query: Why do normal subgroups "work"?
positive_passages.docid: 5024
positive_passages.text: Why do we define quotient groups for normal subgroups only?
negative_passages.docid: 141043
negative_passages.text: $H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
negative_passages.docid: 112263
negative_passages.text: an example for $\pi$-quasinormal subgroup but is not normal
negative_passages.docid: 54911
negative_passages.text: The preimage of a normal subgroup under a group homomorphism is normal
negative_passages.docid: 38095
negative_passages.text: The Commutator Subgroup is Normal.
negative_passages.docid: 49666
negative_passages.text: Generalization of index 2 subgroups are normal
negative_passages.docid: 70373
negative_passages.text: Intersection of normal subgroups proof
negative_passages.docid: 58205
negative_passages.text: $ N $ normal in a finite group $ G $, $ |N| = 5 $ and $ |G| $ odd. Why is $ N \subseteq Z(G) $?
negative_passages.docid: 48273
negative_passages.text: If a maximal subgroup is normal, it has prime index
negative_passages.docid: 156463
negative_passages.text: The complement of a proper subgroup generates the whole group
negative_passages.docid: 100974
negative_passages.text: Showing H is a normal subgroup by calculating left and right coset
negative_passages.docid: 123610
negative_passages.text: How to show that the commutator subgroup is a normal subgroup
negative_passages.docid: 125244
negative_passages.text: Equivalent definitions of normal subgroups
negative_passages.docid: 80381
negative_passages.text: Groups with "few" subgroups
negative_passages.docid: 109236
negative_passages.text: Subgroups containing kernel of group morphism to an abelian group are normal.
negative_passages.docid: 137691
negative_passages.text: What are the self-normalizing subgroups of $S_n$?
negative_passages.docid: 35602
negative_passages.text: If a subgroup has smallest prime index, then it is normal