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 Aω := A indexed by the set Ω. The elements of K can be seen as arbitrary sequences (aω) 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 WrΩ H of A by H is the semidirect product K ⋊ H. The subgroup K of A WrΩ H is called the base of the wreath product.
The restricted wreath product A wrΩ 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 (aω) of elements in A indexed by Ω of which all but finitely many aω 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.
Read more about this topic: Wreath Product
Famous quotes containing the word definition:
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)