Answer to Question #133617 in Abstract Algebra for PRATHIBHA ROSE C S

Question

Find the number of elements of order 9 in Z3 (product ) Z9

Answer 1

We know that

The order of an element in a direct product of finite number of finite groups is the least common multiple of the order of components of the element. In symbols,

$|(g_1,g_2,.........,g_n)|=$ lcm $(|g_1|,|g_2|,..........,|g_n|)$

Therefore, We may count the number of elements $(a,b)$ in $\Z_3 \bigoplus \Z_9$ with the property that ,

$9=|(a,b)|=$ lcm $(|a|,|b|)$ .

Clearly, this requires either $|a|=1$ and $|b|=9$ or $|a|=3$ and $|b|=9$ .

Case:1 $|a|=1$ and $|b|=9$

Here there are 1 choice for $a$ and 6 choices for $b$ ( namely 1,2,4,5,7 and 8).

This gives $1×6=6$ element of order 9.

Case 2. $|a|=3$ and $|b|=9$

This time 2 choices for $a$ (namely, 1, and 2) and as before 6 choices for $b$ .

This gives $2×6=12$ elements.

Therefore the total number of elements of order 9 is 6+12=18.