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\begin{center}
  {\large Mathematics 111}\\Spring 2011\\Homework 7
\end{center}

\begin{enumerate}

\item (\#3 p. 852)  Show that the degree one representations of a finite
  group $G$ are in bijective correspondence with the degree one
  representations of the abelian group $G/G'$, where $G'$ is the
  commutator subgroup of $G$.

\item  (\#15, p 853) Exhibit all one-dimensional complex
  representations of a finite cyclic group, and determine which are
  inequivalent.

\item (\#16, p 853) Exhibit all one-dimensional complex representations of a
  finite abelian group.  Deduce that the number of inequivalent degree
  one complex representations of a finite abelian group equals the
  order of the group. Hint:  First decompose the abelian group into a
  direct sum of cyclic groups and use the previous problem.

\item \#20, p 854) Prove that the number of degree one complex
  representations  of any finite group $G$ equals the index $[G : G']$
  where $G'$ is the commutator subgroup.

\item (based upon \#19, p 854) The goal is to prove that any
  finite-dimensional complex representation of a finite abelian group
  is the direct sum of one-dimensional (necessarily irreducible)
  representations without the use of Maschke's theorem.  In the
  process, we prove a number of interesting results in linear algebra.

  Let $k$ be a field and $V$ an $n$-dimensional vector space over $k$.
  The first goal is to prove that if $\S$ is a set of commuting,
  diagonalizable linear operators in $GL(V)$, then there is a basis
  for $V$ with respect to which every element of $\S$ has a diagonal
  matrix representation, that is the elements of $\S$ are
  simultaneously diagonalizable.
  \begin{enumerate}
  \item Consider an element $T \in \S$.  We know that $T$ is
    diagonalizable iff its minimal polynomial $m_T$ is the product of
    distinct linear factors: $m_T = (x-\lambda_1) \cdots
    (x-\lambda_r)$.  In our consideration of the Jordan form we wrote
    $V = V(x-\lambda_1) \oplus \cdots \oplus V(x-\lambda_r)$, where
    $V(x-\lambda_i) = \{ v \in V \mid (T - \lambda_i I)^k(v) = 0
    \mbox{ for some } k \ge 1\}$.  Show that $V(x-\lambda_i)$ is the
    $\lambda_i$-eigenspace $E_{\lambda_i}$.

\item To proceed you may want to consider a proof by induction on
  $n$.  Let $T \in \S$ and write $V = \oplus_\lambda E_\lambda$.
  Explain why it is possible (wlog)  to choose $T$ so that there are
  at least two eigenspaces $E_\lambda$.

\item  Show that for all $S \in \S$, $S$ is invariant on each
  $E_\lambda$. Show that $S$ is diagonalizable on each $E_\lambda$.

\item  Show that any set of elements in $\S$ can be
  simultaneously diagonalized.

\item Explain why whether $\S$ is a finite or infinite set doesn't matter.

\item  Finally show that any
  finite-dimensional complex representation of a finite abelian group
  is the direct sum of one-dimensional (necessarily irreducible)
  representations without the use of Maschke's theorem. 
  \end{enumerate}


\end{enumerate}






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