Question. Determine the subgroup of $S_4$ generated by $\sigma=(1\ 2\ 3\ 4)$ e $\tau = (2\ 4)$. Show that $\left<\sigma, \tau\right> <S_4$ is isomorphic to the group of square symmetries.
Attempt to solve. I found that, $\left<\sigma, \tau\right>=\{e, \sigma, \sigma^2, \sigma^3, \tau, \sigma \circ \tau, \tau \circ \sigma, \sigma^2 \circ \tau\}$. We have that, the group of symmetries of square $D_4$ has $8$ elements. Consider the application $$f : \left<\sigma, \tau\right> \to D_4$$ I need help defining this application and showing that there is an isomorphism.