Exotic Definitions of Groups

Question

Inspired by this question I was wondering, whether there are alternative definitions of groups, namely ones different from the usual 4 axioms. I already suspected that the category theorists have one and indeed, a group can also be defined as a groupoid with only one object. Do you know of any other?

And if you happen to know any weird ones, I also welcome exotic definitions of algebraic objects other than groups :)

Answer 1

Let $S$ be a semigroup. Then $S$ is a group if and only if

$$aS=S=Sa$$

for all $a\in S$, where $aS=\{ as\in S\mid s\in S\}$ and $Sa=\{ sa\in S\mid s\in S\}$.

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A semigroup $S$ is a group if there exists an $e$ in $S$ such that for all $a$ in $S$, $ea=a$ and for all $x$ in $S$ there exists a $y$ in $S$ such that $yx=e$.

Answer 2

Here is another funny definition of a group. (This was an old American Mathematical Monthly problem but I've lost the reference.)

Proposition. A semigroup $S$ is a group if there are elements $a,b\in S$ such that:
(1) for each $x\in S$ there is a unique $y\in S$ such that $xy=a$;
(2) for each $x\in S$ there is a unique $y\in S$ such that $yx=b$.

Proof. Let $ac=a$. Then the equation $xy=c$ is equivalent to $axy=a$ Hence, for each $x\in S$ there is a unique $y\in S$ such that $xy=c$; that is, the hypothesis holds with $a$ replaced by $c$. Since $cc=c$, we can assume that $aa=a$; likewise we can assume that $bb=b$.

Let $a=bu$ and $b=va$; then $ba=bbu=bu=a$ and $ba=vaa=va=b$, so $a=b$.

For any $x\in S$ there is an element $y\in S$ such that $yx=b=a$; then $yxa=a=yx$, so $xa=x$. Thus $a$ is a right identity. Since every element has a right inverse with respect to $a$, the semigroup $S$ is a group.