Here is the question:
Just in case:If is a homomorphism of groups then and for all . Show by example that the first conclusion may be false if G, H are monoids that are not groups.
I have constructed a couple of monoids with homomorphisms, but I'm not getting anywhere. In addition I have a proof of that does not use the inverses relationship , so I don't see where this would enter the problem. Additionally the proof does not seem to give me any clues as to how might not be equal to .A monoid is a set G with a binary operation and contains a two-sided identity element such that .
The proof is short so I will include it. Perhaps someone may be able to see a loophole that is escaping me.
Let G be a set with a binary relation * and let there be an element of G, such that for all g in G. Let H be another set with a binary relation and let there be an element such that for all h in H. Further let f be a homeomorphism .
. Since by definition for all h, must be .