ABSTRACT ALGEBRA · THE LEARNING STUDIO

Explore a pattern.
Build an argument.

Groups, rings, fields and representations. Start with a concrete object; learn exactly what makes its mathematics work.

TRY A SYMMETRY

Same square · eight possible moves

Does order matter?

Rotate, then reflect. Reset and reverse the order. Track the labelled vertices.

Current transformation

e

Vertex images: A ↦ A · B ↦ B · C ↦ C · D ↦ DRecent moves in time order: none

$r^4=s^2=e,\qquad sr=r^{-1}s$
All moves preserve adjacency and distances. Composition applies the rightmost factor first; the move list records time order. Testing these eight symmetries illustrates the relations; the lesson justifies them.

YOUR NEXT STEP

Start with the foundations.

Functions and equivalence explain what a rule must do before it can define an algebraic structure.

Your demonstrated competencies

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ANOTHER WAY IN

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