query: Quotient group $G/H$ isomorphic to $G$
positive_passages.docid: 27436
positive_passages.text: Does $G\cong G/H$ imply that $H$ is trivial?
negative_passages.docid: 56245
negative_passages.text: Let H be a subgroup of a group G. Prove that the following statements are equivalent.
negative_passages.docid: 87766
negative_passages.text: Using Cayley's Theorem to find an isomorphic subgroup.
negative_passages.docid: 23579
negative_passages.text: If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
negative_passages.docid: 150854
negative_passages.text: Identifying a quotient group (NBHM-$2014$)
negative_passages.docid: 140962
negative_passages.text: Counterexamples for $G \times H$ isomorphic to $G$ implies $H$ trivial.
negative_passages.docid: 94090
negative_passages.text: How to show that for any abelian group $G$, $\text{Hom}(\mathbb{Z},G)$ is isomorphic to $G$
negative_passages.docid: 129452
negative_passages.text: Groups $G,H$ with surjective homomorphisms that are quotient of each other
negative_passages.docid: 151727
negative_passages.text: If $|G/H|=4$ then $G$ is union of three proper subgroups
negative_passages.docid: 15276
negative_passages.text: $G \times H \cong G \times K$ , then $ K \cong H$
negative_passages.docid: 25439
negative_passages.text: f(a) = inverse of a is an isomorphism iff a group G is Abelian
negative_passages.docid: 112993
negative_passages.text: Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?
negative_passages.docid: 105910
negative_passages.text: How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
negative_passages.docid: 65744
negative_passages.text: How can I Prove that H=G or K=G
negative_passages.docid: 137739
negative_passages.text: Prove that the group isomorphism $\mathbb{Z}^m \cong \mathbb{Z}^n$ implies that $m = n$
negative_passages.docid: 157915
negative_passages.text: Isomorphism between the group $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$
negative_passages.docid: 37351
negative_passages.text: Group of Order $p^2$ Isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_{p}$ $\times$ $\mathbb{Z}_{p}$