Title: Commutativity theorems for groups and semigroups,
Abstract: In this note we prove a selection of commutativity theorems for various
classes of semigroups. For instance, if in a separative or completely regular
semigroup $S$ we have $x^p y^p = y^p x^p$ and $x^q y^q = y^q x^q$ for all
$x,y\in S$ where $p$ and $q$ are relatively prime, then $S$ is commutative. In
a separative or inverse semigroup $S$, if there exist three consecutive
integers $i$ such that $(xy)^i = x^i y^i$ for all $x,y\in S$, then $S$ is
commutative. Finally, if $S$ is a separative or inverse semigroup satisfying
$(xy)^3=x^3y^3$ for all $x,y\in S$, and if the cubing map $x\mapsto x^3$ is
injective, then $S$ is commutative.