Cyclicity of (Z/nZ)*: Primitive Roots Only for n = 1, 2, 4, p^k, 2p^k — E8 Intelligence Research

FINDING: The multiplicative unit group (Z/nZ)* is cyclic (has primitive roots) only for n ∈ {1, 2, 4, p^k, 2p^k} with p odd prime — this cyclicity condition is the arithmetic gatekeeper for which n admit a single generator whose powers sweep all reduced residues, directly governing rotational symmetry orders realizable in modular lattices. MATH: - (Z/nZ)* = {a mod n : gcd(a,n)=1}, |(Z/nZ)*| = φ(n) - Cyclic ⟺ ∃ primitive root g with ord_n(g) = φ(n); then (Z/nZ)* = ⟨g⟩ ≅ Z/φ(n)Z - Cyclicity theorem: (Z/nZ)* cyclic ⟺ n = 1, 2, 4, p^k, 2p^k (p odd prime) - φ(p^k) = p^(k-1)(p−1); φ(2p^k) = p^(k-1)(p−1) - Non-cyclic for n = 8, 12, 15, 16, 24, … (e.g. (Z/8Z)* ≅ Z/2 × Z/2, Klein four-group) - Structure: (Z/2^kZ)* ≅ Z/2 × Z/2^(k−2) for k ≥ 3 - CRT: (Z/mnZ)* ≅ (Z/mZ)* × (Z/nZ)* for gcd(m,n)=1 CONNECTION: - Cyclic group Z/nZ is the abstract rotation group C_n — the symmetry group of a regular n-gon. Primitive roots enumerate the generators of C_n, i.e., the rotations by k/n turns with gcd(k,n)= Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com