Operational Mathematics of Group Operations: Extending the Iteration Count to the Complex Domain

This paper systematically transplants the core methodology of Operational Mathematics—the extension of the repetition count of fundamental operations from natural numbers to integers, rational numbers, real numbers, and ultimately complex numbers onto a new class of binary operations: the group operation σGn(g,z) and its inverse operation σG−1n (g,z). By virtue of the defining algebraic properties of a group—associativity, the existence of a unique identity element, and the uniqueness of inverses—the theory acquires a profound simplification and a rich algebraic–analytic structure absent in all previous Operational Mathematics frameworks. Acomplete system of seven axioms is established; integer-order, fractional-order, real order, and complex-order iterations are rigorously defined, and the existence of iterative roots at each level is proved by means of an explicitly solvable Schröder equation (conjugated to the logarithm map) and an Abel equation naturally provided by the exponential map of Lie groups. Uniqueness theorems under natural regularity conditions (a group adapted logarithmic convexity) are provided. The singularity structure of complex-order group iterations is analyzed in depth, revealing a fundamentally novel phenomenon determined by the torsion subgroup and the fundamental group: branch points are of mixed algebraic type (from finite-order torsion elements) and logarithmic type (from infinite-order elements and non-trivial coverings), together with rational poles in quantum group representations. The natural boundary phenomenon is completely characterized via the rank of the period lattice. A fundamental structural discovery is rigorously proved: the group operational hierarchy collapses completely for all levels n ≥ 2, leaving only the base operations at level n = 1 and the collapsed family at level n = 2. In the weighted parameterisation, a necessary and sufficient condition for breaking this collapse is established, giving rise to a strictly increasing hierarchy whose structure is governed by a differential Galois group. Fractional calculus and the calculus of variations with a group kernel are shown to be special cases of the group operational framework, thereby unifying discrete group hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of group operations is established, yielding a field isomorphism between the group hyperfield and the complex numbers. The theory is extended to quantum groups, p-adic Lie groups, arithmetic groups with modular forms, and infinite-dimensional Kac–Moody groups. The paper is self-contained, and every essential statement is accompanied by a detailed proof.