query: The groups $G_1, G_2$ and the normal subgroups $N_1,N_2$ are isomorphic. Dis/prove $G_1/N_1\cong G_2/N_2$
positive_passages.docid: 2254
positive_passages.text: Finite group with isomorphic normal subgroups and non-isomorphic quotients?
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: 22477
negative_passages.text: If H is a subgroup of a group G and K is a normal subgroup of G the show that K is a normal subgroup of HK.
negative_passages.docid: 23279
negative_passages.text: Group Theory Isomorphism
negative_passages.docid: 103423
negative_passages.text: Normal subgroup where G has order prime
negative_passages.docid: 37379
negative_passages.text: Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$.
negative_passages.docid: 113357
negative_passages.text: Groups of order $pqr$ and their normal subgroups
negative_passages.docid: 115204
negative_passages.text: Show that SO(n) is a normal subgroup of O(n)
negative_passages.docid: 65968
negative_passages.text: Prove that any two cyclic groups of the same order are isomorphic?
negative_passages.docid: 50706
negative_passages.text: $|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
negative_passages.docid: 58707
negative_passages.text: Prove Quotient Group Isomorphism
negative_passages.docid: 101079
negative_passages.text: Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$
negative_passages.docid: 25404
negative_passages.text: Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
negative_passages.docid: 123610
negative_passages.text: How to show that the commutator subgroup is a normal subgroup
negative_passages.docid: 91774
negative_passages.text: Existence of group $G$ with $|G|>2$ and automorphism group of odd order
negative_passages.docid: 78553
negative_passages.text: A simple finite group $G$ with $n$ p-Sylows is isomorphic to a subgroup of $\mathbb A_n$
negative_passages.docid: 47179
negative_passages.text: Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?