Calculations with permutations: show that $(1,2,3)^2(5,7)^2=(1,3,2)$ How can I show $(1,2,3)^2(5,7)^2=(1,3,2)$? And, specifically, what does $(1,2,3)^2$ and $(5,7)^2$ equal individually?

GAP shows  that you could have other powers in which $$(1, 2, 3)^i \cdot (5, 7)^j = (1, 3, 2)$$ machinery. I examined this by it and you see that $i$ and $j$ could be some other positive integers than $i=j=2$:
gap> S7:=SymmetricGroup(IsPermGroup,7);;
     for i in [1..5] do
      for j in [1..5] do
       if (1,2,3)^i*(5,7)^j=(1,3,2) then Print("i=",i,"  ","j=",j,"\n");
       fi;
      od;
     od;


i=2  j=2
i=2  j=4
i=5  j=2
i=5  j=4

