query: If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
positive_passages.docid: 145511
positive_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: 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: 157819
negative_passages.text: Every group of order $56$ has a proper non - trivial normal subgroup?
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: 75590
negative_passages.text: How many normal subgroups does a non-abelian group $G$ of order $ 21$
negative_passages.docid: 14666
negative_passages.text: What are the non-trivial normal subgroups of $O(3)$?
negative_passages.docid: 104080
negative_passages.text: Group with order $pq$ has subgroups of order $p$ and $q$
negative_passages.docid: 1220
negative_passages.text: Have $\alpha:G \to H$. If a subgroup $U$ is normal in $H$ prove the pre-image of $U$ is normal in $G$.
negative_passages.docid: 90449
negative_passages.text: Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
negative_passages.docid: 106779
negative_passages.text: How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
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: 58205
negative_passages.text: $ N $ normal in a finite group $ G $, $ |N| = 5 $ and $ |G| $ odd. Why is $ N \subseteq Z(G) $?
negative_passages.docid: 125498
negative_passages.text: Check whether subgroup is normal in $ S_4$
negative_passages.docid: 15706
negative_passages.text: "Every Group of order $216$ contains a normal subgroup of order $27$ or $9$
negative_passages.docid: 25404
negative_passages.text: Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
negative_passages.docid: 12667
negative_passages.text: How many subgroups of order $n$ does $D_n$ have?