Answer to Question #138066 in Abstract Algebra for Ram

Question

Write down all elements of quotient group Z18/<6>. Is any element of order 5?

Answer 1

Obviously

$^{\mathbb{Z}_{18}}\!/\!_{<6>}= \mathbb{Z}_6=\{0,1,2,3,4,5\}$

There are no elements of order 5 here because $ord(\mathbb{Z}_6)=6$ and order of any element must be a divisor of the group order, but $5\nmid6$