1. Determine whether the given subset of the set of positive rational numbers is a subgroup of the group Q+ under multiplication. H = { 3" | n E Z} = {...,3-3, 3-2,3-1, 3º, 3', 3², 3³, .} 1 1 1 = {..., 33' 32' 3 33 32 31, 3, 3?, 3³, ...} 1 1 1 - = {..., 1,3, 9, 27, ...} 27' 9'3

1. Determine whether the given subset of the set of positive rational numbers is a subgroup of the group Q+ under multiplication. H = { 3" | n E Z} = {...,3-3, 3-2,3-1, 3º, 3', 3², 3³, .} 1 1 1 = {..., 33' 32' 3 33 32 31, 3, 3?, 3³, ...} 1 1 1 - = {..., 1,3, 9, 27, ...} 27' 9'3

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 12E: Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.

See similar textbooks

Related questions

Q: (d) Show that Theorem 1 does not hold for n 1 and n = 2. That is, show that the multiplicative…

Q: The following is a Cayley table for a group G. The order of 3 is: 1 2 3. 4 2 3. 4 5 1 2 3 4 5 2 3 4…

A: In given Cayley Table : * 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1…

Q: The following is a Cayley table for a group G. The order of 4 is: 2 3 5 2 3 4 5 3 4 1 2 4 2 1 3 2 3…

A: According to our company's guidelines I can only answer first question since you have asked multiple…

Q: (d) Show that Theorem 1 does not hold for n = 1 and n = 2. That is, show that the multiplicative…

Q: 3. Define an operation on G = R\{0} x R as follows: (a, b) (c,d) = (ac, bc + d) for all (a, b),…

A: 3. Define an operation * on G=ℝ\{0} ×ℝ as follows: (a,b)*(c,d)=(ac, bc+d) for all (a,b), (c,d) ∈G…

Q: (a) Following the same approach as in the proof of Proposition 2, show that for an integer m> 1, the…

A: Since part b,part are independent of part a,as per the guidelines I am answering the part a only.…

Q: B. Write at least five elements of each of the following cyclic groups. 1. 25Z under multiplication…

A: 25Z under multiplication: {-50, -25, 25, 50, 75}

Q: (a) Following the same approach as in the proof of Proposition 2, show that for an integer m > 1,…

Q: 13. Which of the following is a subgroup of (R+, *) where a*b = (ab)/2? (Q, *) A. B. (Z, *) C. (Q+,…

A: Given that ℝ+, * is a group where…

Q: We will now consider three irreducible representations for the permutation group P(3): E A B I : (1)…

Q: 12. For which integers n such that 3 ≤ n ≤ 11 is there only one group of order n (up to isomorphism)…

A: Given: 12. n is an integer such that 3≤n≤11. To find: For which value of n there is only one group…

Q: 12. Prove that the following groups are not cyclic: (a) Z2 x Z2 (b) Z2 x Z (c) Z x Z.

A: To show the following groups are not cyclic. If a group is cyclic the there exit an element in that…

Q: Let G be a group and a be an element of this group such that a^6=e. The possible orders of a are: *…

A: First option is correct.

Q: a) List all the subgroups of Z, e Zz. b) Is the groups Z, ® Zz and Z, isomorphic? (why?)

A: We use the fact that for distinct prime p and q Zp x Zq is isomorphic to Zpq.

Q: Let G be the set of nonnegative integers. Is a group, |a - b| for all a, b, EG? where a * b =

A: We will check whether it is group or not.

Q: The following is a Cayley table for a group G. The order of4 is: 2 1 3 3 4 1 4 1 4 1 5. 2 512 m 45…

A: By observing the Cayley table :

Q: Let G be a group and a be an element of this group such that a^6=e. The possible orders of a are: *…

Q: For each of the following groups G and subgroups H, how many distinct left cosets of H in G are…

A: The given group is G and H≤ G. To find: How many distinct left cosets of H in G.

Q: The following is a Cayley table for a group G. The order of 3 is: 2 4 2 4 2 3 4 1 3 4 5 2 4 ere to…

A: Given Cayley table

Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

A: To identify the given group is cyclic or not.

Q: 15. Suppose that N and M are two normal subgroups of a group G and that NO M = {e}. Show that for…

Q: Show that the groups (Z/4, +4) and (Z/5 – {[0]}, x5) are isomorphic.

Q: 50. How many proper subgroups are there in a cyclic group of order 12? A 4 в з с 2

A: see 2nd step

Q: 1. Prove or disprove the following. (a) The set of all subsets of R form a group under the operation…

Q: 3. List all elements of the cyclic subgroup of Z12 generated by 5

A: Solving

Q: ?Which of the following statement is true The largest order among the orders of all the cyclic…

A: We go through each options one by one to choose the correct.

Q: 13. Find groups that contain elements a and b such that |al = \b| = 2 and a. Jab| = 3, b. Jab| = 4,…

Q: Let G be a group and a be an element of this group such that a^63e. The possible orders of a are: *…

Q: One of the following is a p-sylow subgroup of the group Z, × Zg H={(0,0),(0,4)} O…

Q: 1. There is no simple group of order 200.

A: Solution:-As per guidelines I submit first question only

Q: 15.Consider the group (A, *) where *is defined by the table below A B C D ABA D C BA B C D C D C A B…

Q: 2. Suppose M and D are isomorphic groups. Prove that Aut(M) is isomorphic to Aut(D).

