As I know quotient group $\Bbb Q/\Bbb Z$ is isomorphic to $n$-th roots of unity in $\Bbb C^* $, so every finite subgroup of $\Bbb Q/\Bbb Z$ is cyclic . Now $\Bbb Q/\Bbb Z$ also has infinite proper non-cyclic subgroups because of subgroup $2^n$-th roots for $n=0,1,\cdots$ of $\Bbb C^* $. Am in right? I am confused because one of my local book it is written as every proper subgroup of $\Bbb Q/\Bbb Z$ is cyclic. Please suggest. Thanks you.
Subgroups of Quotient group $\Bbb Q/\Bbb Z$ of infinite order .
group-theoryquotient-group
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Best Answer
The book claim is indeed false; $2^n$-th roots of unity correspond to dyadic rationals in $\mathbb Q/\mathbb Z$. $\frac13$ is not in the group because it has an infinite binary expansion. Similarly, those elements with a terminating base-$b$ expansion are non-cyclic proper subgroups of $\mathbb Q/\mathbb Z$.