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\title{Math 122: Homework 4}
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  \item Section 18.3 \# 1,2a,4,5,11,20
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{\textbf{Section 18.3 \#1}}. Prove that $\operatorname{tr} (A B) = \operatorname{tr} (B A)$
for $n \times n$ matrices $A$ and $B$ with entries in any commutative ring.

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{\textbf{Section 18.3 \#2a.}} Let $\varphi$ be the degree 2 representation
of $D_{10}$ desribed in Example~6 in the second set of examples in Section 1
(here $n = 5$) and show that $\| \varphi \|^2 = 1$. Deduce that $\varphi$ is
irreducible.

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{\textbf{Section 18.3 \#4.}} Prove that if $N$ is any irreducible
$\mathbb{C}G$-module and $M = N \oplus N$ then $M$ has infinitely many direct
sum decompositions into two copies of $N$.

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{\textbf{Section 18.4 \#5.}} Prove that a class function is a character if
and only if it is a positive integral linear combination of irreducible
characters.

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{\textbf{Section 18.4 \#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 $1$ in $\mathbb{C}$.
[{\textbf{Hint:}} Use Schur's Lemma.]

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{\textbf{Section 18.4 \#20}}. Prove that elements $x$ and $y$ of $G$ are
conjugate in $G$ if and only if $\chi (x) = \chi (y)$ for all irreducible
characters $\chi$ of $G$.

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