Abstract Algebra Lab

Parameters

Group size n: 4

Modulus n: 7

Rotation: 45°

Row a: 1

Col b: 2

Abstract Algebra Explorer – Groups, Rings & Fields

Abstract algebra studies mathematical structures defined by sets and operations that satisfy specific axioms, generalizing arithmetic to far broader contexts. In this explorer you can construct a group from a chosen set and binary operation, verify closure, identity, inverses, and associativity, and visualize the structure through its Cayley table. Moving from groups to rings and fields reveals how each layer of additional axioms unlocks new mathematical power.

What you can do in this simulation

Build a group from integers mod n and verify all four group axioms interactively
Generate and read the Cayley table for any finite group you construct
Identify cyclic subgroups and explore Lagrange's theorem on subgroup orders
Construct rings by adding a second operation and test distributivity
Compare the symmetry groups of regular polygons side by side

Concepts covered

abstract algebra · group theory · Cayley table · rings · fields · symmetry groups

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