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\title{Math 122: Homework 7}
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	\item Section 19.3 \# 2a,b,c
    \item ``Problem A''
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{\textbf{Problem 19.3 \#2 a,b,c}}. In each part a character of a subgroup
$H$ of $G$ is specified. Compute the values of the induced character
$\operatorname{Ind}_H^G (\psi)$ on all conjugacy classes of $G$ and use the character
tables of Section~1 to write $\operatorname{Ind}_H^G (\psi)$ as a sum of irreducible
characters.

(a) $\psi$ is the unique nonprinciple character of the subgroup $\langle (12)
\rangle$ of $S_3$.

(b) $\psi$ is the degree 1 character of the subgroup $\langle r \rangle$ of
$D_8$ such that $\psi (r) = i$ where $i = \sqrt{- 1}$.

(c) $\psi$ is the degree 1 character of the subgroup $\langle r \rangle$ of
$D_8$ defined by $\psi (r) = - 1$.

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{\textbf{Problem A.}} Let $G$ be the following nonabelian group of order 21:
\[ G = \langle x, y|x^7 = y^3 = 1, y x y^{- 1} = x^2 \rangle . \]
Find the conjugacy classes of $G$ and compute the character table. Construct
the characters of degree 3 as induced characters.

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