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
SolutionSolution

If GG contains element gg of order nn , then GG contains cyclic subgroup <g><g> of order nn .


Then G<g>=G<g>=nn=0(G<g>∣G∖<g>∣=∣G∣−∣<g>∣=n−n=0 (G∖<g> is complement <g><g> in GG ), that is G=<g>G=<g> is a cyclic group. So if GG is a non-cyclic group of order nn , then GG has no element of order nn .


Z2Z2Z_2​⊕Z_2 is a non-cyclic group with cyclic proper subgroups


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


Z2Z2Z_2​⊕Z_2​ does not have other proper subgroups, because


(1,0),(0,1)=(1,0),(1,1)=(1,1),(0,1)=Z2Z2⟨(1,0),(0,1)⟩=⟨(1,0),(1,1)⟩=⟨(1,1),(0,1)⟩=Z_2⊕Z_2​


So every proper subgroup of Z2Z2Z_2⊕Z_2​ ​ is a cyclic group.




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