By mapping the generators $s_{i}$ into $S_{n}$ appropriately, find a well-defined epimorphism $\theta :G_{n}\rightarrow S_{n}$ .

Question

So $G_{n}$ is the group with presentation

$\left<s_{1},...,s_{n-1}\mid s_{i}^{2}=1, s_{i}s_{j}=s_{j}s_{i} \text{ if } \left | i-j \right |\geq 2, s_{i}s_{j}s_{i}=s_{j}s_{i}s_{j} \text{ if } \left | i-j \right |=1\right>$.

What is a good way to solve this problem?

Thanks a lot.

Answer 1

Define \begin{align} \theta: \ & G_{n} \to S_{n}\\ & s_i \longmapsto (i,i+1) \tag{1} \end{align}

The mapping defined in (1) is an isomorphism.