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Group Theory: an interactive course in symmetry

Published 8 Nov 20266 min read

Group Theory is an interactive group theory course that starts with turning and flipping a square and builds up to Lagrange's theorem. Its 24 short sections cover symmetries, the four group axioms, Cayley tables, modular arithmetic, the order of an element, cyclic groups and generators, dihedral groups, subgroups, Lagrange's theorem and permutations, with an animation or practice question in nearly every one.

Group theory is the mathematics of symmetry and the first topic of most abstract algebra courses. This course makes it concrete: every definition is shown on shapes, clocks or shuffled cards before it is written in symbols.

What the group theory course covers

Sections	Topic	What you do
1 to 5	Symmetries, groups and the four axioms	Turn and flip a numbered square, then check the axioms for sets and operations
6 to 7	Cayley tables	Watch a table fill in, then complete one like a sudoku
8 to 10	Clock arithmetic and the integers mod n	Add on a clock, then find sums and inverses
11 to 14	The order of an element, cyclic groups and generators	Count steps back to the identity, then pick every generator
15 to 17	Dihedral groups, and why order matters	See the six symmetries of a triangle, then combine symmetries of a square
18 to 21	Subgroups and Lagrange's theorem	Build subsets on a clock, then use Lagrange to rule out impossible orders
22 to 23	Permutations and the symmetric group	Shuffle three cards and match each shuffle to its cycle notation
24	Groups everywhere	See groups in the Rubik's Cube, crystals, particle physics and cryptography

From symmetry to axioms

A square has eight symmetries: four turns and four flips. Doing one, then another, always gives a third. That observation becomes the definition of a group: a set with a way of combining that is closed, associative, has an identity and gives every element an inverse. The course is careful about one point that trips people up: order is not a rule, so ab and ba can differ.

Two families of examples

Most of the course moves between two kinds of group. The integers mod n, written ℤₙ, are clock arithmetic and always cyclic. Dihedral groups, the symmetries of regular polygons, show what happens when order matters: on the square, a turn then a flip is not the same as the flip then the turn.

Structure

The final third finds groups inside groups. A subgroup passes three checks: it contains the identity, is closed and contains every inverse. Lagrange's theorem then says a subgroup's size always divides the group's size, so a group of 12 elements can have subgroups of size 1, 2, 3, 4, 6 or 12, but never 5.

Try a question from the course

To add mod n, add as normal and keep the remainder after dividing by n. In ℤ₇, 5 + 4 = 9, and 9 leaves remainder 2, so 5 + 4 = 2. The inverse of 5 is the number that brings it back to 0: 7 − 5 = 2, because 5 + 2 = 7, which wraps to 0.

Now find a sum or an inverse in a new group, then press Mark to check your answer in the course.

Press Mark to have your answer marked in the Group Theory course.

Who this course is for

University students starting abstract algebra or a discrete maths course, who want intuition before the proofs.
Sixth formers and high school students preparing for a maths degree who want a head start.
Curious learners who have heard group theory explains the Rubik's Cube or particle physics and want to see how.

What you need to know first

The course page lists Algebra as the course to take first. You need to be comfortable with letters standing for things and with remainders when dividing. No proofs are required, though the course shows the reasoning behind each result.

How long it takes

There are 24 sections and most take a few minutes. Learners without a plan can open two sections of a course a day, in order, so the whole course takes 12 days. A plan removes the daily limit and lets you go straight to any topic, such as Lagrange's theorem.

Guided courseCryptocurrenciesSee groups at work: elliptic curve cryptography, the maths behind digital signatures, uses a group where combining is easy but undoing is hard.

Common questions

Is this group theory course free?

You can take every section without paying, two sections a day, in order. A plan removes the daily limit and opens any section straight away.

What are the four group axioms?

Closure (combining two elements gives an element of the set), associativity ((ab)c = a(bc)), identity (an element e with ae = ea = a) and inverses (every a has an element that combines with it to give e).

Is group theory hard?

It is abstract, which is the hard part for most people. Working with concrete groups, such as the symmetries of a square and clock arithmetic, makes the definitions much easier to hold on to, and that is how this course is built.

What is a Cayley table?

A grid showing every combination in a group, with the result of a then b in row a and column b. In a group, every row and every column contains each element exactly once.

What does Lagrange's theorem say?

In a finite group, the number of elements in any subgroup divides the number of elements in the group. It follows that the order of any element divides the order of the group.

What is group theory used for?

Describing symmetry wherever it appears: solving puzzles such as the Rubik's Cube, classifying crystals, describing particles in physics, and building the cryptography that secures data online.

Revision cards

Say the answer first, then flip the card to check it.

Where to go next

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Group Theory
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