Abstract Algebra

David S. Dummit, Richard M. Foote

Chapter 1

Introduction to Groups - all with Video Answers

Educators

Section 1

Basic Axioms and Examples

Problem 1

Determine which of the following binary operations are associative:
(a) the operation $\star$ on $\mathbb{Z}$ defined by $a \star b=a-b$
(b) the operation $\star$ on $\mathbb{R}$ defined by $a \star b=a+b+a b$
(c) the operation $\star$ on $\mathbb{Q}$ defined by $a \star b=\frac{a+b}{5}$
(d) the operation $\star$ on $\mathbb{Z} \times \mathbb{Z}$ defined by $(a, b) \star(c, d)=(a d+b c, b d)$
(e) the operation $\star$ on $\mathbb{Q}-\{0\}$ defined by $a \star b=\frac{a}{b}$.

Nick Johnson

Nick Johnson

Numerade Educator

Problem 2

Decide which of the binary operations in the preceding exercise are commutative.

Nick Johnson

Numerade Educator

Problem 3

Prove that addition of residue classes in $\mathbb{Z} / n \mathbb{Z}$ is associative (you may assume it is well defined).

Nick Johnson

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Problem 4

Prove that multiplication of residue classes in $\mathbb{Z} / n \mathbb{Z}$ is associative (you may assume it is well defined).

Nick Johnson

Numerade Educator

Problem 5

Prove for all $n>1$ that $\mathbb{Z} / n \mathbb{Z}$ is not a group under multiplication of residue classes.

Nick Johnson

Numerade Educator

Problem 6

Determine which of the following sets are groups under addition:
(a) the set of rational numbers (including $0=0 / 1$ ) in lowest terms whose denominators are odd
(b) the set of rational numbers (including $0=0 / 1$ ) in lowest terms whose denominators are cven
(c) the set of rational numbers of absolute value $<1$
(d) the set of rational numbers of absolute value $\geq 1$ together with 0
(e) the set of rational numbers with denominators equal to 1 or 2
(f) the set of rational numbers with denominators equal to 1,2 or $3 .$

Nick Johnson

Numerade Educator

Problem 7

Let $G=\{x \in \mathbb{R} \mid 0 \leq x<1\}$ and for $x, y \in G$ let $x \star y$ be the fractional part of $x+y$ (i.e., $x \star y=x+y-[x+y]$ where $[a]$ is the greatest integer less than or equal to $a$ ). Prove that $\star$ is a well defined binary operation on $G$ and that $G$ is an abelian group under (called the real numbers $\bmod I$ ).

Nick Johnson

Numerade Educator

Problem 8

Let $G=\left\{z \in \mathbb{C} \mid z^{n}=1\right.$ for some $\left.n \in \mathbb{Z}^{+}\right\}$
(a) Prove that $G$ is a group under multiplication (called the group of roots of unity in C).
(b) Prove that $G$ is not a group under addition.

Nick Johnson

Numerade Educator

Problem 9

Let $G=\{a+b \sqrt{2} \in \mathbb{R} \mid a, b \in \mathbb{Q}\}$
(a) Prove that $G$ is a group under addition.
(b) Prove that the nonzero elements of $G$ are a group under multiplication. ["Rationalize the denominators" to find multiplicative inverses.]

Nick Johnson

Numerade Educator

Problem 10

Prove that a finite group is abelian if and only if its group table is a symmetric matrix.

Nick Johnson

Numerade Educator

Problem 11

Find the orders of each element of the additive group $\mathbb{Z} / 12 \mathbb{Z} .$

Nick Johnson

Numerade Educator

Problem 12

Find the orders of the following elements of the multiplicative group $(\mathbb{Z} / 12 \mathbb{Z})^{\times}: \overline{1}, \overline{-1}$, $\overline{5}, \overline{7}, \overline{-7}, \overline{13}$

Nick Johnson

Numerade Educator

Problem 13

Find the orders of the following elements of the additive group $\mathbb{Z} / 36 \mathbb{Z}: \overline{1}, \overline{2}, \overline{6}, \overline{9}, \overline{10}, \overline{12}$,
$-1, \overline{-10}, \overline{-18}$

Nick Johnson

Numerade Educator

Problem 14

Find the orders of the following elements of the multiplicative group $(\mathbb{Z} / 36 \mathbb{Z})^{x}: \overline{1}, \overline{-1}$, $\overline{5}, \overline{13}, \overline{-13}, \overline{17}$

Nick Johnson

Numerade Educator

Problem 15

Prove that $\left(a_{1} a_{2} \ldots a_{n}\right)^{-1}=a_{n}^{-1} a_{n-1}^{-1} \ldots a_{1}^{-1} \quad$ for all $a_{1}, a_{2}, \ldots, a_{n} \in G$.

Rashmi Sinha

Numerade Educator

Problem 16

Let $x$ be an element of $G$. Prove that $x^{2}=1$ if and only if $|x|$ is either 1 or 2 .

Amy Jiang

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Problem 17

Let $x$ be an element of $G .$ Prove that if $|x|=n$ for some positive integer $n$ then $x^{-1}=x^{n-1}$.

