query: Proving that if left coset multiplication is well-defined then $H$ is a normal subgroup
positive_passages.docid: 5024
positive_passages.text: Why do we define quotient groups for normal subgroups only?
negative_passages.docid: 97928
negative_passages.text: Normal subgroup of a normal subgroup
negative_passages.docid: 45727
negative_passages.text: If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
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: 101079
negative_passages.text: Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$
negative_passages.docid: 23579
negative_passages.text: If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
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: 72150
negative_passages.text: Union of the conjugates of a proper subgroup
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 22444
negative_passages.text: Normal subgroups of dihedral groups
negative_passages.docid: 49527
negative_passages.text: Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
negative_passages.docid: 58251
negative_passages.text: To show every sylow p-subgroup is normal in G
negative_passages.docid: 98542
negative_passages.text: Prove that $H$ is a subgroup of an abelian group $G$
negative_passages.docid: 77160
negative_passages.text: How to prove the cosets are distinct?
negative_passages.docid: 125244
negative_passages.text: Equivalent definitions of normal subgroups
negative_passages.docid: 54911
negative_passages.text: The preimage of a normal subgroup under a group homomorphism is normal
negative_passages.docid: 95534
negative_passages.text: If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$