From: magidin@math.berkeley.edu (Arturo Magidin)
Subject: Re: wreath product
Date: 20 May 2000 15:50:11 GMT
Newsgroups: sci.math
Summary: [missing]

In article <8g5u3i$1as$1@shiva.neobee.net>,
Vladimir Lazic <lazicv@verat.net> wrote:
>Would anyone be so kind to tell me what the wreath product is? I know only
>it is some kind of generalization of Cartesian product. Thanks in advance.

For groups, or semigroup, or what?

For groups:

Take two groups, N and K. Consider the direct product of |K| copies of
N (indexed by the elements of K). Equivalently, consider the
collection of all (set) maps from K to N.

This is a group by coordinatewise multiplication; i.e., if f and g are
set maps from K to N, then fg is a set map from K to N, sending k to
f(k)g(k) (the product being done in N).

Now K acts on the set maps of K to N. Given a k in K and a set map
f:K->N, we let f^k be the set map that sends k' to f(kk'). If you
think of N^|K|, you are shifting your coordinates by k.

Since you have an action of K on N^|K|, you can form the semidirect
product of these groups. That's the standard (unrestricted) wreath
product. It contains as subgroups any extension of N by K.

If instead of taking all set maps you take all set maps that are
trivial almost everywhere, you obtain the restricted wreath product.

You can generalize it and instead of taking set maps from K to N,
consider any K-set A (a set with an action of K on it), and consider
the semidirect product of N^A (set maps from A to N) and K, via a
similar action.

You can see all the gory details in a book on group theory, for
example Rotman's "Introduction to the Theory of Groups."

======================================================================
"It's not denial. I'm just very selective about
 what I accept as reality."
    --- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@math.berkeley.edu