Suppose and .
I'm having trouble understanding why . Here's my reasoning:
Let , then .
Where I used since . What is wrong with this 'proof'?
what is the context in which this takes place (what group is GN a subset of)?
note that if N is a subgroup of G, |GN| = |G||N|/|G∩N| = |G||N|/|N| = |G|, so that in fact, GN = G.
usually, one considers two subgroups S,N normal in G. and the reason SN/N ≠ S/N is that there is no reason to suppose S/N even makes sense
(N may be "too big" to fit inside S, or may only intersect S trivially).
if one isn't considering G,N to both be subgroups of a larger group, then GN makes no sense, because it is not clear what the product should be.
on the other hand, GxN/({e}xN) is isomorphic to G, not G/N.