Group Theory

Abstract Algebra & Group Theory Visualizer

Parameters

Group preset

n-gon sides n: 4

Modulus n a b

Lagrange's Theorem

(a+b) mod n—

(a·b) mod n—

a⁻¹ mod n—

aᵇ mod n—

Group Theory & Symmetries Handbook

Mathematical Principles & Theorems

A group \((G, \cdot)\) satisfies Closure, Associativity, Identity \(e\), and Inverses \(g^{-1}\). If \(ab=ba\), the group is Abelian. Lagrange's Theorem: For any finite group \(G\) and subgroup \(H \le G\), the order \(|H|\) divides \(|G|\). Explores Cyclic groups \(\mathbb{Z}_n\), Dihedral groups \(D_n\) of order \(2n\) (symmetries of regular \(n\)-gons), and Symmetric permutation groups \(S_n\) of order \(n!\). Cayley tables display the full binary operation matrix.

Operating Instructions

Select group family: Cyclic \(\mathbb{Z}_n\), Dihedral \(D_n\), Klein 4-Group \(V_4\), or Symmetric \(S_3\).
Set generator parameter \(n\) using the precise numeric value box.
Examine the computed Cayley color table showing group multiplication structures.
Inspect subgroup generator orbits, element orders, and inverse pairs in the analysis cards.