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.

(…)

References

Some general treatments

• W. Charles Holland, The characterization of generalized wreath products, J. Alg. 13 (1989) 152–172 pdf

Important special cases/applications:

• G. Baumslag, Wreath products and extensions, Math Z 81, 286–299 (1963) doi
• I. G. Macdonald, Polynomial functors and wreath products, J. Pure Appl. Alg. 18 (1980) 173–204 pdf

For semigroups:

• Charles Wells, Some applications of the wreath product construction, Amer. Math. Monthly 83:5 (1976) doi

category: algebra