Elementary Properties
- The order of a direct product G × H is the product of the orders of G and H:
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- | G × H | = | G | | H |.
- This follows from the formula for the cardinality of the cartesian product of sets.
- The order of each element (g, h) is the least common multiple of the orders of g and h:
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- | (g, h) | = lcm( | g |, | h | ).
- In particular, if | g | and | h | are relatively prime, then the order of (g, h) is the product of the orders of g and h .
- As a consequence, if G and H are cyclic groups whose orders are relatively prime, then G × H is cyclic as well. That is, if m and n are relatively prime, then
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- ( Z / mZ ) × ( Z / nZ ) ≅ Z / mnZ.
- This fact is closely related to the Chinese remainder theorem.
Read more about this topic: Direct Product Of Groups
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