Quotients of skew morphisms of cyclic groups

A skew morphism of a finite group B is permutation φ that preserves the identity element and has property for every ∈ there exists positive integer ia such φ(ab) = φ(a)φia(b) all b B. The problem classifying morphisms cyclic groups notoriously hard, with no classification available up to date. Each Zn closely related specific Z|⟨φ⟩|, called quotient φ. In this paper, we use relationship other observations prove new theorems about groups. particular, classify order 2em e {0, 1, 2, 3, 4} m odd square-free. We also develop an algorithm finding groups, implement in MAGMA obtain census 161. During preparation paper noticed few flaws Section 5 Cyclic complements [J. Algebra 453 (2016), 68–100]. propose weaker versions problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 Corollary 5.7), show our modifications can be used fix consequent proofs (in aforementioned paper) at least one those assertions.