Three-factor decompositions of ${\mathbb{U}}_n$ with the three generators in arithmetic progression

Irrespective of whether n is prime, prime power with exponent >1, or composite, the group U_n of units of Z_n can sometimes be obtained as the direct product of cyclic groups generated by x, x+k and x+2k, for x, k in Z_n. Indeed, for many values of n, many distinct 3-factor decompositions of this type exist. The circumstances in which such decompositions exist are examined. Many decompositions have additional interesting properties. We also look briefly at decompositions of the multiplicative groups of finite fields.