Prove that every non-trivial sub group of cyclic group has finite index. Hence prove that (Q,+) is not cyclic.
Solution : 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.
Comments