Prove that every non-trivial subgroup of a cyclic group has finite index. Hence
prove that (Q, +) is not cyclic.
Let us prove that every non-trivial subgroup of a cyclic group has finite index. If a cyclic group is finite, then all its subgroup finite and have finite index. If a cyclic group is infinite, then it is isomorphic to the additive group of integers. Taking into account that all subgroups of a cyclic group are cyclic, we conclude that every non-trivial subgroup of a cyclic group is of the form for The quotient group is of order , and hence every non-trivial subgroup of a cyclic group has finite index.
The group contains non-trivial subgroup of infinite continuum index: Hence the group is not cyclic.
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