query: Show that if a group $G$ has odd order, any subgroup $H$ of index 3 in $G$ is normal in $G$?
positive_passages.docid: 7108
positive_passages.text: Normal subgroup of prime index
negative_passages.docid: 20386
negative_passages.text: Any subgroup of index $p$ in a $p$-group is normal.
negative_passages.docid: 133298
negative_passages.text: Assume $|G|=40$. Show that the subgroup of order $8$ is normal and unique.
negative_passages.docid: 111375
negative_passages.text: Normality of subgroups and Sylow groups under homomorphisms...
negative_passages.docid: 4883
negative_passages.text: Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
negative_passages.docid: 146760
negative_passages.text: Prove that H is a subgroup of G
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: 141265
negative_passages.text: Finding a normal and not normal subgroup of $S_3$
negative_passages.docid: 44152
negative_passages.text: If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
negative_passages.docid: 94999
negative_passages.text: A subgroup with an index of 2 is normal to the group.
negative_passages.docid: 7973
negative_passages.text: Groups of order $pq$ have a proper normal subgroup
negative_passages.docid: 80186
negative_passages.text: How to show that $K\leq S_4$ is a normal subgroup?
negative_passages.docid: 103423
negative_passages.text: Normal subgroup where G has order prime
negative_passages.docid: 116786
negative_passages.text: normal subgroups of Alternating group
negative_passages.docid: 1360
negative_passages.text: Every group of order $150$ has a normal subgroup of order $25$
negative_passages.docid: 131204
negative_passages.text: A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
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 $n!$