# nLab wreath product of groups

## Definition

Let $H\leq Sym(X)$ and $G\leq Sym(Y)$ be permutation groups. Then their wreath product $H \wr G$ is defined as the semidirect product $H^Y \rtimes G$ where $G$ operates on $H^Y$ by permuting the components. Note that this is itself a permutation group, acting on $X\times Y$ by letting $G$ act trivially on the first and naturally on the second factor and letting $H^Y$ act on $X\times Y$ such that the $y$-th component of $H^Y$ permutes $X\times\{y\}$ naturally and fixes everything else pointwise.

## Examples

• Every group $G$ is a permutation group on its underlying set $|G|$ (the regular representation aka the Cayley-Yoneda embedding $G\hookrightarrow Sym(|G|)$) and often only this special case is considered so that people speak of “the wreath product $H \wr G$” for abstract groups $G$ and $H$ without mentioning their permutation representations.
• $Dih(4)$ the dihedral group of order $2\cdot 4=8$ is the wreath product $C_2 \wr C_2$ (in this sense of regular representations).
• The sylow-$p$-subgroups $P_n$ of the symmetric groups $Sym(n)$ can be recursively described as the wreath product $C_p \wr P_a$ where $C_p$ is the cyclic group of order $p$ and $n=ap+r$ with $0\leq r \lneq p$.
• The sylow-$\ell$-subgroups of $GL_n(q)$ for $gcd(q,\ell)=1$ can be recursively described as a wreath product of the sylow-$\ell$-subgroup of a strictly smaller $GL_{m}(q)$ and a sylow-$\ell$-subgroups of a suitable symmetric group.

## Relation to the Borel construction

The homotopy quotient described in the Borel construction,

$(X //G)_\bullet \simeq_{iso} X \times_G (E G)_\bullet \,,$

has the wreath product $\pi_1(X)\wr G$ for fundamental group.

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