I've read in three different books about groups, subgroups and homomorphism but I just can't get the concept. I don't find the examples clear enough. For example, why isn't $$\Phi: \mathbb{Z} \rightarrow \mathbb{Q}, x \rightarrow x^2$$ a homomorphism?
I have to the do the following exercise:
Let $(G, \bullet)$ and $(H, *)$ be groups, where the neutral elements are $e_G$ and $e_H$. Further let $\Phi : G \rightarrow H$ be a homomorphism. The kernel from $\Phi$ is the set
$$Kernel \Phi = \{g \in G |\Phi(g) = e_H\}$$
a)
Show that the kernel $\Phi$ is a subgroup from $G$.
$\Phi$ is injective, when kernel $\Phi$ = $\{e_G\}$ is true.
What I have so far:
At number 1, isn't this true because the image from the function $\Phi$ from every element from $G$ is the neutral element? At number 2, if kernel $\Phi$ = $\{e_G\}$, then the homomorphism is true, this means $\Phi$ is injective.
b) Show that the set $G \times H$ with the operation $$(g_1, h_1)\star (g_2,h_2)=(g_1\bullet g_2, h_1*h_2)$$ is a group. Further, show that the function $$\Phi: (G \times H, *)\rightarrow (G, \bullet), (g,h)\rightarrow g,$$ ist a homomorphism. Define the kernel from $\Phi$.
I'm completely lost here. :(
Thanks A LOT in advance for any tips!