Group Elements a,b,c,d,e: Inverse Operation?

[Punkyc7]

Lets say i have elements a,b,c,d,e in some group.


is abcde always = ab(d^-1c^-1)e. My question is and you change elements in the middle of an operation by using the inverse?

 

[jbunniii]

Certainly this is not true. Take your equation and cancel the a, b, and e by multiplying on the appropriate side by the appropriate inverse in each case. You are left with

cd = (d^-1 c^-1)

and the right hand side is equal to (cd)^-1.

Obviously in general, cd does not equal its own inverse, so the equation is false.