In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol ⋉. There are two closely related concepts of semidirect product:

在數學中，特別抽象代數裏的群論中，半直積（英語： semidirect product ）是從其中一個是正規子群的兩個子群形成一個群的特定方法。半直積是直積的推廣。半直積是作為集合的笛卡爾積，但帶有特定的乘法運算。

Nov 23, 2023 · A semi-direct product is a particular case of an extension of a group $B$ by a group $A$ (cf. Extension of a group); such an extension is called split. The semi-direct product …

在数学中，特别是叫做群论的抽象代数领域中，半直积(semidirect product)是从其中一个是正规子群的两个子群形成一个群的特定方法。 半直积是直积的推广。

A semidirect product is a groups having two complementary subgroups one of which is normal. Definition 10.1. Two subgroups H; K · G are called complementary (to each other) if. For example, if G is a group of order paqb then the p-Sylow subgroup P and q-Sylow subgroup Q are complementary.

Mar 5, 2025 · Semidirect Product A "split" extension of groups and which contains a subgroup isomorphic to with and (Ito 1987, p. 710). Then the semidirect product of a group by a group , denoted (or sometimes ) with homomorphism is given by

Mar 19, 2016 · The semi-direct product is a simple way to really mix two groups together. Let's consider matrices. If we have two groups of square matrices (with the same dimension), say $G$ and $H$, then if we were to multiply elements, we'd have $g_1 h_1 g_2 h_2$.

正如上篇笔记所说，构造群（一般指从已知的、较小的群构造较大的群）的两种办法，其中一个是构造群的直积，而另一个是半直积(semidirect product)，这是仅出现在群这一结构中的方法，但在构造群时的要求较弱，且与…

Thus we have a semidirect product G= HoK: Which semidirect product is it? It can, of course be the direct product (this always exists, and corresponds to the case where the homomorphism ˚: K!Aut(H) is the trivial homomorphism). This gives G˘=Z pq. But if a non-trivial homomorphism ˚: K!Aut(H) exists then we also get a non-abelian group G= Ho

What is the difference between a direct product and a semi-direct product in group theory? Based on what I can find, difference seems only to be the nature of the groups involved, where a direct product can involve any two groups and the semi-direct product only allows a normal subgroup $N$ of some group $G$ and another subgroup of $G$ that ...