Joshua Eastwood

Joshua Eastwood

Numerade Educator

Problem 18

Let $x$ and $y$ be elements of $G$. Prove that $x y=y x$ if and only if $y^{-1} x y=x$ if and only if $x^{-1} y^{-1} x y=1$

Rukhmani Jain

Numerade Educator

Problem 19

Let $x \in G$ and let $a, b \in \mathbb{Z}^{+}$
(a) Prove that $x^{a+b}=x^{a} x^{b} \quad$ and $\quad\left(x^{a}\right)^{b}=x^{a b}$.
(b) Prove that $\left(x^{a}\right)^{-1}=x^{-a}$.
(c) Establish part (a) for arbitrary integers $a$ and $b$ (positive, negative or zero).

James Chok

Numerade Educator

Problem 20

For $x$ an element in $G$ show that $x$ and $x^{-1}$ have the same order.

Wendi Zhao

Numerade Educator

Problem 21

Let $G$ be a finite group and let $x$ be an element of $G$ of order $n .$ Prove that if $n$ is odd, then $x=\left(x^{2}\right)^{k}$ for some $k$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 22

If $x$ and $g$ are elements of the group $G$, prove that $|x|=\left|g^{-1} x g\right| .$ Deduce that $|a b|=|b a|$ for all $a, b \in G$.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 23

Suppose $x \in G$ and $|x|=n<\infty .$ If $n=s t$ for some positive integers $s$ and $t$, prove that $\left|x^{s}\right|=t$

Nick Johnson

Numerade Educator

Problem 24

If $a$ and $b$ are commuting elements of $G$, prove that $(a b)^{n}=a^{n} b^{n}$ for all $n \in \mathbb{Z}$. [Do this by induction for positive $n$ first.]

Wendi Zhao

Numerade Educator

Problem 25

Prove that if $x^{2}=1$ for all $x \in G$ then $G$ is abelian.

Nick Johnson

Numerade Educator

Problem 26

Assume $H$ is a nonempty subset of $(G, \star)$ which is closed under the binary operation on $G$ and is closed under inverses, i.e., for all $h$ and $k \in H, h k$ and $h^{-1} \in H .$ Prove that $H$ is a group under the operation $\star$ restricted to $H$ (such a subset $H$ is called a subgroup of $G$ ).

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 27

Prove that if $x$ is an element of the group $G$ then $\left\{x^{n} \mid n \in \mathbb{Z}\right\}$ is a subgroup (cf. the preceding exercise) of $G$ (called the cyclic subgroup of $G$ generated by $x$ ).

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 28

Let $(A, \star)$ and $(B, \circ)$ be groups and let $A \times B$ be their direct product (as defined in Example
6). Verify all the group axioms for $A \times B:$
(a) prove that the associative law holds: for all $\left(a_{i}, b_{i}\right) \in A \times B, i=1,2,3$ $\left(a_{1}, b_{1}\right)\left[\left(a_{2}, b_{2}\right)\left(a_{3}, b_{3}\right)\right]=\left[\left(a_{1}, b_{1}\right)\left(a_{2}, b_{2}\right)\right]\left(a_{3}, b_{3}\right)$

Wendi Zhao

Numerade Educator

Problem 29

Prove that $A \times B$ is an abelian group if and only if both $A$ and $B$ are abelian.

Nick Johnson

Numerade Educator

Problem 30

Prove that the clements $(a, 1)$ and $(1, b)$ of $A \times B$ commute and deduce that the order of $(a, b)$ is the least common multiple of $|a|$ and $|b|$

Wendi Zhao

Numerade Educator

Problem 31

Prove that any finite group $G$ of even order contains an element of order $2 .$ [Let $t(G)$ be the set $\left\{g \in G \mid g \neq g^{-1}\right.$ \}. Show that $t(G)$ has an even number of elements and cvery nonidentity clement of $G-t(G)$ has order $2 .]$

Wendi Zhao

Numerade Educator

Problem 32

If $x$ is an element of finite order $n$ in $G$, prove that the clements $1, x, x^{2}, \ldots, x^{n-1}$ are all distinct. Deduce that $|x| \leq|G|$.

Wendi Zhao

Numerade Educator

Problem 33

Let $x$ be an clement of finite order $n$ in $G$.
(a) Prove that if $n$ is odd then $x^{i} \neq x^{-i}$ for all $i=1,2, \ldots, n-1$.
(b) Prove that if $n=2 k$ and $1 \leq i<n$ then $x^{i}=x^{-i}$ if and only if $i=k$.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 34

If $x$ is an clement of infinite order in $G$, prove that the elements $x^{n}, n \in \mathbb{Z}$ are all distinct.

Wendi Zhao

Numerade Educator

Problem 35

If $x$ is an element of finite order $n$ in $G$, use the Division Algorithm to show that any integral power of $x$ equals one of the clements in the set $\left\{1, x, x^{2}, \ldots, x^{n-1}\right\}$ (so these are all the distinct clements of the cyclic subgroup (cf. Exercise 27 above) of $G$ gencrated by $x)$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 36

Assume $G=\{1, a, b, c\}$ is a group of order 4 with identity 1 . Assume also that $G$ has no clements of order 4 (so by Exercise 32, every clement has order $\leq 3$ ). Use the cancellation laws to show that there is a unique group table for $G$. Deduce that $G$ is abelian.

Wendi Zhao

Numerade Educator