Homework 07

Read: Section 2.6.

• There is a lot to read in 2.6—try to go slowly and carefully.

Turn in: 2.68, 2.69, 2.71, 2.72(a,b,c,d,e), 2.73

• For 2.72, you first need to draw a vertex for each element of your group. To do this, you need to choose a way to represent the elements, and there are often (equally good) choices to make. Sometimes there is an “obvious” choice like $R_4 = \{e,r,r^2,r^3\}$, but you could also choose $R_4 = \{ r,r^2,r^3,r^4 \}$ if you want. For $D_3 = \langle r,s\rangle$, you will probably choose $D_3 = \{e,r,r^2,s,sr,sr^2\}$ or $D_3 = \{e,r,r^2,s,rs,r^2s\}$ (keeping in mind that $sr = r^2s$ and $sr^2 = rs$), and for $D_3 = \langle s,s’\rangle$, you will probably choose $D_3 = \{e,s,s’,ss’,s’s,ss’s\}$ or $D_3 = \{e,s,s’,ss’,s’s,s’ss’\}$ (since $ss’s = s’ss’$). I recommend making your choice based on how the elements were labeled when you made your group table in the previous section, and remember that the book provides group tables for $D_3$ on page 28 and for $V_4$ on page 30.

  Now, once you make a vertex for each element of the group, you need to draw an arrow coming out of each vertex for each element of the generating set. When you do this, remember that the arrow corresponds to “multiplying” on the left by the element corresponding to the arrow. For example, when working with $D_3 = \langle r,s\rangle$ you will have six vertices and one of the vertices will be labeled $s$. Then, you should draw an $r$-arrow going from the vertex $s$ to the vertex $rs$, and you should draw an $s$-arrow going from the vertex $s$ to the vertex $ss = e$. Now repeat this for the other 5 vertices. The group tables from before should help with this.

  Note that the Cayley diagram for $D_3 = \langle r,s\rangle$ is actually drawn in the book a couple of pages later, so you can use that to check your work.

• For 2.73, you can just draw the portion of the Cayley diagram that includes the vertices from $-6$ to $6$.

Extra practice: 2.72(f,g), 2.74–2.82