Lecture Notes on Bayesian Estimation and Classification
《Lecture Notes on Bayesian Estimation and Classification》 · 第 32—33 页 · Section 1.6「Permutation Groups and Group Actions」，小节「The Wreath Product of Permutation Groups」
Let $H$ and $K$ be permutation groups acting on sets $X$ and $Y$ respectively. We shall describe a very important way of constructing a new permutation group called the wreath product of $H$ and $K$. This is to act on the set product $Z=X\times Y$. If $\gamma\in H$, $y\in Y$, and $\kappa\in K$, define permutations $\gamma(y)$ and $\kappa^*$ of $Z$ by the rules $\gamma(y):(x,y)\mapsto(x\gamma,y)$, $(x,y_1)\mapsto(x,y_1)$ if $y_1\neq y$, and $\kappa^*:(x,y)\mapsto(x,y\kappa)$. […] The functions $\gamma\mapsto\gamma(y)$, with $y$ a fixed element of $Y$, and $\kappa\mapsto\kappa^*$ are monomorphisms from $H$ and $K$ to $\operatorname{Sym}Z$: let their images be $H(y)$ and $K^*$, respectively. Then the wreath product of $H$ and $K$ is the permutation group on $Z$ generated by $K^*$ and all the $H(y)$, $y\in Y$.