Cyclic implies abelian

Statement
Any cyclic group (i.e., any group generated by only one element) is an abelian group (i.e., any two elements in it commute).

Using weaker hypotheses to deduce abelianness

 * Locally cyclic implies abelian
 * Residually cyclic implies abelian
 * Generating set in which any two elements commute implies abelian
 * Cyclic over central implies abelian

Converse of sorts

 * Finite abelian and aut-abelian implies cyclic (also, cyclic implies aut-abelian)