The Identity Transform of a Permutation and its Applications

Starting from a Theorem by Hall, we define the identity transform of a permutation π as C(π) = (0+π(0), 1+π(1), ⋯, (n-1)+π(n.1)), and we define the set Cn = (C(π) : π ∈ Sn, where Sn is the set of permutations of the elements of the cyclic group ℤn. In the first part of this paper we study the set Cn: we show some closure properties of this set, and then provide some of its combinatorial and algebraic characterizations and connections with other combinatorial structures. In the second part of the paper, we use some of the combinatorial properties we have determined to provide a different algorithm for the proof of Hall's Theorem.