= Solution
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A <transposition> in the <symmetric group> $S_n$ exchanges two distinct symbols and fixes all others. A <permutation cycle> $(a_1\ a_2\ \cdots\ a_k)$ sends $a_j$ to $a_{j+1}$, sends $a_k$ to $a_1$, and fixes the other symbols. Its entries are distinct. For $k=1$ this is the identity <permutation>.
A <group element> has <order of a group element> $k$ if $k$ is the least positive integer for which its $k$th power is the <identity element>. For the displayed <permutation cycle>, its $j$th power sends $a_i$ forward $j$ places, with subscripts read modulo $k$. Hence the $k$th power fixes every symbol, whereas for $0<j<k$ the image of $a_1$ is different from $a_1$. Thus its <order of a group element> is exactly $k$, including $k=1$.
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