and the Group Axioms

### What Is a Group?

Before writing down an axiom, consider what we want to capture. The symmetries of an equilateral triangle are the six transformations that map the triangle to itself: three rotations ($0°$, $120°$, $240°$) and three reflections. These transformations can be *composed* — "rotate, then reflect" is itself a transformation of the triangle. This composition is associative, there is a "do nothing" transformation, and every transformation can be undone. The six symmetries, together with composition, form the dihedral group $D_6$. What makes this interesting is not the triangle, but the algebraic structure of the six-element set itself — and a group is precisely the abstraction of that structure.

[definition:Group] A **group** is a triple $(G, \cdot, e)$ where $G$ is a set, $\cdot : G \times G \to G$ is a binary operation, and $e \in G$ is an element, satisfying: \begin{align*} &\text{(Associativity)} \quad (a \cdot b) \cdot c = a \cdot (b \cdot c) \quad \text{for all } a, b, c \in G, \\ &\text{(Identity)} \quad a \cdot e = e \cdot a = a \quad \text{for all } a \in G, \\ &\text{(Inverses)} \quad \text{for all } a \in G \text{ there exists } a^{-1} \in G \text{ with } a \cdot a^{-1} = a^{-1} \cdot a = e. \end{align*} [/definition]

We usually suppress the operation and write $ab$ for $a \cdot b$. Observe that neither the identity nor inverses are assumed to be unique in the definition — but both turn out to be. If $b$ is also an identity, then $e = eb = b$; if $b$ is also an inverse of $a$, then $b = be = b(aa^{-1}) = (ba)a^{-1} = ea^{-1} = a^{-1}$. So the structure is more rigid than it first appears.

[definition:Abelian Group] A group $G$ is **abelian** if $ab = ba$ for all $a, b \in G$. [/definition]

The distinction between abelian and non-abelian groups is fundamental: in an abelian group, the order of composition is irrelevant, while in a non-abelian group it is not. The symmetry group of the equilateral triangle is non-abelian: rotating by $120°$ and then reflecting across a fixed axis gives a different result than reflecting first and then rotating.

[example:Foundational Examples of Groups] The following are the core examples that will recur throughout the course. *Additive number groups.* The [sets](/page/Set) $(\mathbb{Z}, +, 0)$, $(\mathbb{Q}, +, 0)$, $(\mathbb{R}, +, 0)$, $(\mathbb{C}, +, 0)$ are all abelian groups. The key point is that subtraction is always possible: $a + (-a) = 0$. *Symmetric group.* The **symmetric group** $S_n$ is the group of all bijections $\{1, 2, \ldots, n\} \to \{1, 2, \ldots, n\}$ under composition. It has order $n!$. We write permutations in disjoint cycle notation: $(1\ 2\ 3)(4\ 5)$ denotes the permutation sending $1 \mapsto 2 \mapsto 3 \mapsto 1$ and $4 \mapsto 5 \mapsto 4$, with $6, 7, \ldots$ fixed. Since permutations are functions, composition is right-to-left: to compute $(1\ 2\ 3) \circ (1\ 2)$, first apply $(1\ 2)$, then $(1\ 2\ 3)$. Tracing each element: $1 \overset{(1\ 2)}{\mapsto} 2 \overset{(1\ 2\ 3)}{\mapsto} 3$, then $2 \overset{(1\ 2)}{\mapsto} 1 \overset{(1\ 2\ 3)}{\mapsto} 2$, then $3 \overset{(1\ 2)}{\mapsto} 3 \overset{(1\ 2\ 3)}{\mapsto} 1$. The composition sends $1 \mapsto 3$, $2 \mapsto 2$, $3 \mapsto 1$, which is the transposition $(1\ 3)$. This right-to-left convention is essential to keep in mind. *General linear group.* The set $\mathrm{GL}_n(\mathbb{R})$ of invertible $n \times n$ real matrices forms a group under matrix multiplication. It is non-abelian for $n \geq 2$. *Cyclic group.* For $n \geq 1$, the cyclic group $C_n = \mathbb{Z}/n\mathbb{Z}$ is the group of integers modulo $n$ under addition. It is abelian of order $n$. [/example]

### Subgroups

Not every subset of a group is itself a group under the inherited operation. We want those subsets that are closed under the operation and under taking inverses — that is, subsets which are groups in their own right.

[definition:Subgroup] Let $(G, \cdot, e)$ be a group. A subset $H \subseteq G$ is a **subgroup**, written $H \leq G$, if: \begin{align*} &\text{(i) } e \in H, \\ &\text{(ii) } a, b \in H \implies ab \in H, \\ &\text{(iii) } a \in H \implies a^{-1} \in H. \end{align*} [/definition]

A convenient criterion collapses these three conditions into one. If $H$ is non-empty and closed under the operation $h_1 h_2^{-1}$, then $H$ is a subgroup: taking $h_1 = h_2$ gives $e \in H$; then $h_1 = e$ gives $h_2^{-1} \in H$ for any $h_2 \in H$; and combining, $H$ is closed under products. This "subgroup lemma" is used constantly.

[example:A Subgroup Computation in $S_3$] Consider $S_3$, the symmetric group on three elements. Its six elements in cycle notation are \begin{align*} e,\ (1\ 2),\ (1\ 3),\ (2\ 3),\ (1\ 2\ 3),\ (1\ 3\ 2). \end{align*} Let $H = \{e, (1\ 2\ 3), (1\ 3\ 2)\}$. We verify $H \leq S_3$. The set is non-empty. Computing products: $(1\ 2\ 3)(1\ 2\ 3) = (1\ 3\ 2)$ and $(1\ 2\ 3)(1\ 3\ 2) = e$, so $H$ is closed under products. Inverses: $(1\ 2\ 3)^{-1} = (1\ 3\ 2)$ and vice versa. So $H \leq S_3$. This is the alternating group $A_3 \cong C_3$. Now let $K = \{e, (1\ 2)\}$. Then $K \leq S_3$ as well, since $(1\ 2)^{-1} = (1\ 2)$ (a transposition is its own inverse) and $(1\ 2)(1\ 2) = e \in K$. But $\{(1\ 2), (1\ 3)\}$ is not a subgroup, since $(1\ 2)(1\ 3) = (1\ 3\ 2) \notin \{(1\ 2), (1\ 3)\}$. [/example]

## Cosets and the Counting Principle

Lagrange's theorem is the first genuinely surprising result of group theory. It says that the order of any subgroup must divide the order of the whole group — a powerful constraint that immediately rules out many potential subgroup orders. The proof is elegant and rests on a single observation about how a subgroup partitions its parent group.