AATA Exercises

Exercises 3.5 Exercises

1.

Find all $x \in Z$ satisfying each of the following equations.

$3 x \equiv 2 (\mod 7)$
$5 x + 1 \equiv 13 (\mod 23)$
$5 x + 1 \equiv 13 (\mod 26)$
$9 x \equiv 3 (\mod 5)$
$5 x \equiv 1 (\mod 6)$
$3 x \equiv 1 (\mod 6)$

(a) $3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}\text{;}$ (c) $18 + 26 \mathbb Z\text{;}$ (e) $5 + 6 \mathbb Z\text{.}$

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2.

Which of the following multiplication tables defined on the set $G = {a, b, c, d}$ form a group? Support your answer in each case.

$\begin{array}{ccccc} \circ & a & b & c & d \\ a & a & c & d & a \\ b & b & b & c & d \\ c & c & d & a & b \\ d & d & a & b & c \end{array}$
$\begin{array}{ccccc} \circ & a & b & c & d \\ a & a & b & c & d \\ b & b & a & d & c \\ c & c & d & a & b \\ d & d & c & b & a \end{array}$
$\begin{array}{ccccc} \circ & a & b & c & d \\ a & a & b & c & d \\ b & b & c & d & a \\ c & c & d & a & b \\ d & d & a & b & c \end{array}$
$\begin{array}{ccccc} \circ & a & b & c & d \\ a & a & b & c & d \\ b & b & a & c & d \\ c & c & b & a & d \\ d & d & d & b & c \end{array}$

Hint.

(a) Not a group; (c) a group.

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3.

Write out Cayley tables for groups formed by the symmetries of a rectangle and for $(Z_{4}, +) .$ How many elements are in each group? Are the groups the same? Why or why not?

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4.

Describe the symmetries of a rhombus and prove that the set of symmetries forms a group. Give Cayley tables for both the symmetries of a rectangle and the symmetries of a rhombus. Are the symmetries of a rectangle and those of a rhombus the same?

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5.

Describe the symmetries of a square and prove that the set of symmetries is a group. Give a Cayley table for the symmetries. How many ways can the vertices of a square be permuted? Is each permutation necessarily a symmetry of the square? The symmetry group of the square is denoted by $D_{4} .$

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6.

Give a multiplication table for the group $U (12) .$

Hint.

\begin{equation*} \begin{array}{c|cccc} \cdot & 1 & 5 & 7 & 11 \\ \hline 1 & 1 & 5 & 7 & 11 \\ 5 & 5 & 1 & 11 & 7 \\ 7 & 7 & 11 & 1 & 5 \\ 11 & 11 & 7 & 5 & 1 \end{array} \end{equation*}

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7.

Let $S = R ∖ {- 1}$ and define a binary operation on $S$ by $a * b = a + b + a b .$ Prove that $(S, *)$ is an abelian group.

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8.

Give an example of two elements $A$ and $B$ in $G L_{2} (R)$ with $A B \neq B A .$

Hint.

Pick two matrices. Almost any pair will work.

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9.

Prove that the product of two matrices in $S L_{2} (R)$ has determinant one.

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10.

Prove that the set of matrices of the form

(\begin{matrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix})

is a group under matrix multiplication. This group, known as the Heisenberg group, is important in quantum physics. Matrix multiplication in the Heisenberg group is defined by

(\begin{matrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{matrix}) (\begin{matrix} 1 & x^{'} & y^{'} \\ 0 & 1 & z^{'} \\ 0 & 0 & 1 \end{matrix}) = (\begin{matrix} 1 & x + x^{'} & y + y^{'} + x z^{'} \\ 0 & 1 & z + z^{'} \\ 0 & 0 & 1 \end{matrix}) .

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11.

Prove that $det (A B) = det (A) det (B)$ in $G L_{2} (R) .$ Use this result to show that the binary operation in the group $G L_{2} (R)$ is closed; that is, if $A$ and $B$ are in $G L_{2} (R),$ then $A B \in G L_{2} (R) .$

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12.

Let $Z_{2}^{n} = {(a_{1}, a_{2}, \dots, a_{n}) : a_{i} \in Z_{2}} .$ Define a binary operation on $Z_{2}^{n}$ by

(a_{1}, a_{2}, \dots, a_{n}) + (b_{1}, b_{2}, \dots, b_{n}) = (a_{1} + b_{1}, a_{2} + b_{2}, \dots, a_{n} + b_{n}) .

Prove that $Z_{2}^{n}$ is a group under this operation. This group is important in algebraic coding theory.

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13.

Show that $R^{*} = R ∖ {0}$ is a group under the operation of multiplication.

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14.

Given the groups $R^{*}$ and $Z,$ let $G = R^{*} \times Z .$ Define a binary operation $\circ$ on $G$ by $(a, m) \circ (b, n) = (a b, m + n) .$ Show that $G$ is a group under this operation.

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15.

Prove or disprove that every group containing six elements is abelian.

Hint.

There is a nonabelian group containing six elements.

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16.

Give a specific example of some group $G$ and elements $g, h \in G$ where $(g h)^{n} \neq g^{n} h^{n} .$

Hint.

Look at the symmetry group of an equilateral triangle or a square.

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17.

Give an example of three different groups with eight elements. Why are the groups different?

Hint.

The are five different groups of order 8.

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18.

Show that there are $n!$ permutations of a set containing $n$ items.

Hint.

