TITLE:
Comprehensive Analysis of Complex Products in Groups: From Elementary Combinatorics to Advanced Structural Theory
AUTHORS:
Anton Goncear
KEYWORDS:
Complex Products, Group Theory, Commutator Analysis, Product Polytopes, Computational Complexity, Permutation Invariance
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.15 No.11,
November
21,
2025
ABSTRACT: This extensive research paper provides a thorough investigation of the algebraic structure of complex products in groups, with comprehensive analysis of both commutative and non-commutative settings. We begin with fundamental definitions and progress to advanced theorems connecting product combinatorics with subgroup structure, representation theory, and computational complexity. The work includes detailed proofs of classical results, several original contributions including: 1) a complete characterization of permutation invariance through commutator analysis, 2) new quantitative measures of non-commutativity via commutator discrepancies, 3) geometric interpretations through product polytopes, and 4) polynomial-time algorithms for product equivalence problems, along with connections to open problems in modern algebra. Each section builds systematically upon previous results, creating a cohesive mathematical narrative that demonstrates deep understanding of group-theoretic structures.