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\title{Math 122: Homework 3}
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\begin{itemize}[noitemsep]
  \item Section 18.1 \# 3,11,15,16,20
  \item Section 18.2 \# 3
\end{itemize}

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Let $G$ be a finite group. The {\textit{commutator subgroup}} $G'$, also called
the {\textit{derived subgroup}} is the subgroup generated by all commutators
$[x, y] = x y x^{- 1} y^{- 1}$. The word ``generated by'' is important since
the product of commutators may not be a commutator. See Dummit and Foote
Proposition~7 on page~169.

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{\textbf{Section 18.1 \#3.}} Prove that the degree 1 representations of $G$
are in bijection with the degree~1 representations of $G / G'$.

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{\textbf{Section 18.1 \#11.}} Let $\varphi : S_n \rightarrow \operatorname{GL}_n
(F)$ be the matrix representation given by the permutation module described in
Example~3 in the second set of examples, where the matrices are computed with
respect to the basis $e_1, \cdots, e_n$. Prove that $\det (\varphi (\sigma)) =
\varepsilon (\sigma)$ for all $\sigma \in S_n$, where $\varepsilon (\sigma)$
is the sign of the permutation~$\sigma$. [{\textbf{Hint:}} Check this on
transpositions.]

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{\textbf{Section 18.1 \#15.}} Exhibit all 1-dimensional complex
representations of a finite cyclic group; make sure to decide which are
inequivalent.

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{\textbf{Section 18.1 \#16.}} Exhibit all 1-dimensional complex
representations of a finite abelian group; deduce that the number of
inequivalent degree~$1$ complex representations of a finite abelian group
equals the order of the group. [First decompose the abelian group into a
direct product of cyclic groups, then use the preceding exercise.]

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{\textbf{Section 18.1 \#20.}} Prove that the number of degree 1 complex
representations of any finite group equals $[G : G']$, where $G'$ is the
commutator subgroup of $G$. [{\textbf{Hint:}} use Exercises 3 and 16.]

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{\textbf{Section 18.2 \#3.}} Prove that (4) implies (3) in Wedderburn's
Theorem. [{\textbf{Hint:}} Let $N$ be a nonzero $R$-module. First show that
$N$ contains simple submodules by considering a cyclic submodule. Then use
Zorn's Lemma applied to the set of direct sums of simple submodules
(appropriately ordered) to show that $N$ contains a maximal completely
reducible submodule $M$. If $M \neq N$ let $M_1$ be teh complete preimage in
$N$ of a simple module in $N / M$ and contradict the maximality of $M$.]

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