semidirekti

Semidirekti is a term used in mathematics, specifically in group theory, to describe a particular way of constructing a new group from two existing groups. This construction is a generalization of the direct product of groups. Let G and H be two groups, and let there be a homomorphism from H to the automorphism group of G. This homomorphism, often denoted by a function $\phi$, defines how elements of H act on elements of G. The semidirect product of G and H, denoted by $G \rtimes_{\phi} H$, is a set of pairs $(g, h)$ where $g \in G$ and $h \in H$. The group operation is defined as follows: for two elements $(g_1, h_1)$ and $(g_2, h_2)$ in the semidirect product, their product is given by $(g_1, h_1) \cdot (g_2, h_2) = (g_1 \cdot \phi_{h_1}(g_2), h_1 \cdot h_2)$. Here, $\phi_{h_1}(g_2)$ represents the action of the element $h_1 \in H$ on the element $g_2 \in G$ as defined by the homomorphism $\phi$.

The existence of the homomorphism $\phi$ is crucial for the semidirect product to form a group. This homomorphism must satisfy certain compatibility conditions with the group operations of G and H. A key feature of semidirect products is that they are not necessarily commutative, even if the original groups G and H are. If the homomorphism $\phi$ is trivial (i.e., $\phi_h$ is the identity automorphism for all $h \in H$), then the semidirect product reduces to the direct product $G \times H$. Semidirect products are important in the study of group extensions and have applications in various areas of mathematics and physics. For example, the Poincaré group in physics can be expressed as a semidirect product.