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3.5: Exercises

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1

Find all xZ satisfying each of the following equations.

  1. 3x2(mod7)
  2. 5x+113(mod23)
  3. 5x+113(mod26)
  4. 9x3(mod5)
  5. 5x1(mod6)
  6. 3x1(mod6)

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.

  1. abcdaacdabbbcdccdabddabc
  2. abcdaabcdbbadcccdabddcba
  3. abcdaabcdbbcdaccdabddabc
  4. abcdaabcdbbacdccbaddddbc

3

Write out Cayley tables for groups formed by the symmetries of a rectangle and for (Z4,+). How many elements are in each group? Are the groups the same? Why or why not?

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?

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 D4.

6

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

7

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

8

Give an example of two elements A and B in GL2(R) with ABBA.

9

Prove that the product of two matrices in SL2(R) has determinant one.

10

Prove that the set of matrices of the form

(1xy01z001)

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

(1xy01z001)(1xy01z001)=(1x+xy+y+xz01z+z001).

11

Prove that det in GL_2({\mathbb R})\text{.} Use this result to show that the binary operation in the group GL_2({\mathbb R}) is closed; that is, if A and B are in GL_2({\mathbb R})\text{,} then AB \in GL_2({\mathbb R})\text{.}

12

Let {\mathbb Z}_2^n = \{ (a_1, a_2, \ldots, a_n) : a_i \in {\mathbb Z}_2 \}\text{.} Define a binary operation on {\mathbb Z}_2^n by

(a_1, a_2, \ldots, a_n) + (b_1, b_2, \ldots, b_n) = (a_1 + b_1, a_2 + b_2, \ldots, a_n + b_n)\text{.} \nonumber

Prove that {\mathbb Z}_2^n is a group under this operation. This group is important in algebraic coding theory.

13

Show that {\mathbb R}^{\ast} = {\mathbb R} \setminus \{0 \} is a group under the operation of multiplication.

14

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

15

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

16

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

17

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

18

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

19

Show that

0 + a \equiv a + 0 \equiv a \pmod{ n } \nonumber

for all a \in {\mathbb Z}_n\text{.}

20

Prove that there is a multiplicative identity for the integers modulo n\text{:}

a \cdot 1 \equiv a \pmod{n}\text{.} \nonumber

21

For each a \in {\mathbb Z}_n find an element b \in {\mathbb Z}_n such that

a + b \equiv b + a \equiv 0 \pmod{ n}\text{.} \nonumber

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\text{.}

23

Show that addition and multiplication mod n are associative operations.

24

Show that multiplication distributes over addition modulo n\text{:}

a(b + c) \equiv ab + ac \pmod{n}\text{.} \nonumber

25

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

26

Let U(n) be the group of units in {\mathbb Z}_n\text{.} If n \gt 2\text{,} prove that there is an element k \in U(n) such that k^2 = 1 and k \neq 1\text{.}

27

Prove that the inverse of g _1 g_2 \cdots g_n is g_n^{-1} g_{n-1}^{-1} \cdots g_1^{-1}\text{.}

28

Prove the remainder of Proposition 3.21: if G is a group and a, b \in G\text{,} then the equation xa = b has a unique solution in G\text{.}

29

Prove Theorem 3.23.

30

Prove the right and left cancellation laws for a group G\text{;} that is, show that in the group G\text{,} ba = ca implies b = c and ab = ac implies b = c for elements a, b, c \in G\text{.}

31

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

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\text{.}

33

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

34

Find all the subgroups of {\mathbb Z}_3 \times {\mathbb Z}_3\text{.} Use this information to show that {\mathbb Z}_3 \times {\mathbb Z}_3 is not the same group as {\mathbb Z}_9\text{.} (See Example 3.28 for a short description of the product of groups.)

35

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

36

Compute the subgroups of the symmetry group of a square.

37

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

38

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

39

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

40

Let G consist of the 2 \times 2 matrices of the form

\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}\text{,} \nonumber

where \theta \in {\mathbb R}\text{.} Prove that G is a subgroup of SL_2({\mathbb R})\text{.}

41

Prove that

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

is a subgroup of {\mathbb R}^{\ast} under the group operation of multiplication.

42

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

H = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a + d = 0 \right\}\text{.} \nonumber

Prove that H is a subgroup of G\text{.}

43

Prove or disprove: SL_2( {\mathbb Z} )\text{,} the set of 2 \times 2 matrices with integer entries and determinant one, is a subgroup of SL_2( {\mathbb R} )\text{.}

44

List the subgroups of the quaternion group, Q_8\text{.}

45

Prove that the intersection of two subgroups of a group G is also a subgroup of G\text{.}

46

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

47

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

48

Let G be a group and g \in G\text{.} Show that

Z(G) = \{ x \in G : gx = xg \text{ for all } g \in G \} \nonumber

is a subgroup of G\text{.} This subgroup is called the center of G\text{.}

49

Let a and b be elements of a group G\text{.} If a^4 b = ba and a^3 = e\text{,} prove that ab = ba\text{.}

50

Give an example of an infinite group in which every nontrivial subgroup is infinite.

51

If xy = x^{-1} y^{-1} for all x and y in G\text{,} prove that G must be abelian.

52

Prove or disprove: Every proper subgroup of a nonabelian group is nonabelian.

53

Let H be a subgroup of G and

C(H) = \{ g \in G : gh = hg \text{ for all } h \in H \}\text{.} \nonumber

Prove C(H) is a subgroup of G\text{.} This subgroup is called the centralizer of H in G\text{.}

54

Let H be a subgroup of G\text{.} If g \in G\text{,} show that gHg^{-1} = \{ghg^{-1} : h\in H\} is also a subgroup of G\text{.}


This page titled 3.5: Exercises is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Thomas W. Judson (Abstract Algebra: Theory and Applications) via source content that was edited to the style and standards of the LibreTexts platform.

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