Answer to Question #128662 in Abstract Algebra for Jai

Question #128662
Show that if g is a non cyclic group of order n then g has no elements of order n. Further give an example with justification of a non cyclic group all of whose proper subgroups are cyclic
1
Expert's answer
2020-08-06T16:25:56-0400
"Solution"

If "G" contains element "g" of order "n" , then "G" contains cyclic subgroup "<g>" of order "n" .


Then "\u2223G\u2216<g>\u2223=\u2223G\u2223\u2212\u2223<g>\u2223=n\u2212n=0 (G\u2216<g>" is complement "<g>" in "G" ), that is "G=<g>" is a cyclic group. So if "G" is a non-cyclic group of order "n" , then "G" has no element of order "n" .


"Z_2\u200b\u2295Z_2" is a non-cyclic group with cyclic proper subgroups


{"(0,0),(0,0),(0,1),(0,0),(1,0),(0,0),(1,1)" }


"Z_2\u200b\u2295Z_2\u200b" does not have other proper subgroups, because


"\u27e8(1,0),(0,1)\u27e9=\u27e8(1,0),(1,1)\u27e9=\u27e8(1,1),(0,1)\u27e9=Z_2\u2295Z_2\u200b"


So every proper subgroup of "Z_2\u2295Z_2\u200b" ​ is a cyclic group.




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