A semigroup $X$ is a group iff for every $g\in X$, $\exists! x\in X$ such that $gxg = g$

Question

The following could have shown up as an exercise in a basic Abstract Algebra text, and if anyone can give me a reference, I will be most grateful.

Consider a set $X$ with an associative law of composition, not known to have an identity or inverses. Suppose that for every $g\in X$, there is a unique $x\in X$ with $gxg=g$. Show that $X$ is a group.

Note that I’m not asking for a proof (though a really short one would please me!), just some place where this has been published.

Answer 1

I believe this is well-known to those working in regular/inverse semigroups. So well-known in fact that one can even find proofs on the web, e.g. see this proof on PlanetMath. Alas, I cannot recall the history of this result, though I vaguely recall reading something about such decades ago. For completeness, I reproduce below said common proof (slightly edited).

Theorem $\ $ A non-empty semigroup $\,S\,$ is a group if and only if for every $\,x\in S$ there is a unique $y\in S$ such that $\,xyx=x$.

Proof $\ (\Leftarrow)\,$ For each $\,x\in S\,$ let $\,x'$ be the unique element of $\,S\,$ such that $\,xx'x=x.\ $ Note $\,x(\color{#0a0}{x'xx'})x=(xx'x)x'x=x\color{#C00}{x'}x=x,\,$ so, by uniqueness, $\,\color{#0a0}{x'xx'}=\color{#C00}{x}',\,$ hence $\,\color{#0a0}x = \color{#C00}{x}''.$

For all $\,x\in S\,$ the element $\,xx'$ is idempotent $\,(xx')^2=(xx'x)x'=xx'.$ Thus, being nonempty, $S\,$ has at least one idempotent. If $\,i\in S$ is idempotent, then $\,ix =ix\color{#0A0}{(ix)'}ix=ix\color{#C00}{(ix)'i}\:\!ix\,$ so, by uniqueness, $\color{#C00}{(ix)'i}=\color{#0A0}{(ix)'},$ hence $(ix)'=(ix)'(ix)''(ix)'=\color{#C00}{(ix)'i}x(ix)'=\color{#0A0}{(ix)'}x(ix)'$ so, by uniqueness, $x = (ix)''=ix.\,$ So every idempotent $\,i\,$ is a left identity and, by a symmetry, a right identity. Therefore, $S\,$ has at most one idempotent element. Combined with the previous result, this means that $S\,$ has exactly one idempotent element, denoted $\,e.\,$ We have shown that $\,e\,$ is an identity, and that $\,xx'=e\,$ for each $\,x\in S,$ hence $\,S\,$ is a group.

$(\Rightarrow)\,$ if $\,S\,$ is a group then $\,xyx=x\iff xy=1\iff y = x^{-1},\,$ so $\,y\,$ is unique. $\ \bf\small QED$

Answer 2

I am just putting some sources here. Bill Dubuque already did all the work:

This book may be of some interest: Mark V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, at Google Books

I came across this book in this Wikipedia article, which gives some of the history of Inverse Semigroups.

Also, there is this shorter PDF by the author of the book whose link I gave above, and what you state is probably Proposition 2.4 in the notes (although it is stated in a different way) after some other things have already been proved.