How to prove that $\langle\{ (1,2),(1,2,3) \}\rangle=\mathfrak{S}_3$ 
Prove that
  $\{ (1,2),(1,2,3) \}$ Generating set of a symmetric group $(\mathfrak{S}_3,\circ )$

SOlution provided by book 
we 've
$(1,2,3)(1,2)(1,2,3)^2=(2,3)$ and $(1,2,3)^2(1,2)(1,2,3)=(1,3)$
since transposition Generating $(\mathfrak{S}_3,\circ )$ then $\{ (1,2),(1,2,3) \}$ Generating set of $(\mathfrak{S}_3,\circ )$


*

*I didn't understand that solution would someone explain it to me and  how we can show such permutation is generator of a symmetric group and why they need to calculate  $(1,2,3)^2$ at left of $(1,2,3)(1,2)$.. 


You have that $\mathfrak S_3=\{1, (12),(13),(23),(123),(132)\}$.
You wrote $(23)$ and $(13)$ as combinaison of $(12)$ and $(123)$. By the way, $(132)=(13)(32)$, and thus you can also express it as a combinaison of $(12)$ and $(123)$. Therefore, these two elements generate the group.
