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{\Huge 189-235A: Basic Algebra I} \\ \vskip 0.1in
{\Huge Assignment 5} \\ \vskip 0.1in
{\Large Due: Wednesday, November 28 }
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1. Let $f:\Z\lra R$ be a surjective
homomorphism of rings.
Show that $R$ is isomorphic either to $\Z$ or to
the ring $\Z/n\Z$ for a suitable $n\ge 1$.
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2. Let $R$ be a commutative ring and let $I$ be an ideal of $R$.
Prove or disprove the statement that
if $R$ is an integral domain, then so is $R/I$.
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3. Let $R=\Z[x]$, and let $I$ be the ideal $(p,x^2+1)$ generated by the integer
prime $p$ and the polynomial $x^2+1$. Show that $R/I$ is
isomorphic to $\Z/5\Z\times\Z/5\Z$ is $p=5$, and is isomorphic to a field
with $49$ elements if $p=7$.
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4. Let $F$ be a field and let $R=F[[x]]$ denote the ring of formal power series
with coefficients in $F$, i.e., the set of expressions of the form
$$ \sum_{n=0}^\infty a_n x^n,\quad a_n\in F,$$
where the addition and multiplication are performed by
formally expanding out the sums and products
(without worrying about issues of convergence, which
don't make sense in an arbitrary field $F$ anyways!)
Let $I=(x)$ be the ideal generated by the power series $x$. Show that
$R/I$ is isomorphic to $F$. Show that any element of $R$ which does not
belong to $I$ is invertible. Conclude that any non-trivial ideal of $R$ is
contained in $I$.
(A ring with this property is called a {\em local ring}, a terminology
arising from the prototypical example $F[[x]]$, because power series can be
thought of as ``functions defined in an infinitesimal neighbourhood
of the value $x=0$".)
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5. Let $F$ be a field, and define a binary composition law on
$G=F-\{1\}$ by the rule
$$ a * b = a+b -ab.$$
Show that $G$, with this operation, is a group. (In particular,
write down the neutral element for $*$, and give a
formula for the inverse of $a\in G$.)
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6. List all the elements of order $3$ in $S_3$. How many are there?
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7.
Suppose that $G$ is a group in which $x^2=1$, for all $x\in G$. Show
that $G$ is abelian.
Give an example of a {\bf non-abelian } group $G$ of order $27$
in which $x^3=1$ for all $x\in G$. (Hint: try to find such a group in the
group of $3\times 3$ invertible matrices with entries in $\Z/3\Z$.)
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8. Show that the intersection of two subgroups $H_1$ and $H_2$ of
a group $G$ is a subgroup of $G$.
What about unions of subgroups?
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9. If $a$ is an element of a finite group $G$ of cardinality $n$, show
that $a^n=1$.
Apply this general fact to the group $G=(\Z/p\Z)^\times$
(under multiplication)
to give another proof of {\em Fermat's Little Theorem} that $p$ divides
$a^p-a$ for all integers $a$ when $p$ is prime.
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10. Let $S$ be a subset of a group $G$. The centraliser of $S$, denoted
$Z(S)$, is the set of $a\in G$ which commute with every $s\in S$, i.e., such
that
$as=sa$ for all $s\in S$.
Show that $Z(S)$ is a subgroup of $G$.
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11. Let $G_1$ be the group of strictly positive real numbers, under
multiplication, and let
$G_2$ be the group of all real numbers, under addition. Show that
$G_1$ and $G_2$ are isomorphic, by writing down an explicit isomorphism
between the two groups.
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12. Recall that the {\em conjugacy class} of $a$
in a group $G$ is the set of all elements of $G$ which are of the
form $gag^{-1}$ for some $g\in G$.
Show that a normal subgroup of $G$ is a disjoint union of conjugacy classes.
List the conjugacy classes in $S_4$ and use this to give a complete list
of all the normal subgroups of $S_4$. Same question for $S_5$.
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