Dominant Phase Phenomena for p-Subgroups of GLₙ(F_q)

We study the enumeration of subgroups of a finite group G isomorphic to a fixed group H through an energy–normalizer framework. Assigning to each realized subgroup K ≤ G the Lyapunov energy E(K) = log |N_G(K)|, we obtain an exact partition-function identity for the subgroup counting function. We introduce a Dominant Phase Operator showing that when phase-to-phase normalizer ratios vary sharply, the partition sum is controlled by a narrow equilibrium window. The framework is instantiated for cyclic and elementary abelian p-subgroups of GL_n(F_q) in the semisimple regime p divides (q − 1), where phases are indexed by eigenspace multiplicities and dominance occurs at maximal-entropy configurations. Connections to stabilizer rigidity and barrier-style analyses of subgroup growth are briefly discussed.