Re: Base of the wreath product

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"?

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. If the set on which
the extending group acts is finite, then the two
will coincide (just as direct and restricted direct
products coincide when the index set is finite).

Arturo Magidin
.

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