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Appendix: The Basics
it follows easily that G is the direct product of the subgroups Mi in every
possible ordering of these subgroups. Given a finite collection X of normal
subgroups of a group G, therefore, it is meaningful to say that G is the
direct product of the members of X without specifying an ordering.
We have already seen that the external direct product P of groups Gi
is the internal direct product of certain subgroups Ni P, where Ni = G{.
Another connection between external and internal direct products is the
following.
X.23. Lemma. Let G be a group, and suppose that G is the internal direct
product of the normal subgroups Mi for 1 i r. Then G is isomorphic to
the external direct product of the groups Mi.
Proof. Let P be the external direct product of the Mi, and construct a
map 9 from P to G by setting
0((rai, 777-2, ? ^v)) = 7774777, 2 mr .
Then 9 is a bijection since every element of G is uniquely of the form
777,1772 2 * * Tnr with rrii E Mi. To show that 9 is an isomorphism, we need
(xix2 - xr)(y1y2 '-yr) = xiyix2y2 xryr
for Xi,yi G M^, and this is clear since Xi and yj commute when i j .
For an example of how this can be used, suppose that G is an internal
direct product of normal subgroups, each of which has a trivial center. To
prove that Z(G) = 1, it suffices to prove that Z(F) = 1, where P is the
isomorphic group constructed as the external direct product of the factors
of G. It is clear, however, that an r-tuple (mi, 7722,..., rar) is central in P
if and only if each component rrii is central in its respective group, and it
follows that Z(P) = 1.
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