Editing Wreath product (section)

== Definition ==

Let <math>A</math> be a group and let <math>H</math> be a group [[Group action (mathematics)|acting]] on a set <math>\Omega</math> (on the left). The [[Direct product of groups|direct product]] <math>A^{\Omega}</math> of <math>A</math> with itself indexed by <math>\Omega</math> is the set of sequences <math>\overline{a} = (a_{\omega})_{\omega \in \Omega}</math> in <math>A</math>, indexed by <math>\Omega</math>, with a group operation given by pointwise multiplication. The action of <math>H</math> on <math>\Omega</math> can be extended to an action on <math>A^{\Omega}</math> by ''reindexing'', namely by defining

: <math> h \cdot (a_{\omega})_{\omega \in \Omega} := (a_{h^{-1} \cdot \omega})_{\omega \in \Omega}</math>

for all <math>h \in H</math> and all <math>(a_{\omega})_{\omega \in \Omega} \in A^{\Omega}</math>.

Then the '''unrestricted wreath product''' <math>A \text{ Wr}_{\Omega} H</math> of <math>A</math> by <math>H</math> is the [[semidirect product]] <math>A^{\Omega} \rtimes H</math> with the action of <math>H</math> on <math>A^{\Omega}</math> given above. The subgroup <math>A^{\Omega}</math> of <math>A^{\Omega} \rtimes H</math> is called the '''base''' of the wreath product.

The '''restricted wreath product''' <math>A \text{ wr}_{\Omega} H</math> is constructed in the same way as the unrestricted wreath product except that one uses the [[Direct sum of groups|direct sum]] as the base of the wreath product. In this case, the base consists of all sequences in <math>A^{\Omega}</math> with finitely many non-[[identity element|identity]] entries.  The two definitions coincide when <math>\Omega</math> is finite.

In the most common case, <math>\Omega = H</math>, and <math>H</math> acts on itself by left multiplication. In this case, the unrestricted and restricted wreath product may be denoted by <math>A \text{ Wr } H</math> and <math>A \text{ wr } H</math> respectively. This is called the '''regular''' wreath product.