Direct Product of Groups - Elementary Properties

Elementary Properties

• The order of a direct product G × H is the product of the orders of G and H:

| 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:

| (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

( Z / mZ ) × ( Z / nZ ) ≅ Z / mnZ.

This fact is closely related to the Chinese remainder theorem.