Show that if f,g are 2 disjoint cycles, then fg = gf.
Please show steps. Thanks!
Let $\displaystyle f$ be the cycle $\displaystyle (a_1,...,a_i)$ and $\displaystyle g$ be the cycle $\displaystyle (b_1,...,b_j)$ where $\displaystyle a_1,...,a_i,b_1,....,b_j$ are distinct. To show that $\displaystyle fg = gf$ you need to show $\displaystyle fg(x) = gf(x)$ for any $\displaystyle x$. If $\displaystyle x$ is distinct from $\displaystyle a_1,...,a_i,b_1,....,b_j$ then $\displaystyle fg(x) = x = fg(x)$. If $\displaystyle x\in \{a_1,...,a_i\}$, then $\displaystyle g(x) = x$ while $\displaystyle g(f(x)) = f(x)$ since $\displaystyle f(x) \in \{a_1,...,a_i\}$ so $\displaystyle fg(x) = f(x) = g(f(x)) = gf(x)$. The same idea is when $\displaystyle x\in \{b_1,...,b_j\}$ and so we have that $\displaystyle fg=gf$.