Example of a semigroup S with no identity element and a subgroup G of S I need an example of a semigroup S without an identity element and a subgroup G of S.
I have found it easy to find/make semigroups without identities but then making a subgroup from it has not been fruitful. An example or a hint would be much appreciated.

Take your favorite group $G$.  Let $S$ consist of $G$ together with two additional elements $a$ and $b$, and extend the multiplication in $G$ by defining $as = sa = bs = sb = b$ for all $s \in S$.  You can confirm this operation is associative, and clearly it has no identity since any product with $a$ is $b$.  But $G$ is still a subgroup of $S$.
