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Math 3140: Abstract Algebra 1,
Fall 2015
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Homework
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Assignment
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Assigned
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Due
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Problems
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| HW1 |
8/26/15
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9/2/13
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Read
pages 25-31, 37-41, 69-72. (You probably already know a lot of this, so I hope
this reading goes by quickly.)
1.
(a) How many algebras are there of the form
$\langle \{0,1\}; \star\rangle$ if
$\mathbf{arity}(\star)=2$? (You need to count how many different
tables for $\star$ are possible.)
(b) How many isomorphism types of algebras are there of the form
$\langle \{0,1\}; \star\rangle$? (Count algebras as in part (a), but
discard isomorphic copies.)
2. Suppose that $a$ is an identity element for $+$ in
$\langle \{a,b,c\}; +\rangle$, where
$\mathbf{arity}(+)=2$. (This means that $a+x=x = x+a$ for every $x$.)
How many possibilities are there
for such an algebra? How many possibilities if $+$ is
a commutative operation with identity element $a$?
(``Commutative'' means that $x+y=y+x$ for every $x$ and $y$.)
3. Let $\mathbb Z = \langle Z; +, -, 0\rangle$.
Show that if $h\colon \mathbb Z\to \mathbb Z$ is a homomorphism,
and $h(1) = a$, then $h(n) = an$ for every $n\in Z$.
(Hint: First, use induction to prove it for positive $n$.)
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| HW2 |
9/2/15
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9/9/13
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Read
Sections 2.1 and 2.2.
1. Do Exercise 2.1.13.
2. Using the previous result,
do Exercise 2.1.11.
(We did this in class, but we rushed at the end.)
3. Prove the equivalence of (a), (b), (c), (d)
from Exercise 2.1.15.
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| HW3 |
9/9/15
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9/16/15
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Read
Section 2.3.
1. Prove a Cayley Representation for semigroups. That is,
show that any semigroup is embeddable into a semigroup of functions under
composition. (Hint: Let $\mathbb S$ be a semigroup. Show that you can make
a monoid out of it by simply adding $1$. That is, show that $M = S\cup \{1\}$
can be given a multiplication with extends the multiplication on
$S$ to an associative multiplication on $M$
where $1$ is the identity element. There
is only one way to do this. Next show that inclusion is
an embedding $\mathbb S\to \mathbb M$
of $\mathbb S$ into this monoid. Next use the Cayley Representation Theorem
for monoids to derive that $\mathbb M$ is embeddable in a monoid
${\mathcal F}$ of functions.
Finally, compose the embeddings $\mathbb S\to \mathbb M\to{\mathcal F}$.)
2. Prove a Cayley Representation Theorem for groups.
That is, show that any group is embeddable into
the group of all permutations
of some set. (Hint: copy the proof of the Cayley Representation Theorem
for monoids, but use permutations instead of self-maps.)
3. If $\mathbb A$ is an algebraic structure, then an isomorphism from
$\mathbb A$ to itself is called an automorphism of $\mathbb A$.
Show that
the automorphisms of $\mathbb A$ form a group under the operations
of composition, inverse and identity. (This group is called the
automorphism group of $\mathbb A$, and it is written
$\textrm{Aut}(\mathbb A)$.)
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| HW4 |
9/16/15
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9/23/15
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Read
Section 4.1.
1. Recall that $D_n$ denotes the symmetry group of
a regular $n$-gon. Show that if $n$ is divisible by $m$, then $D_n$
has a subgroup isomorphic to $D_m$.
2. An element of order $2$ in a group is called an
involution. Show that $D_n$ is generated by two
involutions. (Hint: find two reflections that generate
everything.)
3. Using linear algebra it is possible to prove
that every rotation in 3-dimensional space has an axis.
Accept this, and identify the possible axes of symmetry of the cube.
Use the answer to argue
geometrically that every rotation of the cube has order
$1, 2, 3$ or $4$. Find all of these, expressed as permutations of
the 8 vertices in cycle notation. (There are 24.)
Group them together according to order.
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| HW5 |
9/25/15
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9/30/15
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1. Do Exercise 2.5.4.
2. Do Exercise 2.5.6.
3. Do Exercise 2.5.8.
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| No HW |
9/30/15
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10/7/15
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No HW.
