= Solution
{c}
With products acting from right to left, a <permutation cycle> satisfies
$$
(a_1\ a_2\ \cdots\ a_k)
=(a_1\ a_k)(a_1\ a_{k-1})\cdots(a_1\ a_2).
$$
The right side sends $a_1$ to $a_2$, each $a_j$ to $a_{j+1}$ for $2\leq j<k$, and $a_k$ to $a_1$, while fixing all other symbols. This verifies the formula. Applying it to every cycle in the <disjoint cycle decomposition> expresses any <permutation> as a product of <transpositions>. The identity is represented by the empty product.
Back to article page
