Find conjugacy classes and normal subgroups of a direct product

Question

How can I find the conjugacy classes and the normal subgroups of the direct product: $\mathbb{Z}_{2} \times \mathbb{Q}_{8}$

($\mathbb{Q}_{8}$ is the quaternion group)

How do I start an exercise like this?

Answer 1

Consider two groups $G$ and $H$ and their direct product $G \times H$. Let $N_1 \unlhd G$ and let $N_2 \unlhd H$. We can see that $N_1 \times N_2$ is a subgroup of $G \times H$. To show that it is a normal subgroup, let $g \in N_1$ and $h \in N_2$, then: $$(x,y)(g,h)(x,y)^{-1}=(xgx^{-1},yhy^{-1}) \in N_1 \times N_2$$ for any $(x,y) \in G \times H$, since $N_1$ and $N_2$ are themselves normal in $G$ and $H$ respectively. We can moreover show using this similar argument that any normal subgroup of $G \times H$ has to be of the form $N_1 \times N_2$. A similar analogy exists for conjugacy classes. Can you proceed from here?