Answer to Question #165251 in Abstract Algebra for K

Question

Prove that every cyclic group is abelian. give an example to show the converse is NOT true.

Answer 1

Let G be a cyclic group and $G=\langle g\rangle$ .

Suppose that $a,b\in G$ . Since $G$ is a cyclic group generated by $g$, there exist integers $m$ and $n$ such that $a=g^n$ and $b=g^m$ .

Then we have $ab=g^ng^m=g^{n+m}=g^mg^n=ba$ .

It proves that group $G$ is abelian.

Klein four group is an example of group that is abelian but not cyclic.

$K_4=\{e,a,b,c\}$

$\langle e\rangle=\{e\}\ne K_4$

$\langle a\rangle=\{e,a\}\ne K_4$

$\langle b\rangle=\{e,b\}\ne K_4$

$\langle c\rangle=\{e,c\}\ne K_4$