1. ## cycle decomposition 1

Show that if f,g are 2 disjoint cycles, then fg = gf.

Let $f$ be the cycle $(a_1,...,a_i)$ and $g$ be the cycle $(b_1,...,b_j)$ where $a_1,...,a_i,b_1,....,b_j$ are distinct. To show that $fg = gf$ you need to show $fg(x) = gf(x)$ for any $x$. If $x$ is distinct from $a_1,...,a_i,b_1,....,b_j$ then $fg(x) = x = fg(x)$. If $x\in \{a_1,...,a_i\}$, then $g(x) = x$ while $g(f(x)) = f(x)$ since $f(x) \in \{a_1,...,a_i\}$ so $fg(x) = f(x) = g(f(x)) = gf(x)$. The same idea is when $x\in \{b_1,...,b_j\}$ and so we have that $fg=gf$.