Examples of Groups

Once you know the rules of the game — a set, an operation, and the four axioms of what a group is (closure, associativity, an identity, and inverses) — the natural next move is to go looking for who actually plays it. The surprise is how many familiar objects turn out to be groups in disguise: the counting numbers you add with, the hours on a clock, the ways you can spin a snowflake so it looks unchanged. This page is a gallery — a walk past the most important examples, chosen so that by the end the abstract definition feels populated with real, memorable inhabitants rather than empty scaffolding.

The single big idea to carry through the whole tour: a group is any set paired with an operation that obeys the axioms — nothing more is required, and nothing more is allowed to sneak in. That freedom is exactly why the same abstract group can show up wearing completely different costumes. The four rotations of a square and the numbers \{0,1,2,3\} added modulo 4 look nothing alike, yet they are the same group. Learning to see through the costume to the group underneath is the whole point of abstract algebra.

1. The integers \mathbb{Z} under addition

The most down-to-earth infinite group is the set of all integers \{\dots,-2,-1,0,1,2,\dots\} with the operation of ordinary addition. Run the axioms past it: adding two integers gives an integer (closure); (a+b)+c = a+(b+c) (associativity); adding 0 changes nothing (identity); and every a has a partner -a that cancels it back to 0 (inverses). All four hold, so (\mathbb{Z},+) is a group.

It is also abelian — a+b = b+a always — and infinite, and it is cyclic: every integer is a whole-number multiple of the single generator 1 (or of -1). Notice what is not a group here: (\mathbb{Z},\times) fails, because 3 has no integer multiplicative inverse — 1/3 isn't an integer. The operation matters as much as the set.

2. The cyclic groups \mathbb{Z}_n — a clock with n hours

Take the integers \{0,1,2,\dots,n-1\} and add them the way a clock adds hours: whenever you reach n, wrap back to 0. This is addition modulo n, and the resulting group is written \mathbb{Z}_n (or \mathbb{Z}/n\mathbb{Z}). It has exactly n elements, its identity is 0, and the inverse of a is n-a (which wraps a back to zero).

For example, in \mathbb{Z}_4 = \{0,1,2,3\}:

2 + 3 = 5 \equiv 1 \pmod 4, \qquad 3 + 3 = 6 \equiv 2 \pmod 4, \qquad -1 \equiv 3.

A real clock face is just \mathbb{Z}_{12} in disguise (with 12 relabelled as 0): four hours after ten o'clock is two o'clock, because 10 + 4 = 14 \equiv 2 \pmod{12}. These \mathbb{Z}_n are the cyclic groups, the simplest finite groups there are, and every one of them is abelian.

3. The multiplicative group (\mathbb{Z}/p\mathbb{Z})^{\times}

Multiplication modulo n is trickier: 0 can never have an inverse, and neither can any number sharing a factor with n. So we throw those out and keep only the residues coprime to n. The survivors form a group under multiplication mod n, written (\mathbb{Z}/n\mathbb{Z})^{\times}. When n = p is prime, every nonzero residue is coprime to p, so the group is \{1,2,\dots,p-1\} and has exactly p-1 elements.

Take p = 5. The group is (\mathbb{Z}/5\mathbb{Z})^{\times} = \{1,2,3,4\} under multiplication mod 5. Its identity is 1, and 2 \times 3 = 6 \equiv 1, so 2 and 3 are inverses. Here is a small miracle worth flagging: this group has 4 elements and is also cyclic — the powers of 2 are 2,4,3,1, sweeping out all four. So (\mathbb{Z}/5\mathbb{Z})^{\times} is secretly the same group as \mathbb{Z}_4 — a multiplicative object and an additive one, identical underneath.

4. Symmetry groups — the dihedral group D_n

The richest examples come not from numbers but from symmetry. A symmetry of a shape is any rigid motion — a rotation or a flip — that leaves the shape looking exactly as it started. Compose two symmetries (do one, then the other) and you get another symmetry, the identity is "do nothing," every motion can be undone, and composition is associative. So the symmetries of any shape form a group, automatically.

