Question #165251

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


1
Expert's answer
2021-02-24T07:48:32-0500

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

Suppose that a,bGa,b\in G . Since GG is a cyclic group generated by gg, there exist integers mm and nn such that a=gna=g^n and b=gmb=g^m .

Then we have ab=gngm=gn+m=gmgn=baab=g^ng^m=g^{n+m}=g^mg^n=ba .

It proves that group GG is abelian.


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

K4={e,a,b,c}K_4=\{e,a,b,c\}

e={e}K4\langle e\rangle=\{e\}\ne K_4

a={e,a}K4\langle a\rangle=\{e,a\}\ne K_4

b={e,b}K4\langle b\rangle=\{e,b\}\ne K_4

c={e,c}K4\langle c\rangle=\{e,c\}\ne K_4



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