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## CHAPTER 8

### Structure theorems of groups

#### 1 Direct products

If a group is isomorphic to the direct product of a family of its subgroups (known as summands) whose structures are known to us, then the structure of the group can generally be determined; these summands are like the "building blocks" of a structure. Theorem 1.1 answers the question, When is a group isomorphic to the direct product of a given finite family of its subgroups?

1.1 Theorem. Let H1, …, Hn be a family of subgroups of a group G, and let H = H1 Hn. Then the following are equivalent:

(i) under the canonical mapping that sends (xi, …, xn) to xixn

(ii) , and every element ...

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