Lecture Notes on Bayesian Estimation and Classification
《Lecture Notes on Bayesian Estimation and Classification》 · 第 326 页 · Section 11.1「Group Extensions and Covering Groups」，Exercise 11
Let $N$ and $G$ be arbitrary groups. For each $x$ in $G$ let $N_x$ be a group isomorphic with $N$ via a map $a\mapsto a_x$. Write $B=\operatorname{Cr}_{x\in G}N_x$, the cartesian product. If $b\in B$ and $g\in G$, define $b^g$ by the rule $(b^g)_x=b_{xg^{-1}}$. Show that this action of $G$ on $B$ leads to a semidirect product $W=G\ltimes B$. (Here $W$ is called the standard complete wreath product $N\wr G$ and $B$ is the base group of $W$.)