wreath product
Let A and B be groups, and let B act on the set Γ.
Define the action of B on the direct product AΓ by
bf(γ):=f(b-1γ), |
for any f∈AΓ and γ∈Γ. The wreath product of A and B according to the action of B on Γ, denoted A≀ΓB, is the semidirect product of groups AΓ⋊.
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 |