query: Let $G$ be a group and $H$ a normal subgroup of order 2. Prove that if $G/H$ is cyclic, then $G$ is abelian.
positive_passages.docid: 23394
positive_passages.text: If $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 134820
negative_passages.text: $p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 124709
negative_passages.text: Suppose G be a non-abelian group and H,K be two abelian subgroups of G. Then must HK be an abelian subgroup of G?
negative_passages.docid: 23533
negative_passages.text: All subgroups normal $\implies$ abelian group
negative_passages.docid: 46342
negative_passages.text: Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism
negative_passages.docid: 76317
negative_passages.text: Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian
negative_passages.docid: 90332
negative_passages.text: Prove that that $U(n)$ is an abelian group.
negative_passages.docid: 98542
negative_passages.text: Prove that $H$ is a subgroup of an abelian group $G$
negative_passages.docid: 87882
negative_passages.text: If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
negative_passages.docid: 41237
negative_passages.text: $H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
negative_passages.docid: 144065
negative_passages.text: Proving $\phi(G)$ is cyclic if $\phi$ : $G \to H$ is group homomorphism and $G$ is cyclic
negative_passages.docid: 156429
negative_passages.text: If $G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian.
negative_passages.docid: 152071
negative_passages.text: Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
negative_passages.docid: 13742
negative_passages.text: Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.
negative_passages.docid: 97565
negative_passages.text: Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
negative_passages.docid: 29996
negative_passages.text: Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$