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\begin{center}
{\large MATH 150A Winter 2020 \ - \ {\bf Problem Set 3}}
% Put your name and section here.
{\large due January 31}
\end{center}
\begin{enumerate}
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\item Let $G$ be a group generated by set $A$. Prove that if $a$ and $b$ commute for all $a,b \in A$, then $G$ is abelian.
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\item Prove that if group $G$ has order 4 then $G$ is cyclic or $G$ is isomorphic to the Klein four group.
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\item For group $G$, $\Aut(G)$ denotes the {\em automorphism group} of $G$, whose elements are all automorphisms $G \to G$ and with composition as the operation.
\begin{enumerate}
\item Prove that $\Aut(G)$ is in fact a group.
\item Let $\gamma:G \to \Aut(G)$ be defined by $g \mapsto \varphi_g$ where $\varphi_g:G\to G$ is the map that conjugates by $g$, $\varphi_g(x) = gxg^{-1}$. Prove that $\gamma$ is a group homomorphism.
\end{enumerate}
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\item (2.5.2) Find all automorphisms of
\begin{enumerate}
\item the cyclic group of order 10,
\item the symmetric group $S_3$.
\end{enumerate}
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\item (2.7.1) Let $G$ be a group and define the relation $\sim$ on $G$ by $a \sim b$ if $b = gag^{-1}$ for some $g \in G$ (in which case we say $a$ and $b$ are {\em conjugates}).
\begin{enumerate}
\item Prove that $\sim$ is an equivalence relation.
\item The equivalence classes of $\sim$ are called {\em conjugacy classes}. For $a \in G$, prove that $a$ is in the center of $G$ if and only if its conjugacy class is $\{a\}$.
\end{enumerate}
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\item Let $H$ be the {\em quaternion group}, which can be represented as the group of matrices
\[ H = \{\pm \mathbf{1}, \pm \mathbf{i}, \pm \mathbf{j}, \pm \mathbf{k} \} \]
where
\[ \mathbf{1} = \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}, \;
\mathbf{i} = \begin{bmatrix} i&0 \\ 0&-i\end{bmatrix}, \;
\mathbf{j} = \begin{bmatrix} 0&1 \\-1&0 \end{bmatrix}, \;
\mathbf{k} = \begin{bmatrix} 0&i \\ i&0 \end{bmatrix}. \]
The elements of $H$ satsify the relations
\[ \mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = -\mathbf{1}, \quad
\mathbf{ij} = -\mathbf{ji} = \mathbf{k}, \quad
\mathbf{jk} = -\mathbf{kj} = \mathbf{i}, \quad
\mathbf{ki} = -\mathbf{ik} = \mathbf{j} .\]
Find the conjugacy classes of $H$, and the center of $H$.
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\item (2.8.4) Let $G$ be a group of order 35.
\begin{enumerate}
\item Prove that $G$ contains an element $a$ of order 5.
\item Prove that $G$ contains an element $b$ of order 7.
\item Prove that $\langle a,b\rangle = G$.
[Hint: show that the elements $a^{n}b^{m}$ with $0 \leq n < 5$ and $0 \leq m < 7$ are distinct.]
\end{enumerate}
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\end{enumerate}
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