Answer to Question #312461 in Abstract Algebra for Bishnu

Question

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

Answer 1

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

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

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