Editing Wreath product (section)

== Properties ==

=== Agreement of unrestricted and restricted wreath product on finite Ω ===
Since the finite direct product is the same as the finite direct sum of groups, it follows that the unrestricted ''A''&nbsp;Wr<sub>Ω</sub>&nbsp;''H'' and the restricted wreath product ''A''&nbsp;wr<sub>Ω</sub>&nbsp;''H'' agree if Ω is finite. In particular this is true when Ω = ''H'' and ''H'' is finite.

=== Subgroup ===
''A''&nbsp;wr<sub>Ω</sub>&nbsp;''H'' is always a [[subgroup]] of ''A''&nbsp;Wr<sub>Ω</sub>&nbsp;''H''.

=== Cardinality ===
If ''A'', ''H'' and Ω are finite, then
:: |''A''≀<sub>Ω</sub>''H''| = |''A''|<sup>|Ω|</sup>|''H''|.<ref>Joseph J. Rotman, An Introduction to the Theory of Groups, p. 172 (1995)</ref>

=== Universal embedding theorem ===
{{Main|Universal embedding theorem}}
''Universal embedding theorem'': If ''G'' is an [[Group extension|extension]] of ''A'' by ''H'', then there exists a subgroup of the unrestricted wreath product ''A''≀''H'' which is isomorphic to ''G''.<ref>M. Krasner and L. Kaloujnine, "Produit complet des groupes de permutations et le problème d'extension de groupes III", [[Acta Sci. Math.]] 14, pp. 69–82 (1951)</ref> This is also known as the ''Krasner–Kaloujnine embedding theorem''. The [[Krohn–Rhodes theorem]] involves what is basically the semigroup equivalent of this.<ref name="Meldrum1995">{{cite book|author=J D P Meldrum|author-link=John D. P. Meldrum|title=Wreath Products of Groups and Semigroups|year=1995|publisher=Longman [UK] / Wiley [US]|isbn=978-0-582-02693-3|page=ix}}</ref>