semidirectly

Semidirectly refers to a type of product or relationship where one entity has a significant but not complete control over another. This concept is often used in mathematics, particularly in group theory, to describe a specific kind of group extension. In a semidirect product, a group G is extended by another group H, where H acts on G through automorphisms. This means that H does not necessarily act trivially on G, but it does not completely determine the structure of G either. The semidirect product is denoted as G ⋊ H, and it is a group that contains both G and H as subgroups. The semidirect product is a generalization of the direct product, where the action of H on G is trivial. Semidirect products are important in various areas of mathematics and have applications in physics and computer science. They provide a way to construct new groups from existing ones and to study the relationships between different groups.