$G$Groups and Fields

A Gallery of Groups

Clocks you add on, clocks you multiply on, a spinning hexagon and a handful of matrices — and the one number that describes every element: its order.

Hunting Subgroups

Find the groups hiding inside groups, see which elements generate everything, and draw a map of every subgroup of a cyclic group.

Shuffles and Cycles

Build permutations with arrows, read them as cycles, compose them, sort them into even and odd, and find out why a magician needs exactly eight perfect shuffles.

Slicing Groups into Cosets

Shift a subgroup around and watch the whole group fall apart into equal slices. One picture proves Lagrange's theorem, Fermat's little theorem and more.

Maps That Keep the Structure

Two groups can be the same group in disguise. Learn to spot the disguise, to prove it with a map, and to unmask an impostor without trying every map.

When Cosets Form a Group

Collapse a group by a subgroup and watch its table shrink into a smaller table. It only works for normal subgroups, and you will see exactly why.

Building Groups from Pieces

Glue groups side by side, walk on a torus to see when the result is cyclic, solve an ancient remainder puzzle, and count every abelian group of a given size.

Counting with Symmetry

Let a group move things around, sort the results into orbits, and count necklaces, bracelets and painted cubes without listing them all.

Elements in Disguise

Conjugation shows you a move from someone else's point of view. Sort elements into look-alike families, find the class equation, and prove things about every group of order 27.

Sylow Detective

Lagrange says which subgroup sizes are allowed. Sylow says which ones must exist, and counting them cracks open groups you have never even seen.

Two Operations at Once

Add AND multiply: rings, the numbers you can divide by, why clock arithmetic becomes a field exactly when the clock has a prime number of hours, and why book codes use 11.

Polynomials That Will Not Factor

Find roots by trying every element of a finite field, hunt for polynomials that refuse to factor, fall into a classic trap, and meet the tests that work over the rationals.

Building New Numbers

Invent a root for a polynomial that has none, get a brand-new field, measure how big it is, and use the tower law to prove numbers can never meet.

Impossible Constructions

Collect every root of a polynomial into one field, meet the roots of unity, and settle three problems the ancient Greeks could not — with one line of arithmetic each.

Fields with Eight Elements

Every finite field has a prime-power size, its nonzero elements march round a single circle, multiplying becomes adding, and your phone uses one with 256 elements.

The Symmetries of Equations

Where the two halves of the course meet: every field extension has a symmetry group, its subgroups ARE the fields in between, and that is why the quintic has no formula.