Wreath Product - Definition

Definition

Let A and H be groups and Ω a set with H acting on it. Let K be the direct product

of copies of A_ω := A indexed by the set Ω. The elements of K can be seen as arbitrary sequences (a_ω) of elements of A indexed by Ω with component wise multiplication. Then the action of H on Ω extends in a natural way to an action of H on the group K by

.

Then the unrestricted wreath product A Wr_Ω H of A by H is the semidirect product K ⋊ H. The subgroup K of A Wr_Ω H is called the base of the wreath product.

The restricted wreath product A wr_Ω H is constructed in the same way as the unrestricted wreath product except that one uses the direct sum

as the base of the wreath product. In this case the elements of K are sequences (a_ω) of elements in A indexed by Ω of which all but finitely many a_ω are the identity element of A.

The group H acts in a natural way on itself by left multiplication. Thus we can choose Ω := H. In this special (but very common) case the unrestricted and restricted wreath product may be denoted by A Wr H and A wr H respectively. We say in this case that the wreath product is regular.