Re: Direct external vs Cartesian

In article <46434cbc$0$37207$4fafbaef@xxxxxxxxxxxxxxxxxxx>,
bluelabel <jhsbadfb@#jkhasdf.it> wrote:
Please help me to understand what follows.
Let {G_a | a in A} a given set of groups. We define the cartesian product

C = Cr G_a
a in A

to be the group (with operations to be defined) whose universe C*

This is usually called the "underlying set", rather than "universe",
but okay...

is the set
of all (g_a) where g_a is in G_a and a in A. In the finite case this means:
{ (g_{a_1},g_{a_2},...,g_{a_n} | a_i in A }.
^

you are missing a ")" after g_{a_n}, but okay.

Yes. In other words: the underlying set of the "cartesian product" of
the groups will be the cartesian product (in its set-theoretic sense)
of the underlying sets of the groups.

We can naturally define an operation (g_a).(h_a)=(g_a.h_a), with
(g_a)^{-1}=(g_a^{-1}), 1_C=(1_a).

I.e., the operation is defined "coordinate-wise."

It is easy to show that C=(C*,.) is a group.

Indeed.

Now, consider the elements (g_a) such that g_a=1_a for "almost all a in A",
that is , with finitely many exceptions.

Sometimes called the "restricted direct product", or "direct sum"
(although in the latter case, usually reserved for the situation where
the G_a are abelian).

This is a subset of C* and we can form the so called "external direct
product". We write

D = Dr G_a .
a in A

I would like to ask:
1) what does it mean "with finitely many exceptions"?

It means that each element of the set has at most finitely many values
which are not equal to the identity. If the index set is finite, this
does not mean anything, but in the case of an infinite index, it does
place a restriction.

For example (from now
on I omit some subscript indexes)

(1,1,.....,g,1,h,1,1,...) --> this is an infinite length "vector",

is it in D?

If "1" represents the identity element of the groups, yes.

2) If A is finite, is the finite vector (a,b,c,d,e,f), with no 1's
occurring, in D?

Yes, because all entries except for finitely many (namely, six), are
equal to 1.

3) is C the same object as D when A is finite?

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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