query: If $G$ is a torsion-free group, such that there exists a subgroup $H \cong \mathbb Z$ of finite index, then $G \cong$?
positive_passages.docid: 28527
positive_passages.text: Torsion-free virtually-Z is Z
negative_passages.docid: 2254
negative_passages.text: Finite group with isomorphic normal subgroups and non-isomorphic quotients?
negative_passages.docid: 154398
negative_passages.text: Show that H is a subgroup of G
negative_passages.docid: 155502
negative_passages.text: Index of a maximal subgroup in a finite group
negative_passages.docid: 107511
negative_passages.text: If $G$ and $H$ are nonisomorphic group with same order then can we say that $Aut(G)$ is not isomorphic to $Aut(H)$?
negative_passages.docid: 93666
negative_passages.text: Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$?
negative_passages.docid: 106277
negative_passages.text: Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
negative_passages.docid: 91021
negative_passages.text: Torsion elements of a group aren't necessarily a subgroup
negative_passages.docid: 86823
negative_passages.text: proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian
negative_passages.docid: 132704
negative_passages.text: If G is not commutative, then is there always a subgroup that is not a normal subgroup?
negative_passages.docid: 94507
negative_passages.text: Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
negative_passages.docid: 155054
negative_passages.text: $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
negative_passages.docid: 153797
negative_passages.text: Is a quotient group a subgroup?
negative_passages.docid: 101345
negative_passages.text: Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is...
negative_passages.docid: 61064
negative_passages.text: Finite index subgroup $G$ of $\mathbb{Z}_p$ is open.
negative_passages.docid: 133001
negative_passages.text: $G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
negative_passages.docid: 35353
negative_passages.text: Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$?