query: $H \lhd N \lhd G$ but $H \ntriangleleft G$
positive_passages.docid: 13523
positive_passages.text: Are normal subgroups transitive?
negative_passages.docid: 106927
negative_passages.text: How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
negative_passages.docid: 16602
negative_passages.text: Proving $M_n(I)\lhd M_n(R)$ and proofs about ideal rings.
negative_passages.docid: 141043
negative_passages.text: $H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
negative_passages.docid: 33397
negative_passages.text: Sobolev space $H^s(\mathbb{R}^n)$ is an algebra with $2s>n$
negative_passages.docid: 110217
negative_passages.text: $GL(n,K)$ is open in $M(n,K)$
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: 41237
negative_passages.text: $H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
negative_passages.docid: 132311
negative_passages.text: T$\mathbb{S}^{n} \times \mathbb{R}$ is diffeomorphic to $\mathbb{S}^{n}\times \mathbb{R}^{n+1}$
negative_passages.docid: 27436
negative_passages.text: Does $G\cong G/H$ imply that $H$ is trivial?
negative_passages.docid: 46355
negative_passages.text: If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
negative_passages.docid: 154117
negative_passages.text: Showing $[(fg)h]_n = [f(gh)_n]$ in the linear algebra $F^\infty$
negative_passages.docid: 35188
negative_passages.text: Counterexample: $G \times K \cong H \times K \implies G \cong H$
negative_passages.docid: 116737
negative_passages.text: An element of a group $G$ is not conjugate to its inverse if $\lvert G\rvert$ is odd
negative_passages.docid: 34724
negative_passages.text: For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
negative_passages.docid: 124309
negative_passages.text: $g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
negative_passages.docid: 154398
negative_passages.text: Show that H is a subgroup of G