query: Why does all order $n$ subgroups of an order $2n$ group have to be normal?
positive_passages.docid: 75568
positive_passages.text: Subgroup of index $2$ is Normal
negative_passages.docid: 86336
negative_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: 19352
negative_passages.text: Normal subgroup of index m contains every element of order n where m and n are relatively prime
negative_passages.docid: 113925
negative_passages.text: Normal Subgroup Counterexample
negative_passages.docid: 142413
negative_passages.text: Normal Subgroups of $SU(n)$
negative_passages.docid: 132704
negative_passages.text: If G is not commutative, then is there always a subgroup that is not a normal subgroup?
negative_passages.docid: 111375
negative_passages.text: Normality of subgroups and Sylow groups under homomorphisms...
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: 138406
negative_passages.text: If N and every subgroup of N is normal in G then G/N is abelian .
negative_passages.docid: 20386
negative_passages.text: Any subgroup of index $p$ in a $p$-group is normal.
negative_passages.docid: 151029
negative_passages.text: The number of non normal subgroups of a group cannot be $1$?
negative_passages.docid: 99309
negative_passages.text: Groups of order $n^2$ that have no subgroup of order $n$
negative_passages.docid: 115204
negative_passages.text: Show that SO(n) is a normal subgroup of O(n)
negative_passages.docid: 88344
negative_passages.text: Order of a group and subsequent subgroups
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: 13523
negative_passages.text: Are normal subgroups transitive?
negative_passages.docid: 48393
negative_passages.text: Abelian groups of order n.