query: Give an example of a non-abelian group G containing a proper normal subgroup N such that $G/N$ is abelian.
positive_passages.docid: 27703
positive_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: 44942
negative_passages.text: A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
negative_passages.docid: 101360
negative_passages.text: Why is a normal subgroup of $G_1\times G_2$ with trivial intersections with $G_1$ and $G_2$ is abelian?
negative_passages.docid: 89382
negative_passages.text: In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.
negative_passages.docid: 33593
negative_passages.text: An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$
negative_passages.docid: 97928
negative_passages.text: Normal subgroup of a normal subgroup
negative_passages.docid: 65926
negative_passages.text: Group of order $1575$ having a normal sylow $3$ subgroup is abelian.
negative_passages.docid: 85508
negative_passages.text: Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
negative_passages.docid: 80849
negative_passages.text: There does not exist group $G$ such that Aut$(G)\cong \mathbb{Z}_n$ (for odd $n$)
negative_passages.docid: 138108
negative_passages.text: If a normal subgroup $N$ of order $p$($p$ prime) is contained in a group $G$ of order $p^n$,then $N$ is in the center of $G$.
negative_passages.docid: 111168
negative_passages.text: Automorphisms of a finite non-abelian group
negative_passages.docid: 47539
negative_passages.text: Find an abelian infinite group such that every proper subgroup is finite
negative_passages.docid: 115204
negative_passages.text: Show that SO(n) is a normal subgroup of O(n)
negative_passages.docid: 60075
negative_passages.text: Does there exist an $n$ such that all groups of order $n$ are Abelian?
negative_passages.docid: 116125
negative_passages.text: A group $G$ with $G'$ abelian and every abelian normal subgroup finite
negative_passages.docid: 142375
negative_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: 58688
negative_passages.text: Example of a group where $o(a)$ and $o(b)$ are finite but $o(ab)$ is infinite