[en] A new class of exact solutions of the Dirac equation with external electromagnetic fields is derived by assuming a set of field-dependent solution matrices which obey an algebra isomorphic to the Pauli matrices. The method of exact solution may be applied to any field having a four-vector potential. (a) If μ is an element of M/sub n/ and the coefficients of μ are not all divisible by p² then n is greater than or equal to p. (b) If μ is an element of M/sub n/ and the coefficients of μ are not all divisible by p then n is greater than or equal to 2p-1. (c) M/sub p/ contains at least p-1 elements which are linearly independent modulo p. Theorem: Let G be a group of exponent 3². Denote the nth term in the lower central series of G by G/sub n/. (a) If g is an element of G/sub n/, n is greater than or equal to 3 then g³ is an element of G/sub n+1/. (b) If g₁,g₂ is an element of G then (g₁,g₂,13) is an element of G₁₅