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| HW6 |
10/4/15
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10/14/15
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1. Do Exercise 2.6.6.
2. Supppose that $G$ is a group, $a\in G$, and
$c_a\colon G\to G\colon x\mapsto axa^{-1}$.
Write down what must be checked to verify that
$c_a$ is an automorphism of $G$, and then check those things.
3. Define
$c\colon G\to \textrm{Aut}(G)$ by $a\mapsto c_a$.
Write down what must be checked to verify that
$c$ is a group homomorphism, and then check those things.
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| No HW |
10/14/15
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10/21/15
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No HW.
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| HW 7 |
10/21/15
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10/28/15
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Read pages 149-154, 157-159.
1. Show that the infinite cyclic group
$\mathbb Z$ cannot factor as $\mathbb Z\cong A\times B$ with both
$A$ and $B$ containing more than one element.
2. Find groups so that $A_1\times A_2\cong B_1\times B_2$
with no $A_i$ isomorphic to any $B_j$.
3. The polar form of a nonzero complex number is
$z = r(\cos(\theta)+i\sin(\theta))$ where $r\in \mathbb R^+$
is the absolute value of $z$ (i.e., if $z=a+bi$, then
$r = |z| = \sqrt{a^2+b^2}$), and $\theta$ is the argument of $z$
(i.e., if $z=a+bi$, then $\tan(\theta)=b/a$). The polar form
is useful for understanding
the multiplication of complex numbers, since if
$z_1 = r_1(\cos(\theta_1)+i\sin(\theta_1))$ and
$z_2 = r_2(\cos(\theta_2)+i\sin(\theta_2))$, then
$z_1z_2 = (r_1r_2)(\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2))$.
By considering the polar form of a complex number, explain why
$\mathbb C^{\times}\cong \mathbb R^+\times T$ where
$\mathbb R^+$ is the multiplicative group of positive real numbers
and $T$
is the multiplicative
subgroup of $\mathbb C^{\times}$ consisting of
elements on the unit circle.
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| HW 8 |
10/28/15
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11/4/15
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Read pages 160-162 (Semidirect Product), 199-215
(Fundamental Theorem of Finite Abelian Groups).
1. Determine the number of
isomorphism types of abelian groups of order 100,
and express each of them in elementary divisor form and also in
invariant factor form.
2. A number $n$ is square free if it is not divisible by
$m^2$ for any integer $m>1$. (So, $6, 10, 15$ are square free,
but $9, 12, 18$ are not.) If $n$ is square free, what can you say about
the number of isomorphism types of abelian groups of order $n$?
3.
The
$\textit{partition function}$ is the function
whose value at $n$ is the number of ways of writing
$n$ as a sum of nonincreasing positive integers.
Thus
$p(1) = 1$ ($1: 1$),
$p(2) = 2$ ($2: 1+1=2$),
$p(3) = 3$ ($3: 1+1+1 = 2+1 = 3$),
$p(4) = 5$ ($4: 1+1+1+1 = 2+1+1 = 2+2 = 3+1 = 4$), etc.
Explain why, if the prime factorization of $n$ is
$p_1^{e_1}\cdots p_k^{e_k}$, the number of isomorphism types
of abelian groups of order $n$ is $p(e_1)\cdots p(e_k)$.
(Hint: try to work out the answer for the case $n = p_1^{e_1}$ first.)
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| HW 9 |
11/4/15
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11/11/15
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Read 242-252.
1.
Do Exercise 5.1.1.
2.
Do Exercise 5.1.5.
3.
Do Exercise 5.2.3.
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| HW 10 |
11/11/15
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11/18/15
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Read 252-262.
1.
Do Exercise 5.2.5.
2.
Find the centralizer and conjugacy class of each element of $D_4$.
3.
Show that the centralizer of
$\alpha = (1\;2\;3\;\ldots\;n)\in S_n$
is
$C=\{e, \alpha, \alpha^2, \cdots, \alpha^{n-1}\}$. (Hint:
Show that elements of $C$ centralize $\alpha$, and show that
$C$ has the right size to be the centralizer of $\alpha$.)
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11/18/15
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12/2/15
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