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Math 3140: Abstract Algebra 1,
Spring 2026
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Homework
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Homework text should be typed and submitted to Canvas in pdf form.
Latex HW template.
HWtemplate.tex,
HWtemplate.zip,
HWtemplate.pdf.
Latex guide
You do not have to use Latex.
Also, you do not have to create digital images.
Rather, you may submit hand-drawn images
to accompany your solutions when convenient and desirable.
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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 |
1/15/26
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Read
pages 16-22, 25-31, 37-41. (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 $n$.)
Solution.tex.
Solution.pdf.
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| HW2 |
1/22/26
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1/28/26
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Read
pages 69-72.
1. In an earlier HW assignment, you determined the
number of 2-element algebras of the form
$\langle \{0,1\}; \star\rangle$ where
$\mathbf{arity}(\star)=2$. You also determined the number
of isomorphism types of such algebras. Let's continue that
investigation by answering the following additional question:
How many $2$-element algebras of the form
$\langle \{0,1\}; \star\rangle$ have an identity element?
First give the number of such algebras
up to equality, then state
the number of such algebras up to isomorphism.
2. Show that if some algebra
$\mathbb{A}=\langle A; \star\rangle$
has an identity element for $\star$, then the identity
element for $\star$ is unique.
3. Suppose that
$\mathbb{A}=\langle A; \star, 1\rangle$ is an algebra
with one binary operation $\star$
and one zeroary operation $1$.
Assume that $1$ is an identity element of
$\mathbb{A}$ with respect to $\star$. If $a\in A$, then
an inverse to $a$ with respect to $\star$
is an element $b\in A$
such that $a\star b = 1$ and $b\star a = 1$.
(a) Show that if $\star$ is an associative operation,
then any $a\in A$ can have at most one inverse.
(b) The purpose of this example is to show
that $a\in A$ may have more than one inverse
if the multiplication $\star$ is not associative.
Give an example of a $3$-element algebra
$\mathbb{A}=\langle \{1,a,b\}; \star, 1\rangle$
where (i) $1$ is an identity element for $\mathbb{A}$,
(ii) every element of
$\mathbb{A}$ has an inverse with respect to $\star$,
but (iii) inverses are
not unique in $\mathbb{A}$. (To “ Give an example ”
it suffices to write down a table for $\star$.)
Solution.pdf.
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| HW3 |
1/29/26
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2/4/26
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Read
Section 2.1. Read Exercises 2.1.3 and 2.1.8-2.1.11.
1. Exercise 2.1.7.
2. Exercise 2.1.13.
3. Exercise 2.1.15.
(Show only the
equivalence of (a), (d), (e).)
(Remember that a group is
called abelian if it satifies the
Commutative Law for multiplication: $(\forall x)(\forall y)(xy=yx)$.)
To solve these problems, you may cite and use anything stated
in the book, provided the cited item occurs earlier in the book
than the assigned problem.
Solution.pdf.
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| HW4 |
2/5/26
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2/11/26
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Read Sections 2.2-2.3.
1. A permutation of cycle type $(i\;j)$
is called a transposition.
Show that the symmetric group $S_n$
is generated by the transpositions it contains.
Equivalently, show that every element of
$S_n$ is a product of transpositions.
(Compare this exercise to Exercises 2.2.16 and 2.2.17.)
2. 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 $D_m$.
3. 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
all elements of $D_n$.)
Solution.pdf.
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| HW5 |
2/11/26
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2/18/26
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Read Section 2.4 up to page 117. Read Section 2.5.
Briefly review Section 2.6.
1. Exercise 2.5.4.
2. Exercise 2.5.6.
3. Exercise 2.5.8. (You do not have to do Exercise 2.5.7
to use it here.)
Solution.pdf.
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| HW6 |
2/19/26
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2/26/26
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1. Let $G$ be a finite group and assume
that $H, K\leq G$ are subgroups. Explain why
- $|H\cap K|$ divides gcd($|H|$, $|K|$).
- $|\langle H\cup K\rangle|$ is divisible by lcm($|H|$, $|K|$).
2. Show that the normality relation is not transitive
by showing that, in $D_4$, we have
$\langle f\rangle\triangleleft
\langle r^2, f\rangle$ and
$\langle r^2, f\rangle\triangleleft
D_4$, but not
$\langle f\rangle{\triangleleft} D_4$.
3. Assume that $G$ is a finite group, $H\leq G$
is a subgroup, and $g\in G$. Show that $gHg^{-1}$
is a subgroup of $G$ and $|H|=|gHg^{-1}|$. (Hint:
First show that the function
$x\mapsto gxg^{-1}$ is an automorphism of $G$.)
Solution.pdf.
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| HW7 |
3/9/26
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3/25/26
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1. Exercise 2.6.5.
2. Exercise 2.7.4.
Hint: Show that the function
$p\colon A\times N\to AN\colon (a,n)\mapsto an$
has these properties:
- It is surjective.
- $|p^{-1}(1)| = |A\cap N|$.
- $\textrm{ker}(p)$ is a uniform equivalence relation.
3. Exercise 2.7.6(b).
Solution.pdf.
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| HW 8 |
3/25/26
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4/1/26
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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.
Solution.pdf.
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| HW 9 |
4/2/26
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4/8/26
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Read pages 160-162.
1. Exercise 3.2.4.
2. Exercise 3.2.6.
3. Show that if $G\cong A$ ⋊ $B$ and $G$ is abelian,
then $G\cong A\times B$.
Solution.pdf.
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HW 10
(Last one!)
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4/9/26
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4/15/26
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Read pages 199-212
(Fundamental Theorem of Finite Abelian Groups).
1. Exercise 3.6.14.
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.
Assume that $G$ is a finite abelian $p$-group.
What is the relationship between the number of elementary divisors
of $G$ and the number of elements of order $p$ in $G$?
Solution.pdf.
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