Classify via fundamental theorem of finitely generated abelian groups. Z×Z⟨3182⟩

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Classify via fundamental theorem of finitely generated abelian groups.  Z×Z⟨3182⟩

Cleveland Walters · Accepted Answer

Step 1  Let f be as defined in the question. We get  Z×Zkerf=∼⟨x,y∣3x+y, 8x+2y, x+y=y+x⟩  =∼⟨x,y∣y=−3x, 8x+2y, x+y=y+x⟩  1) =∼⟨x∣8x−6x, x−3x=−3x⟩  =∼⟨x∣2x, −2x=−2x⟩  =∼⟨x∣2x⟩  =∼Z2  where (1) holds by the Tietze transformation of eliminating y as y can be written in terms of x.

Ethan Sanders · Accepted Answer

Step 1  The algorithmic way of solving problems like this is to compute the Smith normal form of the matrix formed by the ideal generators, which reveals the invariant factors.   In this case the invariant factors are 1,2, which means the group is isomorphic to  Z(1)⊕Z(2)=Z2  The following invertible elementary row/column operations convert the original matrix to Smith normal form:  [3182]→[3120]→[1120]→[0120]→[1002]

Classify via fundamental theorem of finitely generated abelian groups. \frac{\mathbb{Z}\times\mathbb{Z}}{<<(3,1),(8,2)>>}

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