Definitions and group structure -- The duality transformation -- Duality relation for the correlation functions -- Phase transitions with spontaneous symmetry breakdown. Ergodic decomposition -- Series and cluster expansion an application of universality hypothesis a “generalized” droplet-model -- The partial trace transformation. Equation for the correlation functions and representation of the symmetry group -- Invariant equilibrium states and duality transformation for infinite systems -- Asano contraction and group structure. Analyticity properties of the free energy -- Analyticity and uniqueness of the invariant equilibrium state -- Definitions and group structure for systems with constraints -- Expansions for the partition function -- Partial trace method and equilibrium equations -- Duality transformation restricted to finite bonds -- Asano contractions and unicity of state -- General framework of higher spin systems -- Physical implications of the group structure -- Spin 1 lattice systems -- The duality transformation -- Zeroes of the partition function.