This paper asks a question that standard group theory does not: why must the axioms of group theory---and no weaker or alternative structure---govern the symmetry of any system capable of making operational distinctions? Semigroups, monoids, and non-associative magmas are not merely less convenient than groups; they are operationally inadmissible. This paper derives that inadmissibility from first principles. The group axioms (closure, associativity, identity, invertibility) are established as necessary conditions for Πd-consistent operational symmetry under the axiom system A1, A2, A4 of Operatiology, without presupposing any group-theoretic result. Invertibility is forced by the requirement that no operational step induces an irreversible collapse of distinguishability; associativity follows from the inherent associativity of function composition; identity existence follows from the finiteness of the Πd-class structure. A complete explicit construction of the six-class minimal non-commutative Πd-closure is provided from a period-3 and period-2 generator pair, with saturation depth K=2 verified by a non-circular three-step closure consistency check. S₃ emerges as the minimal finite non-commutative operational witness of this necessity. Minimality is established in two structurally independent parts: Theorem A (constructive minimality, Tier-2 internal) provides an existence witness; Theorem B (extensional extension theorem, via established series results) provides universality. This paper forms part of the Operatiology Applied Series, in which group theory is the symmetry layer of the rank-3 minimal operational closure C⁽³⁾_Πd.
T.O. (Tue,) studied this question.