Multiplicative atom decomposition of sets of exceptional units in residue class rings

Abstract Given the multiplicative group Z n ⁎ of units in the ring Z n : = Z / n Z , let Z n ⁎ ⁎ denote the set of exceptional units in Z n , i.e. units u ∈ Z n ⁎ satisfying 1 − u ∈ Z n ⁎ . A subset of a finite group G containing all generators of any (cyclic) subgroup of G is called an atom of G . Let A n ⁎ denote the set of all atoms of Z n ⁎ . By means of A n ⁎ ⁎ : = { A ∈ A n ⁎ : A ⊂ Z n ⁎ ⁎ } , the set Z n ⁎ ⁎ trivially decomposes into atoms, i.e. Z n ⁎ ⁎ = ⋃ A ∈ A n ⁎ ⁎ as a disjoint union. An explicit construction of that atom decomposition is easily obtained if n is a prime power. We characterise so-called tame integers, i.e. odd n > 1 with prime factorisation n = ∏ p i k i , say, for which the atom decomposition of Z n ⁎ ⁎ is obtained by multiplicative composition of the atom decompositions of the Z p i k i ⁎ ⁎ . Moreover, it is shown that the set of tame integers has density zero.