query: Unique order $d$ subgroup of $\Bbb Z/n\Bbb Z$ if $d\mid n$
positive_passages.docid: 39938
positive_passages.text: Subgroups of a cyclic group and their order.
negative_passages.docid: 68201
negative_passages.text: Cyclic group of order $n$
negative_passages.docid: 33387
negative_passages.text: Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$
negative_passages.docid: 22444
negative_passages.text: Normal subgroups of dihedral groups
negative_passages.docid: 38153
negative_passages.text: Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .
negative_passages.docid: 37913
negative_passages.text: Prove that $ AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.
negative_passages.docid: 114667
negative_passages.text: $d\mid a,b \iff d\mid\gcd(a,b) \ $ [GCD Universal Property]
negative_passages.docid: 24970
negative_passages.text: Only two groups of order $10$: $C_{10}$ and $D_{10}$
negative_passages.docid: 79055
negative_passages.text: Orders of a symmetric group
negative_passages.docid: 97928
negative_passages.text: Normal subgroup of a normal subgroup
negative_passages.docid: 37928
negative_passages.text: Isomorphism between $D_{2n}/Z(D_{2n})$ and $D_n$
negative_passages.docid: 52565
negative_passages.text: Prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$.
negative_passages.docid: 6134
negative_passages.text: Some Subgroup of Dihedral Group is Normal
negative_passages.docid: 36709
negative_passages.text: subgroup of direct product of two groups
negative_passages.docid: 88344
negative_passages.text: Order of a group and subsequent subgroups
negative_passages.docid: 113357
negative_passages.text: Groups of order $pqr$ and their normal subgroups
negative_passages.docid: 1712
negative_passages.text: $ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property]