Re: Base of the wreath product
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Mon, 2 Jun 2008 03:45:28 +0000 (UTC)
In article <8eafbd0c-9c1f-493c-b3ea-7b1d2d4b06b3@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<newsgr.mail@xxxxxxxxx> wrote:
On 2 Giu, 01:19, "magi...@xxxxxxxxxxxxxxxxx"
<magi...@xxxxxxxxxxxxxxxxx> wrote:
Please look at this page:
http://snipurl.com/2bxz1.
My question is: why the author claims that B is
actually an external
direct product (written with "Dr") and not a
cartesian direct product?
I mean, if H and Y are both infinite this could be
the case, isn't it?
In your nomenclature, what is the difference between a
"direct product" and a "cartesian direct product"?
I mean "external direct product" vs "cartesian product".
The same book, at page 20, says that the external direct product is a
normal subgroup of the cartesian product. The author also calls the
cartesian product as "unrestricted direct product".
And what is the difference between the "external direct product" and
the cartesian product? I'm not going to search through an on-line
version of a book to find it.
In the case that a set of indexes S is infinite, the cartesian product
of the family {G_i} with i in S has its underlying set made of all the
functions
S --> \+/ G_i s.t s |-> g_s in G_s.
In other words, it has as its underlying set the cartesian product of
the G_i. (That ->is<- what the cartesian product of a family of sets
is: it is the collection of all functions from the index set to the
union of the sets with the property that the image of the index i lies
in the set indexed by i).
Instead, the external direct product should contain all the functions
that are "almost everywhere" = 1.
That is what most people call the "restricted" direct product. Calling
it "external" is definitely a conflict with standard nomenclature.
For ->most<- people, "external" direct product lies in contraposition
to "internal" direct product, not with the unrestricted product: a
group G is the "internal direct product" of two subgroups H and K if
and only if:
(i) G = HK.
(ii) H/\K = {1}
(iii) hk = kh for all h in H and k in K.
Whereas G is the "external direct product" of two groups H and K if
and only if G = H x K = { (h,k) : h in H, k in K}, with multiplication
performed coordinatewise.
The distinction is usually immaterial, because if G is the internal
direct product of subgroups H and K, then G is isomorphic to the
external direct product of H and K. And H x K is the internal direct
product of the subgroups H x {1} and {1} x K.
I have never before heard of anyone refer to the restricted direct
product as "external direct product".
At the moment, I don't know why the wreath product base should
necessarily be an external direct product and not a Cartesian product.
Both definitions can be used: you can use as a base the restricted
direct product (I suggest dropping the nonstandard nomenclature of the
book you are reading, and perhaps getting a better book) or the
unrestricted direct product. You get two (possibly different) wreath
products, as I explained in my post: the restricted wreath product and
the unrestricted wreath product. Most people will pick one to be the
"standard" and call it just "wreath product", and the other one will
be a variation. So you can go with either "wreath product and
unrestricted wreath product", or you can go with "restricted wreath
product and wreath product" as your two standard constructions. It
does not matter.
Standard nomenclature is that the direct product
will have the cartesian product as base set, with
operations being "componentwise"; the "restricted
direct product" is the (normal) subgroup of the direct
product consisting of all elements that are equal
to the identity in all but finitely many entries
(including, of course, the possibility that 0 entries
are distinct from the identity). If you use the
restricted direct product as your base group, you obtain
the restricted wreath product. If you use the direct
product, you get the wreath product.
Ok, I see. So the author was defining what you call the "restricted
wreath product"?
But with pretty lousy notation for his restricted direct product,
yes.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================
Arturo Magidin
magidin-at-member-ams-org
.
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