query: $H$ is a normal subgroup which is not closed. How to prove the quotient group $G/H$ is not Hausdorff.
positive_passages.docid: 154682
positive_passages.text: G/H is Hausdorff implies H is closed (General topology, Volume 1 by N. Bourbaki)
negative_passages.docid: 37904
negative_passages.text: Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.
negative_passages.docid: 153797
negative_passages.text: Is a quotient group a subgroup?
negative_passages.docid: 105139
negative_passages.text: Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center
negative_passages.docid: 62635
negative_passages.text: Prove that the additive group $ℚ$ is not isomorphic with the multiplicative group $ℚ^*$.
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: 29996
negative_passages.text: Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
negative_passages.docid: 129452
negative_passages.text: Groups $G,H$ with surjective homomorphisms that are quotient of each other
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: 75254
negative_passages.text: Properties possessed by $H , G/H$ but not G
negative_passages.docid: 93596
negative_passages.text: If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
negative_passages.docid: 46904
negative_passages.text: Let $G$ be a non-trivial group with no non-trivial proper subgroup. Prove that $G$ cannot be infinite group.
negative_passages.docid: 113013
negative_passages.text: Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
negative_passages.docid: 67383
negative_passages.text: Prove that $G'$ is not planar.
negative_passages.docid: 37971
negative_passages.text: Prove that the center of a group is a normal subgroup
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: 33370
negative_passages.text: Show that the one point compactification of $\mathbb{Q}$ is not Hausdorff