query: $G/N \cong G$ then $N = \{1\}$ with $G$ infinite.
positive_passages.docid: 27436
positive_passages.text: Does $G\cong G/H$ imply that $H$ is trivial?
negative_passages.docid: 50673
negative_passages.text: Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$
negative_passages.docid: 29445
negative_passages.text: Trying to prove that $H=\langle a,b:a^{3}=b^{3}=(ab)^{3}=1\rangle$ is a group of infinite order.
negative_passages.docid: 43004
negative_passages.text: Why are there $|G/G'|$ 1-dimensional representations of $G$?
negative_passages.docid: 17178
negative_passages.text: Order of $\phi(g)$ divides the order of $g$
negative_passages.docid: 123677
negative_passages.text: Normal subgroups of infinite symmetric group
negative_passages.docid: 110708
negative_passages.text: On proving that a finitely generated group has a finite number of subgroups with index $n$.
negative_passages.docid: 11003
negative_passages.text: Proving that $G/N$ is an abelian group
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: 99782
negative_passages.text: $a_{n+1}=|a_n|-a_{n-1} \implies a_n \; \text{is periodic}$
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: 64193
negative_passages.text: Cardinality of Infinite Symmetric group
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
negative_passages.docid: 21673
negative_passages.text: If $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m <n.$
negative_passages.docid: 120988
negative_passages.text: Infinite Series $\sum 1/(n(n+1))$
negative_passages.docid: 86926
negative_passages.text: Why isn't this a well ordering of $\{A\subseteq\mathbb N\mid A\text{ is infinite}\}$?
negative_passages.docid: 120495
negative_passages.text: Does this correctly interpret the "$\cong$" between groups as "$=$" between sets?