A group $G$ is residually finite if, for any two elements $g$ and $g^\prime$ in $G$, there is a finite group $G^\prime$ and a (group) homomorphism $f: G \rightarrow G^\prime$ such that $f(g)$ doesn't equal $f(g^\prime)$. The definition for a semigroup is analagous: just make $G$ and $G^\prime$ semigroups and make $f$ a semigroup homomorphism. I was wondering if there is a good reference which will answer questions like the following:


Is there a group $G$ which is not residually finite as a group but is residually finite as a semigroup (in other words there is a finite semigroup $S$ and a semigroup homomorphism from $G$ to $S$ which separates elements, but there is no finite group $G^\prime$ and a group homomorphism from $G$ to $G^\prime$ which separates elements)?


If $S$ is a residually finite semigroup and $G$ is a subgroup of $S$, then $G$ is residually finite as a semigroup. Is $G$ residually finite as a group?


Thanks!



Residual finiteness of groups versus residual finiteness of semigroups
I first posted this as a comment, but I guess that this is an answer.


If a group is residually finite as a semigroup, it is residually finite as a group. This is an immediate consequence of the following easy fact: if G is a group and φ:G→S is a semigroup homomorphism, then the image φ(G) is a group and φ is a group homomorphism from G to φ(G). I guess that the latter fact is in any textbook on semigroups, though I do not have one at hand.


