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