het $G$ be a group. Recall, for $5, T \subset G, S \cdot T=\{s \cdot t \mid s \in S, t \in T\}$ 1) For $G=D_{4}$, find $\left\{R_{90}, D, R_{0} ight\} \cdot\left\{R_{180), H, V ight\}$. 2) For $G=\mathbb{Z}$, we use $+$ insteal of. Recall $5 \mathbb{2}={{5 n \mid n \in \mathbb{Z}\}$ forms a subgrop of $\mathbb{Z}$ which, since $G$ is abelian, is ambivalent. Sone examples of left cosets of $5 \mathbb{Z}$ are $13+5 \mathbb{Z}, -10+5 \mathbb{Z}$. a) Find all 5 distinct left cosets of $5 \mathbb{Z}$. b) Find $(1+5 \mathbb{Z})+(3+5 \mathbb{Z})$ and express as a left cocet of $5 \mathbb{Z}$. CS.JG.063