Learn abstract algebra by breaking it

No prior algebra needed. Start by clicking on a grid.

Abstract algebra is usually taught as definitions you are asked to accept before you have seen anything they describe. This site does the reverse. You get a small grid of numbers, you change it, and software tells you what you have just built — and what you have just broken.

The whole subject sits on one idea. Pick a few things, and a rule that combines any two of them into a third. Write out what every pair gives you. That grid is the mathematical object; everything else is noticing which patterns the grid does and does not have.

START HERE

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The Lab — change a rule, see what it becomes
A grid showing what every pair of elements combines to. Click any square to change its answer. The panel beside it re-decides, every time, what kind of object you now have — and when a law fails it names the exact case that broke it, rather than just saying no. Begin with rock-paper-scissors, which you already understand, and take it apart.

WHAT YOU WILL LEARN

Five questions, asked of any rule. Each "yes" is a rung on a ladder:

  1. Closed — does combining two of your things always give you back one of your things?
  2. Associative — does it matter where you put the brackets?
  3. Identity — is there one element that leaves everything else alone?
  4. Inverses — can every move be undone?
  5. Commutative — does the order of the two inputs matter?

Those answers have names. A rule that is merely closed is a magma. Add associativity and it is a semigroup; add an identity, a monoid; add inverses, a group — the structure behind symmetry, and the reason this subject shows up in physics, cryptography and machine learning. You will meet each name only after you have already needed it.

Two routes from magma to group, each arrow labelled with the law that earns it
Where you are heading, and the law that earns each step. Two different routes reach a group. This diagram is checked, not drawn — every box holds a real grid, and a program verifies each one lands where the picture claims.

BEING BUILT

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The course — guided, one small step at a time
Read two sentences, edit a line of code, press Run, and have the answer checked — the way SQLZoo teaches SQL. The interpreter and the grading already work (see below); the lessons are being written now.
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Runtime check — the machinery, if you are curious
Not a lesson. It proves the real sw-MLPL interpreter runs inside this page, loads the same algebra library the command-line tests use, and grades submitted answers. Ten checks; watch them pass.