Answered: 3.40. Let G be an abelian group and n e…

Name: How do I do this? 3.40. Let G be an abelian group and n e N. Let H = {
Uploaded: 2024-03-20T06:46:03.000Z
Channel: Bartleby
Description: How do I do this? 3.40. Let G be an abelian group and n e N. Let H = {a e G : a" = e} and K ={a" : a e G}. Show that H and K are subgroups of G.%3D

Math

Advanced Math

3.40. Let G be an abelian group and n e N. Let H = {a e G : a" = e} and K = {a" : a e G}. Show that H and K are subgroups of G. %3D

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.8: Some Results On Finite Abelian Groups (optional)

Problem 18E

See similar textbooks

Related questions

Q: 50

A: From the given information, it is needed to prove or disprove that H is a subgroup of Z:

Q: Find the right cosets of the subgroup H in G for H = ((1,1)) in Z2 × Z4.

A: Let, the operation is being operated with respect to dot product. Elements of ℤ2=0,1 Elements of…

Q: et H ≤ S4 be the subgroup consisting of all permutations σ that satisfy σ(1) = 1. Find at least 4…

A: This is a good exercise in working with cosets. We first find out the subgroup $H$ and then working…

Q: Let G be a cyclic group of order 135. The number of generators of G is The number of elements of…

Q: 10. Let E = Q(V2, V5). What is the order of the group Gal(E/Q)? What is the order of Gal(Q(V10/Q)?

Q: (2) of order 5 is in H. Let G be a group of order 100 that has a subgroup H of order 25. Prove that…

Q: 3. Let G be a group of order 8 that is not cyclic. Show that at = e for every a e G.

A: Concept:

Q: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.

Q: Let H be the subgroup of S3 generated by the transposition (12). That is, H = ((12)) Prove that H is…

A: We know that S3=1, 12, 13, 23, 123, 132. Giventhat H=12 is a subgroup of S3. H=1, 12We have to show…

Q: 6. Let a be an element of order n in a group and let k be a positive la| = integer. Then ª"|-…

Q: When we say xH = Hx where H is a normal subgroup of G and x is an element of G, what exactly does…

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: 10. Let (G, *) be a group, and let H≤ G. Define N(H) = {x € G: x¹ *H* x = H} [Normalizer of H in G].…

Q: 3. am e H for every a E G. Let H be a normal subgroup of a group G, and let m = (G : H). Show that

Q: If H and K are subgroups of G, |H|= 18 and |K|=30 then a possible value of |HNK| is

A: It s given that H and K are subgroups of G, H=18 and K=30. Since H, K are subgroups, H∩K≤H and…

Q: (e) Find the subgroups of Z24-

A: Given that

Q: Let Ø: Z50 → Z,5 be a group homomorphism with Ø(x) = 4x. Ø-1(4) = %3D O None of the choices O (0,…

A: Here we will find out the required value.

Q: I denotes the set of real numbers and (*) is an operation on R such that a * B = a +B+ aß for ali a,…

Q: The cyclic group of order 12 acts on {1,2,..., 12} with the following cycle structure. (1)…

A: Given that, the group of order 12 In this case of necklace, there is no difference between…

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

Q: Let E = Q(√2, √5). What is the order of the group Gal(E/Q)?What is the order of Gal(Q(√10)/Q)?

Q: Let G be a group (written additively), S a nonempty set and M (S, G) the set of all functions from S…

Q: 6. Let H and K be subgroups of G. Suppose |H| = 35 and |K| = 28. Prove %3D that HNK is abelian.

A: The intersection of two subgroups is a subgroup. Also, recall that the prime order subgroup is…

Q: Q2) If G = Z24 Group a) Is a G=Z24 cyclic? Why b) Find all subgroups of G = Z24 c) Find U,(24)

A: Given that G=ℤ24. a) Then G is generated by the element 1. That is, 1=1,2,3...,22,23,0=ℤ24.…

Q: let H be a normal subgroup of G and let a belong to G . if the element aH has order 3 in the group…

Q: Q7. Suppose that the index of the subgroup H in G is two. If a and b are not in H, then ab ∈ H.…

Q: Let G be a group and let a e G have order pk for some prime p, where k ≥ 1. Prove that if there is x…

