query: If $H$ is a subgroup of $G$ having finite index, then $H$ contains a normal subgroup of $G$ of finite index
positive_passages.docid: 86336
positive_passages.text: For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
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: 103423
negative_passages.text: Normal subgroup where G has order prime
negative_passages.docid: 107935
negative_passages.text: If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
negative_passages.docid: 48273
negative_passages.text: If a maximal subgroup is normal, it has prime index
negative_passages.docid: 8841
negative_passages.text: A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
negative_passages.docid: 142894
negative_passages.text: Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
negative_passages.docid: 44165
negative_passages.text: the image of normal subgroups
negative_passages.docid: 80186
negative_passages.text: How to show that $K\leq S_4$ is a normal subgroup?
negative_passages.docid: 61379
negative_passages.text: Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
negative_passages.docid: 145511
negative_passages.text: Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
negative_passages.docid: 93596
negative_passages.text: If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
negative_passages.docid: 99999
negative_passages.text: A question on identifying normal subgroups of $S_4$
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: 64408
negative_passages.text: Subgroup of $\mathbb{Q}$ with finite index
negative_passages.docid: 11825
negative_passages.text: Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
negative_passages.docid: 37971
negative_passages.text: Prove that the center of a group is a normal subgroup