Cyclic implies abelian

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., cyclic group) must also satisfy the second group property (i.e., abelian group)
View all group property implications | View all group property non-implications
Get more facts about cyclic group|Get more facts about abelian group

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).

Related facts

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)