Group Theory

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.