Let
$$
B=\bigoplus_{g\in G}H_g
$$
be the group of finitely supported functions $f:G\to H$ with pointwise multiplication. The left-translation action
$$
(g\cdot f)(x)=f(g^{-1}x)
$$
defines the restricted <wreath product>
$$
\boxed{H\wr G=B\rtimes G.}
$$

Suppose $X$ and $Y$ are finite generating sets for $G$ and $H$. Embed each $y\in Y$ as a lamp supported at the identity of $G$. Conjugating these lamps by words in $X$ produces copies of $Y$ at every coordinate, and these copies generate $B$. Thus $X$ together with the identity-coordinate copy of $Y$ is a finite generating set for $H\wr G$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 5 / a / Solution

{title2=$H\wr G$}
{wiki}

The restricted wreath product of groups $H$ and $G$ is
$$
H\wr G=\left(\bigoplus_{g\in G}H_g\right)\rtimes G,
$$
where $G$ acts on the finitely supported functions $G\to H$ by translation.


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