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{\Large {\bf Homework 8}} \hfill MATH 301/601 \\
{Due Wednesday, April 3, 2024} \\[-2mm]
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\medskip
{\bf Instructions.} \textit{Read the appropriate homework guide (\href{http://qc.edu/~nvlamis/301S24/Homework_Guide_301.pdf}{Homework Guide for 301} or \href{http://qc.edu/~nvlamis/301S24/Homework_Guide_601.pdf}{Homework Guide for 601}) to make sure you understand how to successfully complete the assignment.
All claims must be sufficiently justified. }
\medskip
\begin{Exercise}
Complete the following exercises from \href{http://abstract.ups.edu/aata/cosets-exercises.html}{Section~6.5} in the course textbook:
\# 1, 3, 4, 5, 6, 8, 11, 12, 17, \textbf{*18},
\end{Exercise}
\medskip
\begin{Exercise}
Let \( H \) be a subgroup of a group \( G \).
Fix \( g \in G \), and define \( \varphi_g \co H \to gH \) by \( \varphi_g(h) = gh \).
Prove that \( \varphi_g \) is a bijection.
\end{Exercise}
\medskip
\begin{ExGraded}
Let \( p \in \bn \) be prime. How many subgroups does \( \bz_{2p} \) have? Prove it.
\end{ExGraded}
\medskip
\begin{Exercise}
Let \( G \) be a group.
Define the relation \( \sim \) on \( G \) as follows: \( a \sim b \) if and only if \( b \) is a conjugate of \( a \) (that is, there exists \( g \in G \) such that \( b = gag^{-1} \)).
Prove that \( \sim \) is an equivalence relation.
\end{Exercise}
\medskip
\begin{ExGraded}
Prove that the 3-cycles \( (1 \,\, 2 \,\, 3) \) and \( (1 \,\, 3 \,\, 2) \) are not conjugate in \( A_4 \).
\end{ExGraded}
\subsection*{Double-star problem set up\footnote{See \href{http://abstract.ups.edu/aata/actions-section-groups-acting-on-sets.html}{Section 14.1}}}
\begin{Def}
Let \( G \) be a group, and let \( X \) be a set.
A \emph{group action} of \( G \) on \( X \) is a function \( \phi\co G \times X \to X \) satisfying:
\begin{enumerate}[(i)]
\item \( \phi(e, x) = x \) for all \( x \in X \), and
\item \( \phi(gh,x) = \phi(g,\phi(h,x)) \) for all \( g,h \in G \) and for all \( x \in X \).
\end{enumerate}
Usually the group action is clear from context and we simply write \( gx \) or \( g \cdot x \) instead of \( \phi(g,x) \).
In this notation, (i) says \( e\cdot x = x \) for all \( x \in X \), and (ii) says \( (gh)\cdot x = g\cdot(h\cdot x) \).
Again suppressing the function \( \phi \), we generally write \( G \curvearrowright X \) to denote the fact that the group \( G \) is acting on the set \( X \).
\end{Def}
In the ``real world'', we generally think about a group by the way it acts on some set.
For example, we think about the dihedral groups via their action on regular polygons, and we think of matrix groups via their action on vector spaces.
\begin{Def}
Let \( G \curvearrowright X \).
Given \( x \in X \), the \emph{orbit} of \( x \), denoted \( \mathcal O_x \), is the subset of \( X \) given by
\[ \mathcal O_x = \{ g\cdot x : g \in G \} \]
and the \emph{stabilizer} of \( x \), denoted \( \mathrm{Stab}_G(x) \) is the subgroup\footnote{You should convince yourself that this is indeed a subgroup.} of \( G \) given by
\[ \mathrm{Stab}_G(x) = \{ g \in G : g\cdot x = x \}.\]
\end{Def}
The goal of the next exercise is to prove the following:
\begin{Thm}[Orbit--Stabilizer Theorem]
Let \( G \) be a group acting on a set \( X \).
If \( x \in X \), then \( |G| = |\mathcal O_x|\cdot |\mathrm{Stab}_G(x)| \).
\end{Thm}
The orbit--stabilizer theorem should be viewed as a generalization of Lagrange's theorem (which we will use to prove the orbit--stabilizer theorem).
Indeed, let \( H \) be a subgroup of \( G \), and let \( \mathcal L_H \) be the left cosets of \( H \).
Then \( G \) acts on \( \mathcal L_H \) by \( g\cdot (aH) = (ga)H \), with \( \mathrm{Stab}_G(H) = H \) and \( \mathcal O_H = \mathcal L_H \).
\medskip
\begin{ExChallenge}
Let \( G \) be a group acting on a set \( X \).
Let \( x \in X \).
\begin{enumerate}[(a)]
\item Let \( g, h \in G \).
Prove that \( gx = hx \) if and only if \( h^{-1}g \in \mathrm{Stab}_G(x) \).
\item Let \( \mathcal L \) be the set of left cosets of \( \mathrm{Stab}_G(x) \) in \( G \).
Let \( \psi \co \mathcal L \to \mathcal O_x \) be given by \( \psi(g \mathrm{Stab}_G(x)) = gx \).
\begin{enumerate}[(i)]
\item Prove that \( \psi \) is a well-defined.
\item Prove that \( \psi \) is bijective.
\end{enumerate}
\item The previous part implies that \( |\mathcal O_x| = [G: \mathrm{Stab}_G(x)] \).
Apply Lagrange's theorem to obtain \( |G| = |\mathcal O_x|\cdot |\mathrm{Stab}_G(x)| \).
\item Now suppose \( G \) is a finite group, and let \( G \) act on itself by conjugation, that is, the action is given by \( g\cdot a = gag^{-1} \).
Apply the orbit--stabilizer theorem to show that, for \( a \in G \), the cardinality of the set \( \{ gag^{-1} : g \in G\} \) (that is, the conjugacy class of \( a \)) divides \( |G| \).
\end{enumerate}
\end{ExChallenge}
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