Given $G:=S_7,M:=\{1,...,7\},g:=(1\, 3\, 6)\cdot(3\, 5),x := 5,$ calculate $g\cdot x$.

Question

Given

$$\begin{align} G& := S_7, \\ M& := \{1,...,7\},\\ g &:= (1 \, 3 \, 6) \cdot (3 \, 5), \\ x &:= 5, \end{align}$$

calculate $g \cdot x$.

I would simply do:

$$ ((1 \, 3 \, 6) \cdot (3 \, 5))\cdot(5) = (3 \, 5 \, 6 \, 1) \cdot (5) = (3 \, 5 \, 6 \, 1) $$

But the solution is:

$$ ((1 \, 3 \, 6) \cdot (3 \, 5))\cdot(5) = ((1 \, 3 \, 6)) \cdot (3) = 6 $$

How is it so?

Answer 1

Hint: Recall that a permutation $\sigma\in S_n$ is a bijection on the underlying set $\overline{1, n}=\{1, \dots, n\}$. Thus $\sigma\cdot m$ for $m\in\overline{1, n}$ is $\sigma$ evaluated as a bijection at $m$.

If $\sigma=(1 \, 3 \, 6)$ and $\tau=(3 \, 5)$, then $(\sigma\circ\tau)(5)=\sigma(\tau(5))=\sigma(3)$, since $\tau: 5\mapsto 3$, so that $(\sigma\circ\tau)(5)=\sigma(3)=6$ because $\sigma: 3\mapsto 6$; put in a diagram: $$5\stackrel{\tau}{\mapsto} 3\stackrel{\sigma}{\mapsto} 6.$$