query: filling in the gaps of this proof involving the fundamental theorem of finitely generated abelian groups
positive_passages.docid: 24966
positive_passages.text: Converse of Lagrange's theorem for abelian groups
negative_passages.docid: 93029
negative_passages.text: Prove that when $d$ is in $\mathbb{Q} \setminus\mathbb{Z}$, $\mathbb{Z}[\sqrt{d}]$ is not a finitely generated abelian group.
negative_passages.docid: 51359
negative_passages.text: Prove that every subgroup of an abelian group is a normal subgroup.
negative_passages.docid: 10816
negative_passages.text: Infinitely generated subgroup of a finitely one (weird)
negative_passages.docid: 48585
negative_passages.text: Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 108425
negative_passages.text: Problem on abelian group
negative_passages.docid: 71420
negative_passages.text: Prove that a group is abelian.
negative_passages.docid: 13355
negative_passages.text: Product of elements of a finite abelian group
negative_passages.docid: 126893
negative_passages.text: A group of order 2p (p prime) and other conditions - prove abelian.
negative_passages.docid: 137931
negative_passages.text: Problem from "The Theory of Finite Groups" by Kurzweil and Stellmacher
negative_passages.docid: 85508
negative_passages.text: Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
negative_passages.docid: 84789
negative_passages.text: Order of products of elements in a finite Abelian group
negative_passages.docid: 11000
negative_passages.text: Fundamental group of $\mathbb{R}^2 \setminus (\mathbb{Z} \times \{0\})$
negative_passages.docid: 99346
negative_passages.text: Prove a result on transitive group actions.
negative_passages.docid: 92822
negative_passages.text: Finite number of elements generating the unit ideal of a commutative ring
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: 18182
negative_passages.text: An abelian group $G$ with ${\rm Aut}(G)$ non-abelian