This chapter establishes the central idea of abstract algebra: the study of sets equipped with operations, and the structure-preserving maps between them. The goal is not to memorize vocabulary but to absorb the habit of treating the operation as part of the object.

§1.1 What algebra is about

Remark 1.1 (The core idea)

Abstract algebra studies sets with operations and the structure-preserving maps (homomorphisms, isomorphisms) between them. The key insight: the same underlying set can carry different algebraic structures depending on which operation is imposed.

Example: The set $\mathbb{Z}$ under addition $(\mathbb{Z},+)$ is a group, but $\mathbb{Z}$ under multiplication $(\mathbb{Z},\cdot )$ is not (most elements lack multiplicative inverses). Same set, different structures, fundamentally different behavior.

Figure: the same underlying set can support different algebraic structures.

The point of the figure is that the operation is part of the data; changing the operation changes the object.

Remark 1.2 (Why abstraction matters)

Abstraction is not generalization for its own sake. It serves three purposes:

1. Unification. Seemingly unrelated objects (rotations of a square, residue classes mod $n$, permutations of a set) turn out to satisfy the same axioms, so one proof covers all cases.
2. Structural invariants. Instead of computing with specific elements, one reasons about properties (like element orders or subgroup lattices) that are preserved under isomorphism.
3. Transfer of problems. An isomorphism $(\mathbb{R},+)\cong ({\mathbb{R}}^{+},\cdot )$ via $\exp$ allows one to turn additive problems into multiplicative ones and vice versa.

§1.2 Symmetries as motivation

Definition 1.3 (Symmetry of a geometric object)

A symmetry of a geometric figure is a rigid motion (distance-preserving transformation) of the plane that maps the figure onto itself. The set of all symmetries, under composition, forms a group.

Example 1.4 (Symmetries of an equilateral triangle)

An equilateral triangle has $6$ symmetries:

• $3$ rotations: by $0°$, $120°$, and $240°$ about the center.
• $3$ reflections: across each altitude.

These form the dihedral group $D_3$ (also called $S_3$, the symmetric group on $3$ elements), of order $6$. This is the simplest nonabelian group one encounters naturally.

§1.3 The dihedral group $D_4$: symmetries of the square

Definition 1.5 (Dihedral group $D_4$)

The dihedral group of the square, denoted $D_4$, is the group of all symmetries of a square. It has $8$ elements.

Label the vertices of the square $1,2,3,4$ (clockwise). The elements are:

Group structure: The group is generated by $r$ (rotation by $90°$) and $s$ (one reflection), subject to:

The relation $srs=r^{-1}$ says that conjugating a rotation by a reflection reverses the direction of rotation. This is the defining feature of dihedral groups.

Remark 1.6 ($D_4$ is not abelian)

Since $sr\ne rs$ (in fact $rs=s{r}^3$), the group $D_4$ is nonabelian. This is visible from the geometry: rotating then reflecting is not the same as reflecting then rotating.

Example 1.7 (A computation in $D_4$)

Compute $r^2\cdot sr$ in $D_4$.

Using the relation $sr=r^{-1}s=r^{3}s$:

So $r^2\cdot sr=s{r}^3$. In terms of the table above, the composition of a $180°$ rotation with the diagonal reflection gives the other diagonal reflection.

§1.4 Modular arithmetic as a preview

Definition 1.8 (Integers modulo $n$)

For a positive integer $n$, define the set

with addition modulo $n$: $\bar{a}+\bar{b}=\overline{a+b\mathrm{mod}n}$.

Example 1.9 (${\mathbb{Z}}_6$ addition table, partial)

In ${\mathbb{Z}}_6$: $\bar{4}+\bar{5}=\overline{9}=\bar{3}$ (since $9=1\cdot 6+3$).

Key observations:

• Identity: $\bar{0}$.
• Inverses: The inverse of $\bar{a}$ is $\overline{n-a}$. For instance, in ${\mathbb{Z}}_6$, the inverse of $\bar{4}$ is $\bar{2}$.
• Cyclic: $\bar{1}$ generates all of ${\mathbb{Z}}_n$ by repeated addition.

Remark 1.10 (Why modular arithmetic matters in algebra)

${\mathbb{Z}}_n$ is the simplest family of finite groups. It provides test cases for every definition and theorem in the course: subgroups, cyclic groups, homomorphisms, quotients, and direct products all have clean concrete manifestations in ${\mathbb{Z}}_n$.

§1.5 The same set, different structures

Example 1.11 (A non-standard operation on $\mathbb{Z}$)

On $\mathbb{Z}$, define $a\star b=a+b+1$. Then $(\mathbb{Z},\star )$ is a group:

• Closure: $a+b+1\in \mathbb{Z}$ for $a,b\in \mathbb{Z}$.
• Associativity: $(a\star b)\star c=(a+b+1)+c+1=a+b+c+2=a+(b+c+1)+1=a\star (b\star c)$.
• Identity: $e=-1$, since $a\star (-1)=a+(-1)+1=a$ and $(-1)\star a=-1+a+1=a$.
• Inverses: The inverse of $a$ is $-a-2$, since $a\star (-a-2)=a+(-a-2)+1=-1=e$.

Moreover, $\phi :(\mathbb{Z},\star )\to (\mathbb{Z},+)$ defined by $\phi (n)=n+1$ is an isomorphism:

So $(\mathbb{Z},\star )$ is just the usual additive integers in disguise. This illustrates why algebra studies structures up to isomorphism, not specific element names.

§1.6 Axiom-failure diagnosis

Remark 1.12 (Name the first failure)

When a proposed algebraic structure fails to satisfy the axioms, one should identify the first failing axiom in the logical order: (1) closure, (2) associativity, (3) identity, (4) inverses.

Mastery Checklist

• Explain what extra data distinguishes an algebraic structure from a bare set.
• List all $8$ elements of $D_4$ and perform computations using $r^4=s^2=e$ and $srs=r^{-1}$.
• Describe the symmetries of a regular $n$-gon and why they form a group.
• Compute in ${\mathbb{Z}}_n$ (addition, inverses, orders of elements).
• Give an example of two different group structures on the same underlying set.
• Diagnose precisely why a proposed structure fails to be a group (name the first failing axiom).