A: As per the policy, we are solving first question. Please repost it and specify which question is to…

Q: For each of the following non-Abelian groups G of order 60 find the center Z(G), how many elements…

Q: . Let G be a group, let g e G, and let H - G. Suppose that the element Hg E G/H has order n. Show…

A: We have to prove that given statement:

Q: let H be a normal subgroup of G and let a belong to G. if th element aH has order 3 in the group G/H…

A: H is normal subgroup of G. And a belongs to G. O( aH) = 3 in G/H and O(aH) in G/H divides O(a) in…

Q: Let sulgpoups Haud K have finite indices k aud e, their in a group G. Peove that the index of…

Q: 5. Let H and K be normal subgroups of a group G such that H nK = {1}. Show that hk = kh for all h e…

Q: 4

A: To identify the required cyclic subgroups in the given groups

Q: (4) Let G be a group and H ≤ G. The subgroup H is normal in its normalizer NG(H), this imply that…

Q: Let (Z12, +12) be a group , if we take {0,4,8} for the set H then ({0,4,6}, +12) is evidently a…

A: Let H=0, 4, 6 We know that the operation in ℤ12 is addition. So, the element of left coset is of the…

Q: Suppose that a subgroup H of S5 contains a 5-cycle and a 2-cycle.Show that H = S5.

Q: I'm unsure of how to approach this... Let N be a finite group and let H be a subgroup of N. If |H|…

A: According to the given information, Let N be a finite group and H be a subgroup of N.

Q: Let H = {β ∈ S5 : β(4) = 4}. Prove that H is a subgroup of S5. (Reminder: The group operation of S5…

Q: 8. Is Zg isomorphic to D4? What about Z4 and D4? Can you find a subgroup of D4 isomorphic to Z4?

A: Now we have to answer the above question .

Q: If H and K are subgroups of G, |H|= 16 and |K|=28 then a possible value of |HNK| is

A: It is given that H and K are subgroups of G and H=16, K=28. Since H and K are subgroups of G, H∩K≤H…

Q: 1\Find the inverse for each element in the following mathematical syste A) (Z, *) where defined as…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Suppose that G = (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).

Q: a group and H, K be Subgroups of NG (H) = NGCH) Relate H and K? let G be G Such that %3D

A: Given: Let G be the group and H, K be the subgroups of G such that NG(H)=NG(K)

Q: I Let G be a group and assume that for any a, and b one has (ab)= a?b?. Show ihat G is abelian. jLet…

Q: Which of the following groups are cyclic? Justify. (a) G = U(10) = {k e Z10 : ged(k, 10) = 1} =…

A: We know that 1)Every cyclic group is almost countable 2) Every finite cyclic group is isomorphic…

Q: F. Let a e G where G is a group. What shall you show to prove that a= q?

A: Solution: Given G is a group and a∈G is an element. Here a-1=q

Q: 18. Let peR.o G = Show that G is a group under matrix multiplication.

A: Given: G=a00a|a∈ℝ,a≠0 We need to show that G is a group under matrix multiplication.

Q: Q2.6 Question 1f How many Abelian groups (up to isomorphism) are there of order 36? O 2 O 3 O 4

A: Option D is correct answer

Q: Suppose H and K are subgroups of a group G. If |H| = 12 and |K| = 35, find |H N K|. Generalize. %3D

A: Given that H and K are subgroups of a group G. Also, the order of H is H=12 and the order of K is…

Q: D. Let H be the subgroup of S3 generated by the transposition (12). That is, H = ((12)) Prove that…

Concept explainers

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.

Topic Video

Question

100%

How do I do this?

3.40. Let G be an abelian group and n e N. Let H = {a e G : a" = e} and K =
{a" : a e G}. Show that H and K are subgroups of G.
%3D

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

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Algebraic Operations$

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

6. For each of the following values of , describe all the abelian groups of order , up to isomorphism. b. c. d. e. f.
Exercises 13. For each of the following values of, find all subgroups of the group described in Exercise, addition and state their order. a. b. c. d. e. f.
Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
Show that An has index 2 in Sn, and thereby conclude that An is always a normal subgroup of Sn.
If a is an element of order m in a group G and ak=e, prove that m divides k.
Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
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.
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.
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?
Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40
Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.
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 .