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{\bf Math 307 Abstract Algebra \qquad Homework 2 \hfill Your name}
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\centerline{
Solve the following problems (5 points each).}
\begin{enumerate}
\item Prove that the set of all $2\times 2$ upper triangular matrices with entries from
$\R$ and determinant $1$ is a group under matrix multiplication.
(Extra 2 points if you can prove the results for $n\times n$ upper triangular matrices.)
\item Prove that the set $U(n)$ of elements in $\Z_n$ relatively prime to $n$ form a group
under multiplication $\mod n$.
[Hint: If $a \in \Z_n$ satisfies gcd$(a,n) = 1$, there is $x, y \in \Z$ such that
$ax + ny = 1$.]
\item Prove that a group $G$ is Abelian if and only if $(ab)^{-1}=a^{-1}b^{-1}$.
\item Prove that in any group, an element and its inverse have the same order.
Recall that the order of an element $g$ in a group $G$ is $n \in \{1,2,3,\dots\}$
if $n$ is the smallest positive integer such that $g^n = e$, the identity.
If no such positive number, we say that $g$ has infinite order.
Hint: If $g$ has order $n$, show that $(g^{-1})^n = e$ and no smaller positive number $m$
will satisfy $(g^{-1})^m = e$.
\item Suppose that $H$ is a proper subgroup of $\Z$ under
addition and $H$ contains $18, 30$ and $40$, Determine $H$.
[Hint: Find the smallest positive number in $H$.]
\item Suppose $H_\alpha$ is a subgroup of a group $G$ for every $\alpha \in J$.
Show that $\cap_{\alpha \in J} H_\alpha$ is a subgroup of $G$.
\item Let $H$ and $K$ be subgroups of a group $G$. Show that
$H\cup K \le G$ if and only if $H \le K$ or $K \le H$.
[Hint: The $(\Leftarrow)$ is clear. To prove $(\Rightarrow)$, suppose
$H \cup K \le G$. Assume by contradiction that there is
$h \in H - K$ and $k \in K - H$. Then $hk \in H\cup K$ and ....]
\item (Extra credit) Give an example of $G$ with distinct proper subgroups such that
$H_1, H_2, H_3$, whose union is a subgroup.
\end{enumerate}
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