PlanetMath: wreath product

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wreath product

(Definition)

Let and be groups, and let act on the set $\Gamma$ . Define the action of on the direct product $A^{\Gamma}$ by

$\displaystyle b f(\gamma) := f(b^{-1}\gamma),$

for any $f\in A^{\Gamma}$ and $\gamma\in\Gamma$ . The wreath product of

and

according to the action of

on $\Gamma$ , denoted $A\wr_{\Gamma} B$ , is the semidirect product of groups $A^{\Gamma}\rtimes B$ .

Let us pause to unwind this definition. The elements of $A\wr_{\Gamma}B$ are ordered pairs , where $f\in A^{\Gamma}$ and $b\in B$ . The group operation is given by

$\displaystyle (f,b)(f',b') = (fbf', bb').$

Note that by definition of the action of

on $A^{\Gamma}$ ,

$\displaystyle (fbf')(\gamma) = f(\gamma)f'(b^{-1}\gamma).$

The action of on $\Gamma$ in the semidirect product permutes the elements of a tuple $f\in A^{\Gamma}$ , and the group operation defined on $A^{\Gamma}$ gives pointwise multiplication. To be explicit, suppose $\Gamma$ is an -tuple, and let $(f,b),~(f',b')\in A\wr_{\Gamma} B$ . Let denote $b^{-1}(i)$ . Then

$\displaystyle (f,b)(f',b')$	$\displaystyle =$	$\displaystyle \bigl((f(1),~f(2),~\ldots,~f(n)),~b\bigr) \bigl((f'(1),~f'(2),~\ldots,~f'(n)),~b'\bigr)$
	$\displaystyle =$	$\displaystyle \bigl((f(1),~f(2),~\ldots,~f(n)\bigr)\bigl(f'(b_1),~f'(b_2),~\ldots,~f'(b_n)\bigr),~bb')$ (*)
	$\displaystyle =$	$\displaystyle \bigl((f(1)f'(b_1),~f(2)f'(b_2),~\ldots,~f(n)f'(b_n)),~bb'\bigr).$

Notice the permutation of the indices in (*).

A moment's thought to understand this slightly messy notation will be illuminating, and might also shed some light on the choice of terminology.

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"wreath product" is owned by mps. [ full author list (2) | owner history (1) ]

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Cross-references: indices, multiplication, tuple, group operation, ordered pairs, semidirect product of groups, direct product, action, act on, groups
There is 1 reference to this entry.

This is version 15 of wreath product, born on 2002-10-03, modified 2005-12-31.
Object id is 3505, canonical name is WreathProduct.
Accessed 11867 times total.

Classification:

AMS MSC:

20E22 (Group theory and generalizations :: Structure and classification of infinite or finite groups :: Extensions, wreath products, and other compositions)

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