CYCLE TYPES OF LINEAR SUBSTITUTIONS OVER FINITE COMMUTATIVE RINGS

The problem of describing the lengths of the independent cycles in the indicated substitution reduces to the case where the ring and characteristic polynomial of the corresponding matrix are primary. The concept of a distinguished polynomial over a primary (local) ring R is introduced and studied. These polynomials are used to obtain formulas for the cycle types of linear substitutions that generalize known formulas for the case where R is a field. If R is a principal ideal ring, the formulas are practically computable. In the case where R is a Galois ring, there are given a complete description of the linear substitutions of maximal order and an algorithm for enumerating the cycles in such substitutions. Estimates of the exponents of the full linear group over a local ring and its congruence subgroup are given.Bibliography: 17 titles.