Cyclicity of (Z/nZ)* and Its Role in Lattice Rotational Symmetry — E8 Intelligence Research

FINDING: The multiplicative unit group (Z/nZ)* is cyclic iff n ∈ {1, 2, 4, p^k, 2p^k} (p odd prime); its order is φ(n), and its cyclic structure directly governs the rotational symmetry of the corresponding lattice/root system. MATH: - (Z/nZ)* cyclic ⟺ n = 1, 2, 4, p^k, 2p^k (Gauss). Order = φ(n) = n·∏(1−1/p). - Generator count = φ(φ(n)); e.g. φ(20)=8, (Z/20Z)* ≅ Z₂×Z₄ (not cyclic — 20 = 4·5 fails the 2p^k form since 4 is not 2·odd-prime-power in the required sense). - Subgroup lattice of Z_n is isomorphic to the divisor lattice of n: subgroups ↔ divisors d|n, with index n/d. This is a distributive lattice, order-isomorphic to the divisor poset. - Root lattices: A_n, D_n, E_6, E_7, E_8; Weyl group W(E_8) has order 696,729,600 = 2^14·3^5·5^2·7. |W(E_8)| / |W(F_4)| = 696729600/1152 = 604800 = 7!. - E_8 kissing number = 240 = 2·120; Leech lattice kissing number = 196560 = 240·819 = 240·(2^13·... ) — 196560 = 2^4·3^3·5·7·13. - Monster group order = 2^46·3^20·5^9·7^6·11^2·13^3·17·19·23·29· Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com