homework 15 (due Apr 6)

due date

Tuesday, April 6

instructions

Complete the following exercises (which are not in our textbook):

  1. Consider a square with corners labeled clockwise consecutively 1, 2, 3, and 4 with 1 in the upper left hand corner.
    • Let $r$ be rotation by $90^\circ$ clockwise and let $s_1$ be reflection across the line through corners 2 and 4. Draw the Cayley diagram for $D_4$ generated by $r$ and $s_1$.
    • Let $s_1$ be as above and let $s_2$ be the reflection that swaps corner 2 with corner 3 and swaps corner 1 and 4. Draw the Cayley diagram for $D_4$ generated by $s_1$ and $s_2$.
  2. Consider the group $\mathbb{Z}_4\times \mathbb{Z}_2=\{(0,0),(0,1),(1,0),(1,1),(2,0),(2,1),(3,0),(3,1)\}$. Draw the Cayley diagram for $\mathbb{Z}_4\times \mathbb{Z}_2$ generated by $(1,0)$ and $(0,1)$.
  3. Imagine that you have a penny and a nickel sitting side by side. Consider the group generated by the following two actions. Let $S$ be the action of switching the left and right coin, and let $F$ be the action of flipping over the left coin. It turns out that this group has eight elements (sometimes called $B_2$).
    • Draw the Cayley diagram for the group generated by $S$ and $F$.
    • Which Cayley diagram that you already drew is the same as this one?
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