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\title{Math 122: Homework 6}
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\begin{itemize}[noitemsep]
  \item Section 18.3 \# 11$^*$,12,13
	\item Section 19.1 \# 7$^*$,14,15
\end{itemize}

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($^*$) Problems 18.3 \# 11 and 19.1 \# 7 were previously assigned. If you
are confident in your previous solutions you do not need to do these.

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{\textbf{Section 18.2 \#11.}} Let $\chi$ be an irreducible character of
$G$. Prove that for every element $z$ in the center of $G$ we have $\chi (z) =
\varepsilon \chi (1)$ where $\varepsilon$ is some root of unity
in~$\mathbb{C}$. [{\textbf{Hint:}} Use Schur's Lemma]

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{\textbf{Section 18.2 \#12.}} Let $\psi$ be the character of some
representation $\varphi$ of $G$. Prove that for $g \in G$ the following hole.

(a) If $\psi (g) = \psi (1)$ then $g \in \operatorname{Ker} (\varphi)$;

(b) If $| \psi (g) | = \psi (1)$ and $\varphi$ is faithful then $g \in Z (G)$
(where $| \psi (g) |$ is the complex absolute value of $\psi (g)$.
[{\textbf{Hint:}} Use the method of proof of Proposition~14.]

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{\textbf{Section 18.2 \#13.}} Let $\varphi : G \longrightarrow \operatorname{GL}
(V)$ be a representation and let $\chi : G \longrightarrow
\mathbb{C}^{\times}$ be a degree 1 representation. Prove that $\chi \varphi :
G \longrightarrow \operatorname{GL} (V)$ defined by $(\chi \varphi) (g) = \chi (g)
\varphi (g)$ is a representation (note that multiplication of the linear
transformation $\varphi (f)$ by the complex number $\chi (g)$ is
well-defined.) Show that $\chi \varphi$ is irreducible if and only if
$\varphi$ is irreducible. Show that if $\psi$ is the character afforded by
$\varphi$ then $\chi \psi$ is the character afforded by $\chi \varphi$. Deduce
that the product of any irreducible character with a character of degree~1 is
also an irreducible character.

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{\textbf{Section 19.1 \#7}}. Show that $S_6$ has an irreducible character
of degree~5.

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{\textbf{Section 19.1 \#14}}. Let $n$ be an integer with $n \geqslant 3$.
Show that every irreducible character of $D_{2 n}$ has degree 1 or 2 and find
the number of irreducible characters of each degree. (The conjugacy classes of
$D_{2 n}$ were found in Exercises 31 and 32 of Section~4.3 and its commutator
subgroup was computed in Section~5.4.)

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{\textbf{Section 19.1 \#15.}} Prove that the character table is an
invertible matrix.

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