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\noindent{\bf Math 307 Abstract Algebra \qquad Homework 3 \hfill your name}
\medskip\centerline{Five points for each problem unless specified otherwise.}
\begin{enumerate}
\item Let $H=\{a+bi: a, b\in \R, ab\ge 0\}$.
Prove or disprove that $H$ is a subgroup of $\C$ under addition.
\item Suppose $H$ is a non-trivial subgroup of $(\Z,+)$. Show that $H$ contains a positive
integer, and therefore a smallest positive integer $k$. Then deduce that
$H = k\Z = \{kr: r \in \Z\}$.
{\bf Remark}
This shows that every non-trivial subgroup in $\Z$ is generated an element $k \in \Z^+$.
\item Suppose $H$ is a non-trivial subgroup of $\Z_n = \{0, 1, \dots, n-1\}$
under addition,
and $\overline{k}$ is the smallest positive integer in $H$ when the elements
of $H$ are expressed in the form $\overline{r}$ with $0 \le r < n-1$.
Show the $H = \langle k \rangle$.
{\bf Remark}
This shows that every non-trivial
subgroup in $\Z_n$ is generated an element $\overline{k}\in \Z_n$.
\item Determine all the subgroups of $\Z_8$, and the subgroup lattice of $\Z_8$.
\item Suppose that a group contains elements
$a$ and $b$ such that $|a|=4, |b|=2$, and $a^3b=ba$. Show that $|ab| = 2$.
\item (5 points for each part)
Suppose $G$ is a group with $n$ elements, and $H$ is a subgroup of $G$ with
$m$ elements.
(a) Suppose $H \ne G$ and $g_1 \in G-H$. Let $g_1H = \{g_1 h: h \in H\}$.
\quad Show that $H \cap g_1 H = \emptyset$ so that $|H \cup g_1 H|=2m$.
(Here you need to argue $|g_1H| = m$.)
(b) Suppose $H \cup g_1H \ne G$ and
$g_2 \not \in (H \cup g_1H)$.
\quad Show that $(H\cup g_1 H) \cap g_2 H = \emptyset$
so that $|H \cup g_1 H \cup g_2 H| = 3m$.
(c) Show that $G$ is a disjoint union of
$H \cup g_1H \cup g_2 H \cdots g_kH$ for some $g_1, \dots, g_k \in G$
\quad
so that
$n/m$ is a positive integer.
{\bf Remark}
The set $g_iH$ is called
a left coset of the subgroup $H$.
The fact that $n/m$ is an integer is known as the Lagrange Theorem.
\item (5 points for each part) Use the result in the a last problem to prove the following.
(a) A group of prime order must be cyclic.
Hint: Let $a \in G$ not equal to the identity. Show that $\langle a \rangle = G$.
(b) Let $G$ be a group, and $a, b \in G$. If $|a|$ and $|b|$ are relatively prime,
show that $$H = \langle a\rangle\cap \langle b\rangle=\{e\}.$$
Hint: If $|H| = m > 1$ then ...
\item Suppose that $|G|=24$ and that $G$ is cyclic. If $a^8\not=e$ and $a^{12}\not=e$, show that
$G = \langle a\rangle$.
Hint: Suppose $H = \langle a \rangle$. By the Lagrange Theorem, $|H| = ...$.
\item (Extra credits)
Suppose $G$ is a set equipped with an associative binary operation $*$.
Furthermore, assume that $G$ has an left identity $e$, i.e., $e*g = g$ for all $g \in G$,
and that every $g \in G$ has an left inverse $g'$, i.e., $g'*g = e$.
Show that $G$ is a group.
[Hint: Let $\hat g$ be the left inverse of $g'$, where $g'$ is the left inverse of
$g \in G$. Show that $\hat g = g$ and conclude that $gg' = e$, i.e., the left inverse is
also the right inverse. Then show that the left identity is also the right
identity.]
\item (Extra credits) Let $A$ be a set, and
$\cP(A)$ be its power set.
Show that there is a group $G$ with $|G| = |\cP(A)|$,
Hint: Case 1. $|A|$ is finite.
Case 2. $A$ is infinite. Let $S_A$ be the group of bijections (permutations)
on $A$ under function composition.
Then $|S_A| = |\cP(A)|$.
The Axiom of Choice is needed to show
the $|\cP(A)| = |S_A|$.
\end{enumerate}
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