query: Do there exist groups $X,Y$ such that $X/Y,Y$ are commutative/cyclic but $X$ is not?
positive_passages.docid: 17077
positive_passages.text: Prove or disprove: If $H$ and $G/H$ are cyclic, then $G$ is cyclic.
negative_passages.docid: 44361
negative_passages.text: An automorphism that has no fixed points except for the identity and is its own inverse implies commutativity
negative_passages.docid: 29538
negative_passages.text: Does a non-abelian semigroup without identity exist?
negative_passages.docid: 70298
negative_passages.text: Are quotients of chain groups $C_n(X)/C_n(A)$ still free?
negative_passages.docid: 98145
negative_passages.text: If $G$ is cyclic then $G/H$ is cyclic?
negative_passages.docid: 121810
negative_passages.text: Prove that the product of two infinite cyclic groups is not infinite cyclic.
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: 2458
negative_passages.text: Multiplicative group $(\mathbb R^*, ×)$ is group but $(\mathbb R, ×)$ is not group, why?
negative_passages.docid: 122012
negative_passages.text: The ideal $(x,y)$ is not a free $K[x,y]$-module
negative_passages.docid: 144442
negative_passages.text: Let $R$ be a commutative unital ring. Is it true that the group of units of $R$ is not isomorphic with the additive group of $R$?
negative_passages.docid: 156443
negative_passages.text: if $x$ and $y$ commute with $[x y]$, then $[x y]$ is nilpotent
negative_passages.docid: 74978
negative_passages.text: How the center of a non-cyclic free Group is trivial?
negative_passages.docid: 91021
negative_passages.text: Torsion elements of a group aren't necessarily a subgroup
negative_passages.docid: 147517
negative_passages.text: Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic
negative_passages.docid: 99788
negative_passages.text: Why does a multiplicative subgroup of a field have to be cyclic?
negative_passages.docid: 73248
negative_passages.text: Prove that $\mathbb{R^*}$, the set of all real numbers except $0$, is not a cyclic group
negative_passages.docid: 132386
negative_passages.text: Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic