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MATH 3GR3 Assignment \#2
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Due: Monday, 18 October by 11:59pm
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\begin{enumerate}
\item Show that if $G$ is an abelian group and $a$, $b \in G$ both have finite order, then so does the element $ab$. Find an example of a group $G$ and two elements $a$, $b \in G$ both of finite order such that $ab$ has infinite order.
\item Find all of the subgroups of $\mathbb{Z}_{12}$ and find all of the generators of this group.
\item Produce the Cayley table of the group $U(12)$. Is this group cyclic?
\item Let $G$ be a finite cyclic group. Show that if $H$ is a subgroup of $G$ then $|H|$ divides $|G|$. Conversely, show that if $k$ is a natural number such that $k$ divides $|G|$ then there is a subgroup of $G$ of order $k$.
\item Let $H$ and $K$ be subgroups of the group $G$. Show that $H \cap K$ is a subgroup of $G$. Show that the subset $HK = \{h\cdot k\,:\, h \in H, k \in K\}$ is not necessarily a subgroup, by finding an example that illustrates this.
\item Consider the following two elements of $S_7$:
\[
\sigma = \left(\begin{array}{ccccccc} 1&2&3&4&5&6&7\\ 6&7&4&3&1&5&2 \end{array}\right),
\tau = \left(\begin{array}{ccccccc} 1&2&3&4&5&6&7\\ 1&3&4&5&7&6&2 \end{array}\right).
\]
\begin{enumerate}
\item Decompose $\sigma$ and $\tau$ into cycles.
\item Compute $\sigma\tau$ and $\tau\sigma$.
\item Compute the order of $\sigma$, $\tau$, $\sigma\tau$, and $\tau\sigma$.
\item Determine the signs of $\sigma$, $\tau$, $\sigma\tau$, and $\tau\sigma$.
\end{enumerate}
\item What are the possible orders of elements in the group $S_8$? For each one, find an element of that order.
\item Show that every element $\sigma$ of the group $S_n$ can be written as a product of transpositions of the form $(i, i+1)$, where $1 \le i < n$.
\item Let $G$ be the group of symmetries of the circular disk of radius 1. Show that $G$ contains elements of every finite order and that it contains elements of infinite order.
\end{enumerate}
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Supplementary problems from the textbook \\
(not to be handed in)
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\begin{itemize}
\item From Chapter 3, questions 34, 35, 49, 50
\item From Chapter 4, questions 1, 2, 3, 10, 23, 24, 25, 30, 39
\end{itemize}
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