query: The free group of rank $n$ mod its commutator subgroup is isomorphic to the free abelian group of rank $n$
positive_passages.docid: 35913
positive_passages.text: Abelianization of free group is the free abelian group
negative_passages.docid: 33868
negative_passages.text: A transformation between two free group actions.
negative_passages.docid: 116936
negative_passages.text: Commutator subgroup and subgroup generated by square.
negative_passages.docid: 70904
negative_passages.text: Subgroups of $U(n)$ isomorphic to a direct sum of cyclic groups
negative_passages.docid: 40321
negative_passages.text: $\operatorname{Aut}(\mathbb Z_n)$ is isomorphic to $U_n$.
negative_passages.docid: 48393
negative_passages.text: Abelian groups of order n.
negative_passages.docid: 87882
negative_passages.text: If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
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: 120035
negative_passages.text: Pattern in Isomorphisms between $\mathbb Z_m \times \mathbb Z_n$ and $\mathbb Z_{mn}$
negative_passages.docid: 7575
negative_passages.text: Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$
negative_passages.docid: 155669
negative_passages.text: $\mathrm{Aut}(D_4)$ is isomorphic to $D_4$
negative_passages.docid: 123130
negative_passages.text: Showing an isomorphism between exterior products of a free module
negative_passages.docid: 152603
negative_passages.text: Group isomorphic to its automorphism group
negative_passages.docid: 58307
negative_passages.text: Commutator Subgroup and the Largest Abelian Quotient
negative_passages.docid: 151287
negative_passages.text: Isomorphic abelian groups which are not isomorphic R-modules
negative_passages.docid: 98507
negative_passages.text: $\text{Aut}(F)$ is isomorphic to the multiplicative group of all $n\times n$ matrices over $\mathbb Z$
negative_passages.docid: 4648
negative_passages.text: Proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$