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\begin{center}
{\large MATH 150A Winter 2020 \ - \ {\bf Problem Set 1}}
% Put your name and section here.
{\large due January 17}
\end{center}
\begin{enumerate}
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\item Write the operation table for the union operation $\cup$ on $\mathcal{P}(\{1,2\})$ (the set of all subsets of $\{1,2\}$).
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\item Determine whether each set and binary operation is a group. If no, which properties does it fail? If yes, is it abelian? Find an identity element if one exists.
\begin{enumerate}
\item (The set of positive integers, $+$).
\item $(\CC \setminus \{0\},\; \cdot)$.
\item $(\mathcal{P}(\{1,2\}),\; \cup)$.
\item (The set of functions $\ZZ \to \ZZ$, composition).
\item (The set of bijective functions $\ZZ \to \ZZ$, composition).
\end{enumerate}
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\item (2.1.2) Prove the following properties of inverses.
\begin{enumerate}
\item If an element $a$ has a left-inverse $\ell$ and a right-inverse $r$, i.e. $\ell a = 1$ and $a r = 1$, then $\ell = r$, $a$ is invertible and $r$ is its inverse.
\item If $a$ is invertible, its inverse is unique.
\item If $a$ and $b$ are invertible, then so is $ab$ and its inverse is $b^{-1}a^{-1}$.
\end{enumerate}
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\item (2.2.2) Let $S$ be a set with a binary operation that is associative and has an identity element. Prove that the subset consisting of the invertible elements in $S$ is a group.
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\item (2.2.3) Let $x,y,z,w$ be elements of a group $G$.
\begin{enumerate}
\item Solve for $y$ if $xyz^{-1}w = 1$.
\item Suppose that $xyz = 1$. Does it follow that $yzx = 1$? Does it follow that $yxz = 1$?
\end{enumerate}
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\item The {\em Klein four group} $V$ is the group with 4 elements that can be represented by matrices
\[ \begin{bmatrix} \pm 1 & 0 \\ 0 & \pm 1 \end{bmatrix}. \]
\begin{enumerate}
\item Find the order of each element of $V$.
\item Find all subgroups of $V$.
\end{enumerate}
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\item (2.2.4) In which of the following cases is $H$ a subgroup of $G$? If not, say why not.
\begin{enumerate}
\item $G = \GL_n(\CC)$ and $H = \GL_n(\RR)$. ($\GL_n(K)$ denotes the multiplicative group of invertible $n\times n$ matrices with entries in $K$.)
\item $G = \RR^\times$ and $H = \{-1,1\}$.
\item $G = (\ZZ, +)$ and $H$ is the set of positive integers.
\item $G = \RR^\times$ and $H$ is the set of positive reals.
\item $G = \GL_2(\RR)$ and $H$ is the of matrices $\begin{bmatrix} a & 0 \\ 0 & 0 \end{bmatrix}$, with $a \neq 0$.
\end{enumerate}
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\end{enumerate}
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