query: Show that $G=HN$ for $N$ normal and $H$ subgroup.
positive_passages.docid: 155054
positive_passages.text: $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
negative_passages.docid: 59561
negative_passages.text: Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
negative_passages.docid: 98765
negative_passages.text: Showing that $G/(H\cap K)\cong (G/H)\times (G/K)$
negative_passages.docid: 44165
negative_passages.text: the image of normal subgroups
negative_passages.docid: 116125
negative_passages.text: A group $G$ with $G'$ abelian and every abelian normal subgroup finite
negative_passages.docid: 11731
negative_passages.text: Normal subgroups of $S_4$
negative_passages.docid: 14255
negative_passages.text: Commutator subgroup and cyclic normal subgroup
negative_passages.docid: 87882
negative_passages.text: If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
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: 25832
negative_passages.text: Let $K$ be a normal subgroup of order 2 in group $G$, show that $K$ lies in the centre of $G$
negative_passages.docid: 90332
negative_passages.text: Prove that that $U(n)$ is an abelian group.
negative_passages.docid: 95534
negative_passages.text: If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
negative_passages.docid: 13742
negative_passages.text: Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.
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: 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: 71819
negative_passages.text: Prove that $H$ is normal.
negative_passages.docid: 47179
negative_passages.text: Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?