query: Given $G,H_1,H_2$ finite abelian groups and $G\times H_1 \cong G\times H_2$ then $H_1 \cong H_2$
positive_passages.docid: 18415
positive_passages.text: Direct sum of Abelian groups and Isomorphism
negative_passages.docid: 28210
negative_passages.text: A finite group with exactly $2$ conjugacy classes isomorphic to $\mathbb{Z}_2$
negative_passages.docid: 145159
negative_passages.text: Finite group is abelian if the representatives of its conjugacy classes commute
negative_passages.docid: 101360
negative_passages.text: Why is a normal subgroup of $G_1\times G_2$ with trivial intersections with $G_1$ and $G_2$ is abelian?
negative_passages.docid: 137949
negative_passages.text: If the quotient group $G/N$ be isomorphic to $G_2$, is $G$ isomorphic to $G_1\times G_2$ where $N$ is isomorphic to $G_1$?
negative_passages.docid: 58033
negative_passages.text: $G\times H\cong G$ with $H$ non-trivial
negative_passages.docid: 127996
negative_passages.text: $M_1$ and $M_2$ are subgroups and $M_1/N=M_2/N$. Is $M_1\cong M_2$?
negative_passages.docid: 84789
negative_passages.text: Order of products of elements in a finite Abelian group
negative_passages.docid: 137186
negative_passages.text: Let G be $\mathbb Z_p\times\dots\times \mathbb Z_p$ . Find A(G).
negative_passages.docid: 157840
negative_passages.text: Is it true that $(R\times S)[G]\cong R[G]\times S[G]$?
negative_passages.docid: 135808
negative_passages.text: Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
negative_passages.docid: 106277
negative_passages.text: Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
negative_passages.docid: 18182
negative_passages.text: An abelian group $G$ with ${\rm Aut}(G)$ non-abelian
negative_passages.docid: 88718
negative_passages.text: A group is generated by two elements of order $2$ is infinite and non-abelian
negative_passages.docid: 51352
negative_passages.text: Subgroups of a finite abelian group
negative_passages.docid: 48393
negative_passages.text: Abelian groups of order n.
negative_passages.docid: 45310
negative_passages.text: If $H\leq Z (G)$ and $G/H$ is nilpotent, then $G$ is nilpotent.