Direct Product Theorem
If , , the subgroups commute elementwise, and , then .
Theorem 6.3 (Direct Product Theorem)
If and
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,
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,
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, i.e. for every , there exist such that ,
then .
Proof
Define by . Commutativity of the two subgroups makes a homomorphism, and makes it surjective. If , then lies in , so and is injective.
Related
Stated in
- Theorem 6.3 (Direct Product Theorem)§6 Finite Groups