Read: Re-read 6.2, and start Section 7.1.
Turn in: 6.35(d,e), 6.36(a,b), 6.37, 6.40, 6.42, 6.43, 7.2, 7.3
- For 6.36(a), you could first use 6.33 (which follows from Lagrange’s Theorem) to determine the order of the group $Q_8/ \langle -1 \rangle$. Knowing the order of the group will help you focus in on what familiar group it might be isomorphic to. You will need a little more information to decide, and computing orders of elements of $Q_8/ \langle -1 \rangle$ (like in 6.35(b)) should help.
- For 6.40, you need to work abstractly with $G/H$. To show $G/H$ is abelian, you should take arbitrary $aH,bH \in G/H$ and then prove that $(aH)(bH) = (bH)(aH)$. Remember how the operation $(aH)(bH)$ is defined.
- For 6.43, you are assuming that $G$ is cyclic, and want to prove that $G/H$ is cyclic. You should first try to guess a generator $gH$ for $G/H$ (using what you know about $G$), and then show an arbitrary coset $aH$ can be written as $aH = (gH)^k$ for some $k$.
