Exercises

1. Verify that the ``componentwise'' multiplication given in Definition 9.1.1 is actually a binary operation on $G_1 \times G_2 \times ... \times G_n$ (external). Also verify that this binary operation is associative.

2. Prove Proposition 9.1.2.

3. Verify the first two statements in the proof of Theorem 9.1.6; i.e.,

   (1) $H_1\leq G$,

   (2) the map $a_i\longmapsto (e_1, ..., e_{i- 1}, a_i, e_{i+1}, ..., e_n)$ is an isomorphism of $G_i$ onto $H_i$.

4. Let $H_1\lhd G$, $H_2 \lhd G$ be such that the canonical homomorphism $G \rightarrow G/H_2$ when restricted to $H_1$ gives an isomorphism of $H_1$ onto $G/H_2$. Then prove $G = H_1 \times H_2$ (internal).

5. Let $G$ be an abelian group and $H \leq G$ such that $G/H$ is an infinite cyclic group. Then prove that $G\cong H \times G/H$. (HINT: Use exercise 4 above.)