Exercises

1. Prove that the map in Example 7.1.3 is a homomorphism.
2. Let $G_1$, $G_2$, $G_3$ be groups. Suppose $f_1 : G_1 \rightarrow G_2$ and $f_2 : G_2 \rightarrow G_3$ are homomorphisms. Then show $f_2f_1 : G_1 \rightarrow G_3$ is also a homomorphism.
3. Verify that the mapping defined in Example 7.1.9 is a homomorphism. In the second case, i.e., the map $f : \mathbb {Z}/6\mathbb {Z}\rightarrow \mathbb {Z}/3\mathbb {Z}$, show $f$ is well-defined, $f$ is a hom, find $Ker(f)$, and state the conclusion of the FHT for this $f$.
4. Verify that $f : \mathbb {R}\rightarrow \mathbb {C}$ defined by $f(x) = \cos (2\pi x) + i\sin( 2\pi x)$ (= $e^{2\pi ix}$) is a homomorphism of the additive group of $\mathbb {R}$ onto the group of all complex numbers of absolute value $1$. (Recall if $z = a +bi$, $\vert z\vert=\sqrt{a^2+b^2}$.) What is $Ker(f)$? State the conclusion of the FHT for this map.