query: the unit group of an infinite field cannot be cyclic
positive_passages.docid: 21079
positive_passages.text: Why must a field whose a group of units is cyclic be finite?
negative_passages.docid: 132725
negative_passages.text: Is $\mathbb{Z}/\langle n\rangle$ cyclic and of order $n$?
negative_passages.docid: 91021
negative_passages.text: Torsion elements of a group aren't necessarily a subgroup
negative_passages.docid: 18160
negative_passages.text: How to show that the group of nonzero real numbers under multiplication is not isomorphic to the group of real numbers under addition?
negative_passages.docid: 126426
negative_passages.text: Does every infinite finitely generated group contains an element of infinite period?
negative_passages.docid: 93029
negative_passages.text: Prove that when $d$ is in $\mathbb{Q} \setminus\mathbb{Z}$, $\mathbb{Z}[\sqrt{d}]$ is not a finitely generated abelian group.
negative_passages.docid: 68672
negative_passages.text: Commutator subgroup of rank-2 free group is not finitely generated.
negative_passages.docid: 25050
negative_passages.text: To prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups.
negative_passages.docid: 92677
negative_passages.text: Infinite algebraic extension of a finite field
negative_passages.docid: 37904
negative_passages.text: Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.
negative_passages.docid: 145561
negative_passages.text: About the additive group and the multiplicative group of a field
negative_passages.docid: 35691
negative_passages.text: $\mathbb Z[\sqrt 3]$ contains infinitely many units
negative_passages.docid: 32516
negative_passages.text: show that rational numbers with the multipiciation are not abelian finitely generated group
negative_passages.docid: 104562
negative_passages.text: Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field.
negative_passages.docid: 114662
negative_passages.text: Construction of a specific non-commutative and infinite group (with conditions on the order of the elements)
negative_passages.docid: 112933
negative_passages.text: Show that $Z\times Z$ is not cyclic...
negative_passages.docid: 18182
negative_passages.text: An abelian group $G$ with ${\rm Aut}(G)$ non-abelian