24
$\begingroup$

If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is?

Thanks in advance!

  • 23
    Have you tried proving this yourself? Where do you get stuck? The "explanation" is that an element always commutes with powers of itself.2011-06-12
  • 10
    In fact, not only is every cyclic group abelian, every *quasicylic* group is always abelian. (A group is quasicyclic if given any $x,y\in G$, there exists $g\in G$ such that $x$ and $y$ both lie in the cyclic subgroup generated by $g$). The Prufer $p$-group and the rationals (under addition) are examples of quasicylic groups that are not cyclic.2011-06-12

6 Answers 6