Why isn't there much study on right groups (left groups)? This question is related to and inspired by the question Why are groups more important than semigroups?.
I am curious why I don't see much studies on right groups.
On Pp.37 of Clifford& Preston's Algebraic Theory of Semigroups Vol. I : A semigroup is called a right group if it is right simple and left cancellative. It is equivalent to saying that for any element $a$ and $b$ of a semigroup $S$, there exists one and only one element $x$ of $S$ such that $ax = b$.
A right group is the direct product of a group and a right zero semigroup. A right group is the union of a set of disjoint groups. A periodic semigroup is a right group iff it is regular and left cancellative, etc.
A left group is the dual of a right group.
Most literature I found about right groups is very old (back in 50's and earlier). As far as I know, there is not much research on this subject in the past 30 years or so. Am I wrong about this? Or "right group" are common words so I do not get useful results when I google for it?
But I thought the right group concept could serve as a bridge between groups and semigroups. For example, a permutation group is a set of bijective mappings and a right zero transformation semigroup is a set of constant mappings, you have a right group if you take the direct product of a permutation group and a right zero transformation semigroup. What is this right group look like? Another example, an aperiodic semigroup contains no non-trivial groups, then what is a semigroup which contains no non-trivial right groups? What is a semigroup which contains a non-trivial right group? etc. etc.
 A: You mention in your question some theorems about right groups that can be found in Volume 1 of The Algebraic Theory of Semigroups. However, you don't distinguish between those given as theorems and those given as exercises, and the distinction is important here. The only one of them that's not an exercise is this:

Theorem 1.27. The following assertions concerning a semigroup $S$ are equivalent:
(i) $S$ is a right group.
(ii) $S$ is right simple, and contains an idempotent.
(iii) $S$ is the direct product $G\times E$ of a group $G$ and a right zero semigroup $E.$

(page 38.)
And indeed, this theorem, especially the equivalence between (i) and (iii), seems to explain pretty well what right groups are. It is a structure theorem -- it explains the structure of right groups in terms of "simpler" (or at least other) structures. For right zero semigroups, they're quite clearly very simple structures. Groups definitely aren't but it seems to be a general trend in semigroup theory to push problems towards groups and abandon them when groups are reached. I think P. A. Grillet explained it well on the first pages of his book on semigroups, but I seem to have lost the book.
I'm quite sure some interesting things can still be said about right groups but it's natural that they don't arouse very much interest in presence of such a theorem. Modulo group theory, it explains the structure of right groups pretty well.
