Equivalence Classes of Minimal Zero‐Sequences Modulo a Prime

Let G be a finite abelian group and 𝒰 (G) the set of minimal zero‐sequences on G. If M ₁ and M ₂ ∈ 𝒰 (G), then set M ₁ ∼ M ₂ if there exists an automorphism ϕ of G such that ϕ (M ₁) = M ₂. Let ℰ (M) represent the equivalence class of M. In this paper, we consider problems related to determining the possible cardinalities, 𝒞 (ℰ (M)), of equivalence classes in ℤ _p . We characterize which divisors of p − 1 represent the size of an equivalence class in terms of the cycle decompositions of the automorphisms of ℤ _p and investigate properties of related distinct‐element minimal zero‐sequences. We close by counting the number of equivalence classes and the total number of minimal zero‐sequences in ℤ _p for certain M ∈ 𝒰 (ℤ _p ) with distinct elements.