The reduced notation for irreducible representations of the symmetric group S-n is interpreted in terms of symmetric formal series and vertex operators, and is used to prove a number of properties of reduced Kronecker products and inner plethysms in an n-independent manner. Conditions for self-associativity of Kronecker products and inner plethysms are established. Reduced inner plethysms are developed and applied to the question of non-simple phase groups among the symmetric S-n and alternating A(n) groups.