Assignment for week 1

  1. Get the first example to run on your own computer.

  2. Show explicitly that \(\mathbb{Z}_{12}\) is isomorphic to \(\mathbb{Z}_{4} \oplus \mathbb{Z}_{3}\).

  3. How many abelian groups of size 24 are there?

  4. Find a minimal generating set for the abelian group \(\mathbb{Z}_{14} \oplus \mathbb{Z}_{10} \oplus \mathbb{Z}_{9}\).

  5. Explictly find all the homomorphisms from \(\mathbb{Z}_6\) to \(\mathbb{Z}_6\). How many are isomorphisms?

  6. Explicitly find all homomorphisms from \(\mathbb{Z}\) to \(\mathbb{Z}_2 \oplus \mathbb{Z}_2\).

  7. Each of the following matrices defines a map \(f \colon G \to H\), where both \(G\) and \(H\) are isomorphic to \(\mathbb{Z}^2\). Find H/f(G).

    1. \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\)
    2. \(\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}\)
    3. \(\begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}\)
    4. \(\begin{bmatrix} 24 & 36 \\ 48 & 42 \end{bmatrix}\)