Read: Continue with Section 3.1.
Turn in: 3.17, 3.18, 3.22, 3.23, 3.24(b,d,e,f,i,j)
- You will probably want to use Theorem 3.6 to prove 3.23. There are three things to show.
- You need to prove that $Z(G)$ is nonempty. To do this, prove that $e\in Z(G)$.
- Next you need to show that $Z(G)$ is “closed under inverses”. To do this, assume $z\in Z(G)$, and prove that $z^{-1} \in Z(G)$. Remember, you know that $z^{-1} \in G$ (because $G$ is a group); to show $z^{-1} \in Z(G)$, you need to show that $z^{-1}g = gz^{-1}$ for all $g\in G$ (using your assumption that $z\in Z(G)$).
- You also need to show that $Z(G)$ is closed under the operation of $G$. For this, let $z_1,z_2\in Z(G)$, and prove that $z_1z_2 \in Z(G)$ by showing $z_1z_2g = gz_1z_2$ for all $g\in G$.
- Two remarks about 3.24.
- The book highlights that if $G$ is abelian then $Z(G) = G$. Make sure you believe (and could prove this); it makes many parts of 3.24 easy using what we’ve learned before.
- If a group is not abelian, it may be most efficient to first think about which group elements are not in $Z(G)$. The idea is as follows. Suppose $g$ is an element of your group. If you can find some other group element $h$ such that $gh\neq hg$ then you know $g\notin Z(G)$ (and also $h\notin Z(G)$). For example, using our previous notation for $D_3 = \{e,r,r^2,s,sr,sr^2\}$, notice how $rs\neq sr$ since $rs$ moves the top of the triangle to lower right while $sr$ moves the top of the triangle to lower left. Thus, we see that $r,s\notin Z(D_3)$. On the other hand, we know $e\in Z(D_3)$. So now you just need to determine if each of $r^2,sr,sr^2$ are in $Z(D_3)$ or not.
Extra practice: 3.19, 3.21, 3.24(remaining parts)
- Your proof of 3.21 will likely be short. You are given that $H$ is a subgroup of $G$ and that $G$ is abelian (i.e. the operation is commutative); you then need to prove that $H$ is abelian. The key point is that when we say $H$ is a subgroup of $G$ we are using the same operation for both $H$ and $G$.
