query: For what values of $n$ does every element of the group of units of $\mathbb{Z}_n$ have order dividing 2?
positive_passages.docid: 40131
positive_passages.text: If n is such that every element $(\mathbb{Z}/n\mathbb{Z})^{\times}$ is a root of $x^2-1$. Prove that $n$ divides 24.
negative_passages.docid: 35691
negative_passages.text: $\mathbb Z[\sqrt 3]$ contains infinitely many units
negative_passages.docid: 118849
negative_passages.text: If a cyclic group has an element of infinite order, how many elements of finite order does it have?
negative_passages.docid: 75127
negative_passages.text: Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$
negative_passages.docid: 157806
negative_passages.text: Order of a element on a Symmetric Group
negative_passages.docid: 19884
negative_passages.text: The units of $\mathbb Z_4[x]$
negative_passages.docid: 122125
negative_passages.text: Order of group element divides order of finite group
negative_passages.docid: 81543
negative_passages.text: Finite abelian group contains an element with order equal to the lcm of the orders of its elements
negative_passages.docid: 140144
negative_passages.text: Can you find an element of order 10 in $\mathbb{Z}_{31}^{*}$
negative_passages.docid: 13761
negative_passages.text: Automorphisms in $\mathbb{Z}_n$
negative_passages.docid: 95863
negative_passages.text: Describe the units in $\mathbb{Z}[\sqrt{-d}]$
negative_passages.docid: 104549
negative_passages.text: When $d \mid n$, the number of elements of order $d$ in a cyclic subgroup of order $n$ is $\phi (d)$
negative_passages.docid: 77799
negative_passages.text: Determine the set of units $(\Bbb Z/n\Bbb Z)^\times$ for different values of $n$
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: 101602
negative_passages.text: consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
negative_passages.docid: 37351
negative_passages.text: Group of Order $p^2$ Isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_{p}$ $\times$ $\mathbb{Z}_{p}$
negative_passages.docid: 86807
negative_passages.text: The units of $\mathbb Z[\sqrt{2}]$