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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.