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\begin{document}

\head{Worksheet 10}{Week 10: 2024-10-29}

More groups! \vspace{5mm}

\begin{framed}
  Maryam Mirzakhani (1977-2017) was an Iranian mathematician, and the first woman to receive the
  Fields medal (the mathematical equivalent of a Nobel prize). The award committee cited her
  contributions to the understanding of the ``dynamics and geometry of Riemann surfaces and
  their moduli spaces''---in simpler terms, studying the geometric relationships between all
  possible surfaces that locally behave like the complex plane. Raised in Tehran, her
  mathematical talent was evident from a young age; she earned a perfect score on the
  International Mathematical Olympiad in Toronto. Her doctoral thesis at Harvard was published
  in the three top math research journals: \textit{Annals of mathematics}, \textit{Inventiones},
  and \textit{The Journal of the AMS}. She taught at Princeton and then Stanford, where she
  continued making fundamental contributions to Teichm\"{u}ller theory until her death due to
  breast caner at age 40.

  \noindent\makebox[\textwidth][c]{
  \begin{minipage}[t]{0.5\linewidth}
    \vspace{-2ex}
    \centering
    \includegraphics[width=4cm]{mirzakhani}
  \end{minipage}}
\end{framed}

\begin{problem}
  Suppose that \( G \) is a finite group. Show that every element \( g \in G \) has finite
  order.
\end{problem}

\vspace{5cm}

\begin{problem}
  Suppose that \( G \) is a cyclic group. Show that \( G \) is abelian.
\end{problem}

\vspace{4cm}

\begin{problem}
  Let \( G \) be a group, and \( g, h \in G \) any two elements. Show that
  \( (g h)^{-1} = h^{-1} g^{-1} \). (Note: This is sometimes called the \textit{shoes--socks
    principle}. Can you think of why?)
\end{problem}

\vspace{2cm}

\begin{problem}
  Show that a group \( G \) is abelian if and only if \( (g h)^{-1} = g^{-1} h^{-1} \) for every
  \( g, h \in G \).
\end{problem}

\vspace{2cm}

\begin{problem}
  Draw a Cayley graph for the following groups: \( \ZZ \), \( \ZZ_n \), \( D_3 \), \( D_4 \).
  (Note: you will have to pick a set of generators for each group.)
\end{problem}

\vspace{5cm}

\begin{problem}
  If \( G \) and \( H \) are groups, a function \( \varphi : G \to H \) is called a
  \textit{group homomorphism} if, for every elements \( g, g' \in G \),
  \( \varphi(g g') = \varphi(g) \varphi(g') \). Are the following homomorphisms?
  \begin{enumerate}
  \item \( f : D_3 \to D_3 \) given by \( f(\sigma) = \sigma^{-1} \). (Recall: \( D_3 \) is the
    group of rigid symmetries of the triangle.)
  \item Considering \( (\ZZ, +) \) as a group under addition, \( g : \ZZ \to \ZZ \) given by
    \( g(n) = 2 n \).
  \item Again considering the integers under addition, \( h : \ZZ \to \ZZ \) given by
    \( h(n) = n + 1 \).
  \item Letting \( G = (\RR, +) \) and \( H = (\RR_{>0}, \cdot) \), \( \exp : G \to H \) given
    by \( \exp(x) = e^x \).
  \end{enumerate}
\end{problem}

\vspace{3cm}

\begin{problem}
  If \( \varphi : G \to H \) is a homomorphism of groups, show that \( \varphi(e_G) = e_H \) and
  that, for each \( g \in G \), \( \varphi(g^{-1}) = \varphi(g)^{-1} \).
\end{problem}

\vspace{4cm}

\begin{problem}
  \begin{enumerate}
  \item Let \( H \) be a group and \( h \in H \) any element. Show that if \( h^n = e \), then
    the order of \( h \) divides \( n \). (Hint: use the division algorithm.)
  \item If \( \varphi : G \to H \) is a homomorphism of groups, and \( g \in G \) is an element
    of finite order. Show the order of \( \varphi(g) \) divides the order of \( g \).
  \end{enumerate}
\end{problem}

\vspace{5cm}

\begin{problem}
  Recall that, if \( f : S \to T \) is a function between two sets, then it is injective if and
  only if it admits a left inverse. Compare with the following: let
  \( \varphi : \ZZ_2 \to \ZZ_4 \) be defined by \( \varphi([0]_2) = [0]_4 \) and
  \( \varphi([1]_2) = [2]_4 \).
  \begin{enumerate}
  \item Is \( \varphi \) a group homomorphism?
  \item Is \( \varphi \) injective?
  \item Show that there does not exist any group homomorphism \( \psi : T \to S \) so that
    \( \psi \circ \varphi = \id_S \).
  \end{enumerate}
\end{problem}

\vspace{4cm}

\begin{problem*}{Challenge}
  Can you find a \textit{geometric} explanation for the fact that the group of rotations of the
  cube is isomorphic to the group of rigid symmetries of the tetrahedron?
\end{problem*}

\end{document}