External Direct Product Examples: ℤ2 ⊕ ℤ3, ℤ2 ⊕ ℤ3 ⊕ ℤ5, and ℤ2 ⊕ ℤ3 ⊕ ℤ6

Описание: What are the elements of ℤ2 ⊕ ℤ3 and what are their orders? Is ℤ2 ⊕ ℤ3 a cyclic group? How about ℤ2 ⊕ ℤ3 ⊕ ℤ5? How about ℤ2 ⊕ ℤ3 ⊕ ℤ6? Use the fact that the order of an ordered pair (a,b) is the LCM of the orders of the components a and b, and the order of an ordered triple (a,b,c) is the LCM of the orders of the components a, b, and c.

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(0:00) ℤ2 ⊕ ℤ3: Elements and their orders
(8:27) |(a,b)|=lcm(|a|,|b|)
(13:28) ℤ2 ⊕ ℤ3 ⊕ ℤ5: Number of elements of various orders
(21:34) ℤ2 ⊕ ℤ3 ⊕ ℤ6: Number of elements of various orders
(28:48) Number of cyclic subgroups of order 6 in ℤ2 ⊕ ℤ3 ⊕ ℤ6

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