query: Every subgroup of index $p$ is normal in $G$.
positive_passages.docid: 20386
positive_passages.text: Any subgroup of index $p$ in a $p$-group is normal.
negative_passages.docid: 15706
negative_passages.text: "Every Group of order $216$ contains a normal subgroup of order $27$ or $9$
negative_passages.docid: 70373
negative_passages.text: Intersection of normal subgroups proof
negative_passages.docid: 23991
negative_passages.text: Every finite group is isomorphic to some Galois group for some finite normal extension of some field.
negative_passages.docid: 124179
negative_passages.text: Proof that every metric space is normal
negative_passages.docid: 34902
negative_passages.text: If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
negative_passages.docid: 118337
negative_passages.text: Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$
negative_passages.docid: 58937
negative_passages.text: Finite Index of Subgroup of Subgroup
negative_passages.docid: 57227
negative_passages.text: Maximal normal $\pi$-subgroups and torsion subgroups
negative_passages.docid: 141043
negative_passages.text: $H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
negative_passages.docid: 122335
negative_passages.text: Abelian $p$-group with unique subgroup of index $p$
negative_passages.docid: 42389
negative_passages.text: If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
negative_passages.docid: 35602
negative_passages.text: If a subgroup has smallest prime index, then it is normal
negative_passages.docid: 115204
negative_passages.text: Show that SO(n) is a normal subgroup of O(n)
negative_passages.docid: 48273
negative_passages.text: If a maximal subgroup is normal, it has prime index
negative_passages.docid: 97928
negative_passages.text: Normal subgroup of a normal subgroup
negative_passages.docid: 94999
negative_passages.text: A subgroup with an index of 2 is normal to the group.