query: What are the invariant factors of $G = (\Bbb Z/12\Bbb Z) \times (\Bbb Z/25 \Bbb Z) \times (\Bbb Z/45\Bbb Z)$?
positive_passages.docid: 155051
positive_passages.text: Writing $\left(\mathbf Z/2\oplus\mathbf Z/4\oplus\mathbf Z/8\right)\oplus\left(\mathbf Z/9\oplus\mathbf Z/27\right)$ in a different way
negative_passages.docid: 115876
negative_passages.text: $D_6 \cong D_3\times \mathbb Z_2$
negative_passages.docid: 5223
negative_passages.text: $\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$
negative_passages.docid: 98284
negative_passages.text: show $\Bbb Z_6 \cong \Bbb Z_3 \times \Bbb Z_2 $.
negative_passages.docid: 93291
negative_passages.text: Method for determining irreducibles and factorising in $\mathbb Z[\sqrt{d}]$
negative_passages.docid: 11665
negative_passages.text: Show that $D_{12}$ is isomorphic to $D_6\times C_2$
negative_passages.docid: 71245
negative_passages.text: $2 \times 2 $ Orthogonal Group
negative_passages.docid: 41069
negative_passages.text: Justifying $\,\Bbb{Z}[i]/\langle 2\rangle\cong \Bbb{Z}[X]/\langle X^2+1,2\rangle\cong \Bbb{F}_2[X]/\langle X^2+1\rangle$
negative_passages.docid: 48839
negative_passages.text: Show that the order of $a\times b$ is equal to $nm$ if gcd(n,m)=1
negative_passages.docid: 152306
negative_passages.text: Finding homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{6}$.
negative_passages.docid: 131800
negative_passages.text: How to solve cycle equations in Group Theory? Is my approach correct?
negative_passages.docid: 129789
negative_passages.text: Find all Quotient (or Factor) Groups of $D_4$ (Dihedral Group of order $8$)
negative_passages.docid: 47824
negative_passages.text: Calculation of subgroups of $(Z_{12}, +)$
negative_passages.docid: 101596
negative_passages.text: Determine group G as cyclic
negative_passages.docid: 98765
negative_passages.text: Showing that $G/(H\cap K)\cong (G/H)\times (G/K)$
negative_passages.docid: 11003
negative_passages.text: Proving that $G/N$ is an abelian group
negative_passages.docid: 32278
negative_passages.text: Adjoining 1/2 to $\Bbb{Z}/12\Bbb{Z}$