Symmetric Groups, Permutation Groups, and Dihedral Groups Review

We will now review some of the recent material regarding symmetric groups and dihedral groups.

• Recall from the The Symmetric Groups on n Elements page that for the $n$-element set $\{ 1, 2, ..., n \}$ that the set of all permutations $\sigma$ on $\{ 1, 2, ..., n \}$ is denoted $S_n$ where $\sigma : \{ 1, 2, ..., n \} \to \{ 1, 2, ..., n \}$ is a permutation on this set if $\sigma$ is a bijection.

• We then defined the Symmetric Group on $n$ Elements to be $(S_n, \circ)$ where $\circ$ is the operation of function composition defined for all $f_1, f_2 \in S_n$ by:

(1)

\begin{align} \quad (f_1 \circ f_2)(x) = f_1(f_2(x)) \end{align}

• We also saw that the order of $(S_n, \circ)$ is $n!$, i.e., there exists $n!$ permutations $\sigma$ on the set $\{ 1, 2, ..., n \}$.

• On the The Symmetric Group of a General n-Element Set page we extended the notion of a symmetric group to any $n$-element set $A = \{ x_1, x_2, ..., x_n \}$ analogously above. We denote the set of all permutations on $A$ by $S_A$ and the Symmetric Group on $A$ is defined to be $(S_A, \circ)$.

• We then proved a very simple theorem which says that if $A$ is any finite set and $G_a \subseteq S_A$ is the subset of permutations on $A$ for which $\sigma(a) = a$ then $(G_a, \circ)$ is a subgroup of $(S_A, \circ)$. It was fairly easy to check that $G_a$ is closed under $\circ$ and that for every $\sigma \in G_a$ we have that $\sigma^{-1} \in G_a$.

• On the Permutation Groups on a Set we said that if $X$ is ANY set and $S_X$ denotes the set of all permutations on $X$ then the Permutation Group on $X$ is $(S_X, \circ)$. If $X = \{ 1, 2, ..., n \}$ then $S_X = S_n$, i.e., the symmetric groups on $n$-elements are permutation groups!

• On The Dihedral Groups Dn page we began to look at groups known as Dihedral Groups $(D_n, \circ)$ defined for all integers $n \geq 3$ where $D_n$ is the set of all permutations which are symmetries of the regular $n$-gon (the regular $n$-sided polygon) and $\circ$ is the operation of function composition.

• We then examined some of these dihedral groups on the following pages:
  • The Group of Symmetries of the Equilateral Triangle.
  • The Group of Symmetries of the Square.
  • The Group of Symmetries of the Pentagon.

• On The Group of Symmetries of a Rectangle page we then looked at the group of symmetries of a nonregular polygon - the rectangle. Identifying the symmetries for this group was analogous to that of the regular polygons above.