## More Fun with Groups!

Write up and hand in the following.

- Let $ABCDEF$ be a regular hexgon centered at the origin with point $A$ located at $(1,0)$ and point $B$ in the first quadrant. Let $r$ be the symmetry operation of $ABCDEF$ that rotates it by an angle of $60^\circ$ about the origin in the counter-clockwise direction (i.e., it sends $A$ to $B$, $B$ to $C$, and so on). Let $m$ be the reflection of $ABCDEF$ about the $x$-axis (i.e. it sends $A$ to $A$, $B$ to $F$ and so on). If $a$ and $b$ are symmetry operations, $ab$ is composition of those operations as functions, so that $ab$ means “first do $b$ then do $a$ in that order”. In this notation the dihedral group, $D_6$ consists of the elements $$\{e,r,r^2,r^3,r^4,r^5,m,mr,mr^2,mr^3,mr^4,mr^5\}$$
- Write out the multiplication table for $D_6$ in terms of these 12 elements (i.e. only those 12 elements can appear in your table).
- Determine the order of each of the 12 elements of $D_6$. Note: this is level “Easy” in Scrambler! (at the ToyProofs link below).

**Exercise 7.2 #12.****Exercise 7.2 #18.****Exercise 7.2 #24.***Hint: One transitive chain, man!***Exercise 7.2 #36.***Warning: This problem is fun!*