Assignment for week 1

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

2. Show explicitly that ℤ₁₂ is isomorphic to ℤ₄ ⊕ ℤ₃.

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

4. Find a minimal generating set for the abelian group ℤ₁₄ ⊕ ℤ₁₀ ⊕ ℤ₉.

5. Explictly find all the homomorphisms from ℤ₆ to ℤ₆. How many are isomorphisms?

6. Explicitly find all homomorphisms from ℤ to ℤ₂ ⊕ ℤ₂.

7. Each of the following matrices defines a map f: G → H, where both G and H are isomorphic to ℤ². 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. $\begin{bmatrix} 24 & 36 \\ 48 & 42 \end{bmatrix}$