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\title{Math 122: Homework 7}
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\begin{itemize}[noitemsep]
	\item Section 5.2 \# 4b, 9,
	\item Section 12.1 \# 11,12a,
	\item Section 12.2 \# 3,
	\item Section 19.3 \# 4 and Problem F.
\end{itemize}

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{\textbf{Section 5.2 \#4b.}} Determine which pairs are isomorphic. Here $\{
a_1, \cdots, a_k \}$ denotes the abelian group $Z_{a_1} \times \cdots \times
Z_{a_k}$:
\[ \{ 2^2, 2 \cdot 3^2 \}, \quad \{ 2^2 \cdot 3, 2 \cdot 3 \}, \quad \{ 2^3
   \cdot 3^2 \}, \qquad \{ 2^2 \cdot 3^2, 2 \} . \]
\[ \  \]
{\textbf{Section 5.2 \#9.}} Let $A = Z_{60} \times Z_{45} \times Z_{12}
\times Z_{36}$. Find the number of elements of order 2 and the number of
subgroups of index 2 in~$A$.

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{\textbf{Section 12.1 \#11.}} Let $R$ be a PID. Let $a$ be a nonzero
element of $R$ and let $M = R / (a)$. For any prime $p$ of $R$ prove that
\[ p^{k - 1} M / p^k M \cong \left\{\begin{array}{ll}
     R / (p) & \text{if $k \leqslant n$,}\\
     0 & \text{if $k > n$}
   \end{array}\right. \]
where $n$ is the power of $p$ dividing $a$ in $R$.

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{\textbf{Section 12.1 \#12a}}. Let $R$ be a PID and let $p$ be a prime in
$R$. Let $M$ be a finitely generated torsion $R$-module. Use the previous
exercise to prove that $p^{k - 1} M / p^k M \cong F^{n_k}$ where $F$ is the
field $R / (p)$ and $n_k$ is the number of elementary divisors of $M$ which
are powers $p^{\alpha}$ with $\alpha \geqslant k$.

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{\textbf{Section 12.2 \#3}}. Prove that two $2 \times 2$ matrices over $F$
which are not scalar matrices are similar if and only if they have the same
characteristic polynomial.

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{\textbf{Section 19.3 \#4.}} Let $H$ be a subgroup of $G$, let $\varphi$ be
a representation of $H$ and suppose that $N$ is a normal subgroup of $G$ with
$N \subseteq H$ and $N \subseteq \ker (\varphi)$. Prove that $N$ is also
contained in the kernel of the induced representation of $\varphi$.

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{\textbf{Problem F.}} There are two definitions of Frobenius group. The one
in the posted lecture notes is the usual definition: A Frobenius group $G$ is
a transitive group of permutations of the finite set $X$ such that no element
except the identity fixes more than one element. Frobenius' Theorem asserts
that
\[ K := \left\{ k \in G| \text{$k$ has no fixed points} \right\} \cup \{
   1_G \} \]
is a normal subgroup of $G$. Let $x \in K$. Prove that the centralizer $C_G
(x)$ is contained in $K$. (This proves that a Frobenius group by our
definition is also one by Dummit and Foote's. The converse is also true.)

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