Question #312461

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


1
Expert's answer
2022-03-18T07:21:31-0400

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 kZkZ for kZk\in Z \{0}\{0\} . The quotient group Z/kZ={[0],[1],...[k1]}Z/ kZ=\{[0],[1],...[k-1]\} is of order kk ,and hence every non-trivial subgroup of a cyclic group has finite index.

The group (Q,+)(Q,+) contains non-trivial subgroup(Z,+)(Z,+) of infinite continuum index:

Q/Z=[0,1)=c|Q/Z|=|[0,1)|=c . Hence the group (Q,+)(Q,+) is not cyclic.


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