Solution
= Solution
The <order of a group element> represented by a <permutation> is the <least common multiple> of its disjoint cycle lengths. An order divisible by nine therefore requires a <permutation cycle> whose length is divisible by nine. In $S_9$ that must be a nine-cycle, which uses every letter and has order nine, not eighteen. \b[There is no element of $S_9$ of order eighteen.]