Recall: Symmetric groups
Let $[n] = {1, \dots, n}$, and consider a permutation $\sigma : [n] \longrightarrow [n]$ (one-to-one and onto). The collection of all such, together w/ composition, form the symmetric group $S_n$.
We can represent these in three ways:
- Cycle notation.
- Transpositions: for a cycle $(a_1\ \cdots\ a_n)$ we can write $$(a_1\ \cdots\ a_n) = (a_1\ a_n) \cdots (a_1\ a_3)(a_1\ a_2).$$
- Disjoint cycles. (This was a theorem.)
What's the order of an element?
Start w/ the building blocks: $$|(a_1\ \cdots\ a_n)| = n.$$
What if we have two? $$(a_1\ \cdots\ a_m)(b_1\ \cdots\ b_n)$$ First is $\mathbb{1}$ every $m$, second every $n$. Thus we have:
A cycle of type $(n_1, \dots, n_m)$ has order $\operatorname{lcm}(n_1, \dots, n_m)$.
Show that $A_{10}$ contains an element of order $15$.
Proof
Note that order $15 = \operatorname{lcm}(3, 5)$. Are these in $A_{10}$?
Does $A_8$ contain an element of order $26$?
Proof
No. These are either $26$-cycles or of type $(2, 13)$. Can't happen.
What are all the cycle types in $A_4$?
Solution
- $(1, 1, 1, 1)$
- $(2, 2)$
- $(3)$
Dihedral Groups
Consider $n > 0$ a positive integer, and construct a regular $n$-gon. We can say vertices are at $$\alpha_j = \left(e^{2\pi i / n}\right)^j.$$
There are $2n$ possible symmetries of the $n$-gon.
Why?
There are $n$ choices for the first vertex, and for a given vertex $k$, moving CCW returns either $k+1$ or $k-1$.
The set $D_n = { r, s, rs, \dots }$ is called the dihedral group under composition.
Define $r$ = CCW rotation by $2\pi/n$ rad, $s$ = flipping/reflecting across the $x$-axis. Then $D_n = \langle r, s \rangle$, with $r^n = 1$, $s^2 = 1$, and $srs = r^{-1}$.
Solution
Track the vertices after operations
Is $D_n$ abelian?
Let $n = 3$, consider $r, s$. Triangle vertices are labeled $1, 2, 3$ and we track the labels at the angles $0°, 120°, 240°$.
How do we generalize this argument?
Notice from above that $$srs = r^{-1} \iff sr = r^{-1}s.$$
What do subgroups of the dihedral group look like?