摘要：

The factorization problem of groups, known as the dual counterpart to the renowned extension problem in group theory,has garnered significant interest owing to its profound ties to various fields, including group theory, combinatorics, and Hopf algebra. This presentation aims to provide a succinct overview of these intricate interconnections, placing particular emphasis on the classification of exact factorizations with a cyclic or dihedral factor. Furthermore, we will explore its practical implications in the realms of regular Cayley maps and the theory of skew morphisms.