For a regular n-gon this group is the dihedral group D_n. It contains:

A square (n=4) therefore has |D_4| = 8 symmetries: four rotations (0^\circ,90^\circ,180^\circ,270^\circ) and four reflections (two through opposite edges, two through opposite corners). An equilateral triangle (n=3) has |D_3| = 6 — and the interactive figure below walks through every one of them.

Meet D_3: the symmetries of a triangle

Step through the figure. The triangle's three corners are labelled 1,2,3. A 120^\circ rotation sends 1\to 2\to 3\to 1; a 240^\circ rotation cycles them the other way; and a reflection through any corner swaps the other two. Count them: the identity, two rotations, three reflections — 6 = 2\times 3 symmetries in all.

The same group in many disguises

Return to the headline idea. We have now seen a striking coincidence twice: the rotations of a square are \mathbb{Z}_4, and (\mathbb{Z}/5\mathbb{Z})^{\times} is also \mathbb{Z}_4. Three descriptions — spinning a square, multiplying residues mod 5, adding on a four-hour clock — all name the same abstract group. When two groups match up element-for-element and operation-for-operation like this we call them isomorphic: same skeleton, different flesh.

This is why mathematicians study groups abstractly rather than case by case. Prove something about \mathbb{Z}_4 and you have simultaneously proved it about square rotations, about the fourth roots of unity in the complex plane, and about (\mathbb{Z}/5\mathbb{Z})^{\times} — every disguise at once. One theorem, many payoffs. The costume changes; the group does not.

Rotations as matrices

The rotation part of a symmetry group also has a concrete face in linear algebra. A rotation of the plane by angle \theta about the origin is exactly the linear map given by a rotation matrix:

R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}.

The n rotations of D_n are just R(0), R\!\left(\tfrac{2\pi}{n}\right), R\!\left(\tfrac{4\pi}{n}\right), \dots, and composing rotations corresponds to multiplying these matrices — with the tidy identity R(\alpha)\,R(\beta) = R(\alpha+\beta). So the abstract "rotate then rotate" of the group is nothing but matrix multiplication, and the cyclic subgroup of D_n lives inside the group of 2\times 2 rotation matrices. Reflections are matrices too (determinant -1 instead of +1), which is another way to see why a reflection can never equal a rotation.

Every group so far — \mathbb{Z}, the \mathbb{Z}_n, the (\mathbb{Z}/p\mathbb{Z})^{\times} — has been abelian: order of operation never mattered. It is tempting to assume that's just how groups behave. It is not. For n \ge 3 the dihedral group D_n is non-abelian: doing a rotation and then a reflection generally lands you somewhere different from the reflection followed by the rotation.

Try it with a triangle. Let r be the 120^\circ rotation and s a reflection. Do r then s and compare with s then r — the two results send corner 1 to different places. In symbols sr = r^{-1}s \neq rs. So D_3 is the smallest non-abelian group there is, with just 6 elements. The lesson: never assume ab = ba in a group unless you've been told it's abelian.

Symmetry groups get spectacular once you leave single shapes. A snowflake looks the same after a 60^\circ turn, so its symmetry group is D_6 — six rotations and six mirror lines, order 12. That six-fold symmetry is a direct fingerprint of how water molecules lock into a hexagonal ice crystal.

Now tile the whole plane. The symmetries of a repeating pattern — translations, rotations, reflections, and glide-reflections combined — form a wallpaper group, and it is a remarkable theorem that there are exactly 17 of them, no more. Every wallpaper, tiled floor, and Alhambra mosaic that has ever been or ever will be drawn belongs to one of just seventeen symmetry types. Restrict to patterns that repeat along a single strip — a decorative border or frieze — and the count drops to exactly 7 frieze groups. Group theory doesn't just describe symmetry; it counts it.