Well definedness of factor group multiplication I have been reading on this site and the internet, and I do not quite understand the comments which are being made with respect to the problem of proving that factor group multiplication is well defined. According to John Fraleigh, A First Course in Abstract Algebra, 7th ed (page 137), if we make the following multiplication rule for factor groups,
$$
(xH)(yH) = (xy)H,
$$
we must check whether it is well defined. I have searched on this website and the internet to try and find out what the concern is, and it seems that the problem is that we must make sure that if $xH = x'H$ and $yH = y'H$, then $(xy)H = (x'y')H$. 
First question: Is this Fraleigh's concern?
If the answer to this question is indeed yes. How can it be a problem, since if $xH = x'H$ and $yH = y'H$, then $$(x'y')H = (x'H)(y'H)=(xH)(yH) = (xy)H$$ is automatic? This is my second question.


(x′y′)H=(x′H)(y′H)=(xH)(yH)=(xy)H

This is known as "proof by notation". It's a trap you have to be careful about falling into. Simply because you declare some expression to represent a value, doesn't mean that the value exists or is well-defined. 
For instance, suppose we define f(p) to be the area of triangles with perimeter p. We can "prove" that given a perimeter, all triangles with that perimeter have the same area with the following argument:
Given triangle T1 and T2 with the same perimeter p, the area of T1 is f(p), and the area of T2 is also f(p). Thus, their areas are the same.
