Permutation Groups - Rewrite as disjoint cycles / transpositions

Question

Why is this true? "(1243)(3521): Written as a product of disjoint cycles, we get (354)."

Shouldn't (1243)(3521) = (354)(21)? Where did the (21) go?

Thanks!!

Answer 1

The cycle $(3521)$ sends $3$ to $5$, and the cycle $(1243)$ does nothing to $5$, so the product sets $3$ to $5$. The cycle $(3521)$ sends $5$ to $2$, which $(1243)$ sends to $4$, so the product sends $5$ to $4$. The cycle $(3521)$ leaves $4$ alone, and $(1243)$ sends it to $3$, so the product sends $4$ to $3$. This establishes that the product has a cycle $(354)$.

Now what about $1$ and $2$? The cycle $(3521)$ sends $1$ to $3$, and $(1243)$ sends $3$ right back to $1$, so the product leaves $1$ fixed. The cycle $(3521)$ sends $2$ to $1$, and $(1243)$ sends $1$ right back to $2$, so the product leaves $2$ fixed. Thus, the product is $(1)(2)(354)=(354)$, since we don’t normally write out the $1$-cycles (fixed points) explicitly.