Answer to Question #165251 in Abstract Algebra for K

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=\\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"



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