query: prove $G=HN$, if $N \lhd G$ with prime index, and $H$ is not contained in $N$
positive_passages.docid: 147253
positive_passages.text: Prove for any $K \leq G$, either $K \leq N$ or $KN=G$ and $[K:K \cap N]=p$ where N is normal in G and $[G:N]=p$
negative_passages.docid: 93666
negative_passages.text: Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$?
negative_passages.docid: 85129
negative_passages.text: If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
negative_passages.docid: 4838
negative_passages.text: For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
negative_passages.docid: 94571
negative_passages.text: How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
negative_passages.docid: 122581
negative_passages.text: Prove or disprove that $G_1/H_1 \cong G_2/H_2$
negative_passages.docid: 94507
negative_passages.text: Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
negative_passages.docid: 50706
negative_passages.text: $|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
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: 105910
negative_passages.text: How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
negative_passages.docid: 79826
negative_passages.text: $gHg^{-1}\subset H$ whenever $Ha\not = Hb$ implies $aH\not =bH$
negative_passages.docid: 111024
negative_passages.text: Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$?
negative_passages.docid: 151727
negative_passages.text: If $|G/H|=4$ then $G$ is union of three proper subgroups
negative_passages.docid: 58033
negative_passages.text: $G\times H\cong G$ with $H$ non-trivial
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: 37913
negative_passages.text: Prove that $ AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.
negative_passages.docid: 111703
negative_passages.text: If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple