External wreath product with diagonal action
This article describes a product notion for groups. See other related product notions for groups.
This definition uses the left action convention.
Let be any abstract group and be a group along with a homomorphism for some set (in other words, we are given a permutation representation, or group action, of ). Let be a group along with a homomorphism .
Then, the external wreath product with diagonal action of with as the wreathing group (relative to ) and as the diagonally acting group is defined as the external semidirect product of the group by , where a pair sends to the function .
The group is termed the base of the wreath product with diagonal action.
The wreath product with diagonal action can be viewed as follows: consider the group obtained as a direct product of copies of labeled by the elements of . The wreath product with diagonal action is the semidirect product of by the action of , where acts by automorphisms on each coordinate via , and permutes the coordinates via the action on .