Q: True or False: No group of order 21 is simple.

Q: Can a group of order 55 have exactly 20 elements of order 11? Givea reason for your answer

A: Any element of order 11 made a cyclic subgroup with 11 elements. These are non-identity elements of…

Q: Which of the following is the only trivial sub-group of a group G = {e, a, b, c}? {e}, {e, a, b, c},…

Q: ng to a group. If |a| = 12, |6| = 22, and (a) N (b) # {e}, prove that a® = b'1.

Q: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.

Q: 6. Prove the following groups are not cyclic : (a) Z x Z (b) Z6 × Z (c) (Q+, ·) (Here, Q+ = {q € Q…

Q: Let G be a group and a be an element of this group such that a^6=e. The possible orders of a are: O…

Q: The following is a Cayley table for a group G. 2* 5*4 = 1 2 3 5 2 3 4 3 4 2 3 5 1 4 2 3 4 1 2 4 1.…

A: Cayley table for a group G is given as, The objective is to find 2*5*4 Since, G is a group. Hence,…

Q: any two elements x and y in a group G, there is a unique element z in G such that y = xz.

Q: dicfin et Prove that a group G has exactly 3 6. - subgroups iff G is a ylic grop ef ender på pis…

Q: Which of the following is a group? The set of even integers under addition. The set {1,3,4} under…

A: A set along with a binary operation (+) is a group only if : 1: operation is closed. (for all x and…

Q: Which of the following is a group? O The set of even integers under addition. O The set {1,3,4)…

Q: If G is a finite group with |G|<180 andG has subgroups of orders 10, 18 and 30 then the order of G…

A: Use the fact that order of subgroup divides order of group

Q: Let G be a group, and assume that a and b are two elements of order 2 in G. If ab = ba, then what…

Q: If G is a finite group with |G|<120 and G has subgroups of orders 10, 15 and 20 then the order of G…

Q: Which of the following groups are isomorphic? i) Z/75Z ii) Z/3Z ® Z/25Z iii) Z/15Z @ Z/5Z iv) Z/3Z @…

A: Groups (i) ℤ/75ℤ and (ii) ℤ/3ℤ⊕ℤ/25ℤ are isomorphic. Groups (iii) ℤ/15ℤ⊕ℤ/5ℤ and (iv) ℤ/3ℤ⊕ℤ/5ℤ⊕ℤ/5ℤ…

Q: according to exercise 27 of section 3.1 the nonzero elements of zn form a group g with respect to…

A: We know that a finite group is said to be cyclic if there exists an element of group such that order…

Question

1. Determine whether the given subset of the set of positive rational
numbers is a subgroup of the group Q+ under multiplication.
H = { 3" | n E Z}
= {...,3-3, 3-2,3-1, 3º, 3', 3², 3³, .}
1 1 1
= {...,
33' 32' 3
33 32 31, 3, 3?, 3³, ...}
1 1 1
-
= {...,
1,3, 9, 27, ...}
27' 9'3

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Groups$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.
Let be a subgroup of a group with . Prove that if and only if .
15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .
Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative group . Sec. 16. Prove that each of the following sets is a subgroup of , the general linear group of order over .
In Exercises 3 and 4, let be the octic group in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let be the subgroup of the octic group . Find the distinct left cosets of in , write out their elements, partition into left cosets of , and give . Find the distinct right cosets of in , write out their elements, and partition into right cosets of . Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group of rigid motions of a square The elements of the group are as follows: 1. the identity mapping 2. the counterclockwise rotation through about the center 3. the counterclockwise rotation through about the center 4. the counterclockwise rotation through about the center 5. the reflection about the horizontal line 6. the reflection about the diagonal 7. the reflection about the vertical line 8. the reflection about the diagonal . The dihedral group of rigid motions of the square is also known as the octic group. The multiplication table for is requested in Exercise 20 of this section.
In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its multiplication table requested in Exercise 20 of the same section. Let H be the subgroup e, of the octic group D4. Find the distinct left cosets of H in D4, write out their elements, partition D4 into left cosets of H, and give [D4:H]. Find the distinct right cosets of H in D4, write out their elements, and partition D4 into right cosets of H. Example 12 Using the notational convention described in the preceding paragraph, we shall write out the dihedral group D4 of rigid motions of a square The elements of the group D4 are as follows: 1. the identity mapping e=(1) 2. the counterclockwise rotation =(1,2,3,4) through 900 about the center O 3. the counterclockwise rotation 2=(1,3)(2,4) through 1800 about the center O 4. the counterclockwise rotation 3=(1,4,3,2) through 2700 about the center O 5. the reflection =(1,4)(2,3) about the horizontal line h 6. the reflection =(2,4) about the diagonal d1 7. the reflection =(1,2)(3,4) about the vertical line v 8. the reflection =(1,3) about the diagonal d2. The dihedral group D4=e,,2,3,,,, of rigid motions of the square is also known as the octic group. The multiplication table for D4 is requested in Exercise 20 of this section.
5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:
Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)
Let be a subgroup of a group with . Prove that if and only if
For each of the following subgroups H of the addition groups Z18, find the distinct left cosets of H in Z18, partition Z18 into left cosets of H, and state the index [ Z18:H ] of H in Z18. H= [ 8 ] .
Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.