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The number of elements in the direct product are 2.
What is direct product of groups?
The direct product, often written as G x H, is an operation used in group theory, a branch of mathematics, to create a new group from two existing ones. This operation, one of several crucial direct product concepts in mathematics, is the group-theoretic equivalent of the Cartesian product of sets.
Let, the number of elements of order 1 in [tex]Z_1_6[/tex] and [tex]Z_1_9[/tex] is 1.
Number of elements of order 2 in [tex]Z_1_6[/tex] and [tex]Z_1_9[/tex] are 1 and 0 respectively.
Number of elements of order 4 in [tex]Z_1_6[/tex] and [tex]Z_1_9[/tex] are 2 and 0 respectively.
Consider the table of number of elements of order.
Number of elements of order
1 2 4
[tex]Z_1_6[/tex] 1 1 2
[tex]Z_1_9[/tex] 1 0 0
Let x ∈ [tex]Z_1_6[/tex] x [tex]Z_1_9[/tex]
O(x) = lcm (O(a), O(b))
where x = (a, b)
That is, a ∈ [tex]Z_1_6[/tex] and b ∈ [tex]Z_1_9[/tex]
Here only (4, 1) satisfies it and [tex]Z_1_6[/tex] has 2 elements of order 4 and [tex]Z_1_9[/tex] has one element of order 1.
So, [tex]Z_1_6[/tex] x [tex]Z_1_9[/tex] has 2 x 1 = 2 elements of order 4.
Hence, the number of elements in the direct product are 2.
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