# Wreath product

The wreath product of two groups and is constructed in the following way. Let be the set of all functions defined on with values in . With respect to componentwise multiplication, this set is a group which is the complete direct product of copies of ( denotes the number of elements in ); acts on as a group of automorphisms in the following way: if , , then for . With respect to this operation, one can form the semi-direct product of and , that is, the set of all pairs , where , , with multiplication operation given by The resulting group is called the Cartesian (or complete) wreath product of and , and is denoted by (or , a denotation of Ph. Hall). If instead of one takes the smaller group consisting of all functions with finite support, that is, functions taking only non-identity values on a finite set of points, then one obtains a subgroup of called the wreath product (direct wreath product, discrete wreath product) of and ; it is denoted by (or ). Both wreath products are widely used for constructing various examples of groups.