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\noindent
{\bf Math 307 Abstract Algebra \quad Homework 9}
\bigskip
Five points for each question unless specified otherwise.
\begin{enumerate}
\item Suppose $G$ is an Abelian group. Show that $H = \{g \in G: |g| \hbox{ is finite}\}$
is a subgroup.
\item Suppose $G$ is a finite Abelian group, and $m | |G|$. Show that $G$ has a subgroup of
order $m$.
[Hint: Up to isomorphism, we may assume
$G = \IZ_{p_1^{r_1}} \oplus \cdots \oplus \IZ_{p_k^{r_k}}$, where
$p_1, \dots, p_k$ are primes, so that $n = p_1^{r_1} \cdots p_k^{r_k}$.
If $m | n$, we may assume that $m = p_1^{t_1} \cdots p_k^{t_k}$
with $0 \le t_j \le r_j$ for $j = 1, \dots, k$. Show that there
is a subgroup of the form $H_1 \oplus \cdots \oplus H_k$ in $G = G_1 \oplus
\cdots \oplus G_k$
with order $m$, where $G_1 = \langle (1, 0, \dots, 0) \rangle, G_2 = \langle (0,1,0, \dots, 0)\rangle$, etc.
]
\item (10 points) Compare the number of isomorphic classes of subgroups of an Abelian group
of orders $m$ and $n$ for each of the following if $p,q$ are primes, and $r \in \IN$.
(a) $n = 3^2, m = 5^2$.
(b) $n = 2^4, m=5^4$,
(c) $n = p^r, m = q^r$,
(d) $n = p^r$ and $m = p^rq$,
(e) $n = p^r$ and $m = p^rq^2$.
[Hint: For each part, you only need to decide whether they have the same number of isomoprhism
classes, or twice as many, etc.]
\item (a) Give an example of a subset of a ring that is a subgroup under addition but
not a subring.
(b) Give an example of a finite non-commutative ring.
\item Show that if $m,n$ are integers and $a,b$ are elements in a ring. Then
$(ma)(nb) = (mn)(ab)$. [Note that for positive $m$, $ma = a+ \cdots + a$ ($m$ times);
for $(-m)a = m(-a) = (-a) + \cdots + (-a)$.]
[Hint: You might want to discuss the cases (1) $m,n > 0$, (2) $mn = 0$,
(3) $mn< 0$, (4) $m,n < 0$.]
\item
Let $R$ be a ring.
(a) Suppose $a \in R$. Show that $S = \{x \in R: ax=xa\}$ is a subring.
(b) Show that the center of $R$ defined by
$Z(R) = \{x \in R: ax = xa \hbox{ for all } a \in R\}$ is a subring.
\item
Let $R$ be a ring.
(a) Prove that $R$ is commutative if and only if $a^2-b^2 = (a+b)(a-b)$
for all $a, b \in R$.
(b) Prove that $R$ is commutative if $a^2 = a$ for all $a \in R$.
(Such a ring is called
a Boolean ring.)
[Hint: In (b), think about $a = (x+y), (x-y)$, etc.
\item Give an example of a Boolean ring with 4 elements. Give an example of a Boolean ring with
infinitely many elements.
[Hint: Consider $\IZ_2 \oplus \IZ_2$, and extend the idea.]
\end{enumerate}
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