Group Theory, Disjoint cosets dodgy proof problem
G is a finite group of order g, and N(a) is the normaliser of an element a belonging to G. N(a) is a subgroup.
Using 2 elements x,y belonging to G, 2 cosets of N(a) are constructed: xN(a) and yN(a) where x and y have been chosen such that the two cosets are disjoint.
I have already shown that if b belongs to xN(a), ba(b^-1) = xa(x^-1)
The second part requires me to show that if b belongs to xN(a) and c belongs to yN(a), then ba(b^-1) =/ ca(c^-1) where =/ denotes "doesnt equal".
My first instinct was to assume that the statment was false (ie the two things WERE equal) and show that this led to a contradiction eg that the disjoint cosets had a common element. But after getting to
xa(x^1) = ya(y^-1) i couldn't see how to manipulate this to show that there was such an element.
My second strategy was to assume that there existed a common element between the 2 cosets and sure enough this led me to the statement that xa(x^-1) = ya(y^-1). I guess my main question would be "does assuming a false statement lead to necessarily false deductions?" which would seem to imply that xa(x^-1) =/ ya(y^-1) since there doesn't exist an element common to both cosets. Or am i barking up the wrong tree entirely?