Let

\begin{equation*} \sigma = \begin{pmatrix} 1 & 2 & \cdots & n \\ a_1 & a_2 & \cdots & a_n \end{pmatrix} \end{equation*}

be in $S_n\text{.}$ All of the $a_i$s must be distinct. There are $n$ ways to choose $a_1\text{,}$ $n-1$ ways to choose $a_2\text{,}$ $\ldots\text{,}$ 2 ways to choose $a_{n - 1}\text{,}$ and only one way to choose $a_n\text{.}$ Therefore, we can form $\sigma$ in $n(n - 1) \cdots 2 \cdot 1 = n!$ ways.

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19.

Show that

0 + a \equiv a + 0 \equiv a (\mod n)

for all $a \in Z_{n} .$

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20.

Prove that there is a multiplicative identity for the integers modulo $n :$

a \cdot 1 \equiv a (\mod n) .

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21.

For each $a \in Z_{n}$ find an element $b \in Z_{n}$ such that

a + b \equiv b + a \equiv 0 (\mod n) .

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22.

Show that addition and multiplication mod $n$ are well defined operations. That is, show that the operations do not depend on the choice of the representative from the equivalence classes mod $n .$

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23.

Show that addition and multiplication mod $n$ are associative operations.

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24.

Show that multiplication distributes over addition modulo $n :$

a (b + c) \equiv a b + a c (\mod n) .

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25.

Let $a$ and $b$ be elements in a group $G .$ Prove that $a b^{n} a^{- 1} = (a b a^{- 1})^{n}$ for $n \in Z .$

Hint.

\begin{align*} (aba^{-1})^n & = (aba^{-1})(aba^{-1}) \cdots (aba^{-1})\\ & = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1}\\ & = ab^na^{-1}. \end{align*}

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26.

Let $U (n)$ be the group of units in $Z_{n} .$ If $n > 2,$ prove that there is an element $k \in U (n)$ such that $k^{2} = 1$ and $k \neq 1 .$

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27.

Prove that the inverse of $g_{1} g_{2} \dots g_{n}$ is $g_{n}^{- 1} g_{n - 1}^{- 1} \dots g_{1}^{- 1} .$

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28.

Prove the remainder of Proposition 3.2.14: if $G$ is a group and $a, b \in G,$ then the equation $x a = b$ has a unique solution in $G .$

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29.

Prove Theorem 3.2.16.

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30.

Prove the right and left cancellation laws for a group $G;$ that is, show that in the group $G,$ $b a = c a$ implies $b = c$ and $a b = a c$ implies $b = c$ for elements $a, b, c \in G .$

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31.

Show that if $a^{2} = e$ for all elements $a$ in a group $G,$ then $G$ must be abelian.

Hint.

Since $abab = (ab)^2 = e = a^2 b^2 = aabb\text{,}$ we know that $ba = ab\text{.}$

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32.

Show that if $G$ is a finite group of even order, then there is an $a \in G$ such that $a$ is not the identity and $a^{2} = e .$

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33.

Let $G$ be a group and suppose that $(a b)^{2} = a^{2} b^{2}$ for all $a$ and $b$ in $G .$ Prove that $G$ is an abelian group.

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34.

Find all the subgroups of $Z_{3} \times Z_{3} .$ Use this information to show that $Z_{3} \times Z_{3}$ is not the same group as $Z_{9} .$ (See Example 3.3.5 for a short description of the product of groups.)

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35.

Find all the subgroups of the symmetry group of an equilateral triangle.

Hint.

$H_1 = \{ id \}\text{,}$ $H_2 = \{ id, \rho_1, \rho_2 \}\text{,}$ $H_3 = \{ id, \mu_1 \}\text{,}$ $H_4 = \{ id, \mu_2 \}\text{,}$ $H_5 = \{ id, \mu_3 \}\text{,}$ $S_3\text{.}$

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36.

Compute the subgroups of the symmetry group of a square.

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37.

Let $H = {2^{k} : k \in Z} .$ Show that $H$ is a subgroup of $Q^{*} .$

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38.

Let $n = 0, 1, 2, \dots$ and $n Z = {n k : k \in Z} .$ Prove that $n Z$ is a subgroup of $Z .$ Show that these subgroups are the only subgroups of $Z .$

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39.

Let $T = {z \in C^{*} : | z | = 1} .$ Prove that $T$ is a subgroup of $C^{*} .$

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40.

(\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix})

where $θ \in R .$ Prove that $G$ is a subgroup of $S L_{2} (R) .$

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41.

Prove that

G = {a + b \sqrt{2} : a, b \in Q and a and b are not both zero}

is a subgroup of $R^{*}$ under the group operation of multiplication.

Hint.

The identity of $G$ is $1 = 1 + 0 \sqrt{2}\text{.}$ Since $(a + b \sqrt{2}\, )(c + d \sqrt{2}\, ) = (ac + 2bd) + (ad + bc)\sqrt{2}\text{,}$ $G$ is closed under multiplication. Finally, $(a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)\text{.}$

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42.

Let $G$ be the group of $2 \times 2$ matrices under addition and

H = {(\begin{matrix} a & b \\ c & d \end{matrix}) : a + d = 0} .

Prove that $H$ is a subgroup of $G .$

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43.

Prove or disprove: $S L_{2} (Z),$ the set of $2 \times 2$ matrices with integer entries and determinant one, is a subgroup of $S L_{2} (R) .$

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44.

List the subgroups of the quaternion group, $Q_{8} .$

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45.

Prove that the intersection of two subgroups of a group $G$ is also a subgroup of $G .$

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46.

Prove or disprove: If $H$ and $K$ are subgroups of a group $G,$ then $H \cup K$ is a subgroup of $G .$

Hint.

Look at $S_3\text{.}$

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47.

Prove or disprove: If $H$ and $K$ are subgroups of a group $G,$ then $H K = {h k : h \in H and k \in K}$ is a subgroup of $G .$ What if $G$ is abelian?