query: Show that H is normal subgroup of G.
positive_passages.docid: 75568
positive_passages.text: Subgroup of index $2$ is Normal
negative_passages.docid: 44942
negative_passages.text: A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
negative_passages.docid: 65972
negative_passages.text: Show that a group of order $108$ has a normal subgroup of order $9$ or $27$.
negative_passages.docid: 106927
negative_passages.text: How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
negative_passages.docid: 98739
negative_passages.text: If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
negative_passages.docid: 43651
negative_passages.text: Normal subgroups and factor groups
negative_passages.docid: 154878
negative_passages.text: Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
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: 111375
negative_passages.text: Normality of subgroups and Sylow groups under homomorphisms...
negative_passages.docid: 44934
negative_passages.text: Show that every normal subgroup can be written as an intersection of $\ker \chi$'s for $\chi \in \operatorname{Irr}(G).$
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 7973
negative_passages.text: Groups of order $pq$ have a proper normal subgroup
negative_passages.docid: 85508
negative_passages.text: Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
negative_passages.docid: 97928
negative_passages.text: Normal subgroup of a normal subgroup
negative_passages.docid: 116786
negative_passages.text: normal subgroups of Alternating group
negative_passages.docid: 11957
negative_passages.text: G is the product of its Sylow subgroups
negative_passages.docid: 28108
negative_passages.text: Proving the kernel is a normal subgroup.