proof that all subgroups of a cyclic group are cyclic
Let be a cyclic group and . If is trivial, then , and is cyclic. If is the trivial subgroup, then , and is cyclic. Thus, for the of the proof, it will be assumed that both and are nontrivial.
Let . Then there exists with . Since , we have that . Thus, . Hence, .
This proves the claim. It follows that every subgroup of is cyclic. ∎
|Title||proof that all subgroups of a cyclic group are cyclic|
|Date of creation||2013-03-22 13:30:47|
|Last modified on||2013-03-22 13:30:47|
|Last modified by||Wkbj79 (1863)|