Symmetric Groups, Permutation Groups, and Dihedral Groups Review

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:
• 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.