wreath product
Let and be groups, and let act on the set . Define the action of on the direct product by
for any and . The wreath product of and according to the action of on , denoted , is the semidirect product of groups .
Let us pause to unwind this definition. The elements of are ordered pairs , where and . The group operation is given by
Note that by definition of the action of on ,
The action of on in the semidirect product permutes the elements of a tuple , and the group operation defined on gives pointwise multiplication. To be explicit, suppose is an -tuple, and let . Let denote . Then
Notice the permutation of the indices in (*).
A bit amount of thought to understand this slightly messy notation will be illuminating, and might also shed some light on the choice of terminology.
Title | wreath product |
---|---|
Canonical name | WreathProduct |
Date of creation | 2014-12-30 11:23:20 |
Last modified on | 2014-12-30 11:23:20 |
Owner | mps (409) |
Last modified by | juanman (12619) |
Numerical id | 21 |
Author | mps (12619) |
Entry type | Definition |
Classification | msc 20E22 |
Defines | wreath product |