query: Prove or disprove: If $H$ is normal in $G$ and $H$ and $G/H$ are abelian, then $G$ is abelian.
positive_passages.docid: 142375
positive_passages.text: Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
negative_passages.docid: 27703
negative_passages.text: an example of a non-abelian group $G$ containing a proper normal subgroup $N$ such that $G/N$ is abelian.
negative_passages.docid: 124309
negative_passages.text: $g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
negative_passages.docid: 63136
negative_passages.text: Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic
negative_passages.docid: 135039
negative_passages.text: ${\rm Aut}(G)$ is cyclic $\implies G$ is abelian
negative_passages.docid: 48585
negative_passages.text: Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 121587
negative_passages.text: Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
negative_passages.docid: 106779
negative_passages.text: How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
negative_passages.docid: 93596
negative_passages.text: If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
negative_passages.docid: 108425
negative_passages.text: Problem on abelian group
negative_passages.docid: 17077
negative_passages.text: Prove or disprove: If $H$ and $G/H$ are cyclic, then $G$ is cyclic.
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: 27905
negative_passages.text: Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.
negative_passages.docid: 46355
negative_passages.text: If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
negative_passages.docid: 60208
negative_passages.text: Proof of property of abelian group
negative_passages.docid: 65744
negative_passages.text: How can I Prove that H=G or K=G
negative_passages.docid: 63780
negative_passages.text: to show characteristic subgroup is normal in G