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Section Day 11
This is an outline of the topics we covered in the eleventh day of class. The skeleton notes are in a handout, which can be printed out using the printer icon at the top right of its section of the page for filling in during class. Filled notes for each day will be posted after class to Canvas.
Handout Tuesday 6/23
Objectives: Advanced Learning Outcomes
During our class meeting, we will work on learning the following. Fluency with these is not expected or required before class.
State and instantiate the definition of: orbit, fixed point set, stabilizer
State and apply the Fundamental Theorem of Group Actions
State and instantiate the definition of: class equation
State the following mathematical result: p-Groups have nontrivial centers.
Algebraist of the Day.
Augustin-Louis Cauchy , 1789-1857
French mathematician, known for his work in analysis, number theory, and group theory.
Notable theorems: Cauchy’s Integral Theorem, Cauchy’s Theorem for finite groups, first rigorous proof of Taylor’s Theorem
Was a notoriously bad lecturer at the École Polytechnique
Conflicts with the July Revolution; refused to swear allegiance to the new government and lost his positions
Definition 117 . Group Action Features.
Let \(G\) act on \(X\text{.}\) For each \(x\in X\text{,}\) its orbit is
\begin{equation*}
\Orb_x = \mathcal{O}_x=\{g\cdot x \mid g \in G\} \subseteq X
\end{equation*}
and its stabilizer is
\begin{equation*}
\Stab_x = G_x = \{g\in G \mid g\cdot x = x\} \subseteq G\text{.}
\end{equation*}
We say that \(x\) is a fixed point of the action if \(\Orb_x=\{x\} \Longleftrightarrow \Stab_x=G\text{.}\) We write \(X_G\) for the set of all fixed points of the action.
Example 118 . \(\GL(2,\R)\) acting on \(\R^2\) .
Theorem 119 . Fundamental Theorem of Group Actions.
Let a group \(G\) act on a set \(X\text{.}\) Then
Different orbits are disjoint and partition
\(X\text{.}\)
For each
\(x \in X\text{,}\) \(\Stab_x\) is a subgroup of
\(G\) and
\(\Stab_{g\cdot x}=g\left(\Stab_x\right)g^{-1}\) for all
\(g \in G\text{.}\)
For each \(x \in X\text{,}\) the map \(\Orb_x \to G/\Stab_x\) by
\begin{equation*}
g\cdot x \mapsto g\Stab_x
\end{equation*}
is a bijection. In particular,
\(g\cdot x = g' \cdot x\) if and only if
\(g\Stab_x = g'\Stab_x\)
\(|\Orb_x| = [G:\Stab_x]\) (the
Orbit-Stabilizer Theorem )
Proof.
Corollary 120 . Class Equation.
If \(G\) acts on a finite set \(X\text{,}\) with \(x_1,\dots, x_n\) representing the distinct orbits of size greater than 1, then
\begin{equation*}
|X|=|X_G| + \sum_{i=1}^n |\Orb_{x_i}| = |X_G| + \sum_{i=1}^n [G:\Stab_{x_i}]\text{.}
\end{equation*}
When \(G\) acts on itself by conjugation, we get the class equation :
\begin{equation*}
|G|=|Z(G)|+\sum_{i=1}^n [G: C_G(x_i)]\text{.}
\end{equation*}
Theorem 121 . Nontrivial \(p\) -Groups have Nontrivial Centers.
Let
\(G\) be a finite
\(p\) -group with
\(|G|\gt 1\text{.}\) Then
\(p\) divides
\(|Z(G)|\) and in particular
\(Z(G)\neq \{e\}\text{.}\)
Proof.
Theorem 122 . Fixed Point Congruence.
Let \(G\) be a finite \(p\) -group acting on a finite set \(X\text{.}\) Then
\begin{equation*}
|X| \equiv |X_G| \bmod{p}\text{.}
\end{equation*}
