Multiplication Groups of 2_3_4

This example interactively demonstrates the structure of a multiplication group under modular arithmetic. It focuses on how, for a given modulus, the set of invertible (unit) residues is closed under multiplication and forms a group, making properties such as identity and inverses visible through element-to-element relations. The underlying concepts include modular multiplication, group theory (multiplicative groups modulo n), and the organization of residues into operable subsets that reflect group axioms; auxiliary themes like logarithmic/ratio reasoning can support pattern recognition of generators and reproducible relationships. Construction uses GeoGebra’s Random, Textfield, and RandomElement to generate varying inputs and dynamically pick elements from a prepared pool. The many labeled objects (e.g., A, B, C, D, G1, E, F, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, Z, along with indexed variants such as B1–B3, G2–G3, P1–P3, R1–R3) serve as visual carriers for residues, labels, and the resulting multiplication/connection outcomes. Suitable for high-school advanced learners and undergraduate early abstract algebra or number theory modules, especially for intuitive exploration of “units modulo n form a group.”