Answer to Question #105098 in Abstract Algebra for Sourav Mondal

Question #105098
Show that if G is a non-cyclic group of order n, then G has no element of order
n. Further, give an example, with justification, of a non-cyclic group with all its
proper subgroups being cyclic.
1
Expert's answer
2020-03-10T13:01:05-0400

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

Then "|G\\setminus<g>|=|G|-|<g>|=n-n=0" ("G\\setminus<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\\oplus 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)\\}"

"Z_2\\oplus Z_2" does not have other proper subgroups, because "\\langle (1,0),(0,1)\\rangle=\\langle (1,0),(1,1)\\rangle=\\langle (1,1),(0,1)\\rangle="

"=Z_2\\oplus Z_2"

So every proper subgroup of "Z_2\\oplus Z_2" is a cyclic group.


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Comments

Assignment Expert
02.11.20, 01:00

Dear Pintoo, please use the panel for submitting new questions.

Pintoo
31.10.20, 13:23

If A and B are two sets such that AUB=Fie, then A intersection B=fie

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