/sci/ - Science & Math

Im having a problem with the group of permutations.

Im given two permutations: f=613542 and g=231564

these are just abbreviations of [math]f= \binom {123456} {613542}[/math] and [math]g={123456 \choose 231564}[/math]

I multiplied them by doing g then f and got a permutation y=142563

But then I tried to do [math]f^-1*y*g^-1=f^-1*f*g*g^-1[/math] and got 524631 (instead of the desired 123456)

Further when I tried to do [math]f^-1*(y*g^-1)[/math] I got 524631, where as with [math](f^-1*y)*g^-1[/math] I got 165324.

So clearly I am doing something very very wrong.

How do I compose two permutations? How do I do 3 permutations? What am I doing when I do pic relate (which is the process I go through when I do these compositions)

Mind you when I tried calculating f^-1 and g^-1 it worked just fine and gave me back 123456