The group U(15) is an internal direct product of the cyclic subgroups generated by 7 andby 11,U(15) = (7) × (11). True or false?O TrueO False

We have to check that U(15)= <7>×<11> Or not. Concept: If n =p1.p2 Where p1 and p2…

Answered: The group U(15) is an internal direct…

The group U(15) is an internal direct product of the cyclic subgroups generated by 7 by 11, U(15) = (7) × (11). True or false? O True False

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: Let G = Z8 x Z6, and consider the subgroups H = {(0, 0), (4, 0), (0, 3), (4,3)} and K = ((2, 2)).…

Q: C. Find the number of elements in the indicated cyclic group. 1) The cyclic subgroup of Z30…

A: Given: The cyclic subgroup of 230 generated by 25. To find the number of elements that is indicated…

Q: Explicitly construct the Galois group for r- 4r² + 2 over Q. To which group is this isomorphic?

Q: Explain why a group of order 4m where m is odd must have a subgroupisomorphic to Z4 or Z2 ⊕ Z2 but…

Q: 2- Let (C\{0},.) be the group of non-zero -complex number and let H = { 1,-1, i,-i} prove that (H,.)…

A: To Determine: prove that H,. is a subgroup of a group of non zero complex number under…

Q: The group generated by the cycle (1,2) is a normal subgroup of the symmetric group S3. True or…

A: Given, the symmetric group S3={I, (12),(23),(13),(123),(132)}. The group generated by the cycle (12)…

Q: 11. Find the cyclic subgroup of D4 generated by µp². What is the order of this subgroup?

Q: The group (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine…

A: Consider the group elements, Here the order of K is 6. Consider the order of group, The order of G…

Q: Applying what we discussed in cyclic groups, draw the subgroup lattice diagram for Z36 and U(12).

Q: The symmetry group of a nonsquare rectangle is an Abelian groupof order 4. Is it isomorphic to Z4 or…

Q: Consider the square X = [-1,1]2 = {(x, y)|x > -1, y < 1} and 0 = (0,0). Show that the fundamental…

A: image is attached

Q: [Zp-(0),.] Where p is prime is an abelian group

A: We have to show that [Zp-(0),.] Where p is prime is an abelian group

Q: The group GLQ,R) abelian group is an

Q: 1. Show that H={[0], [2], [4]} is a subgroup of a group (Z6+6). Obtain all the distinct left cosets…

A: Given that H=0,2,4 and let G=ℤ6,+6.

Q: Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

A: Given question: Is S3 x S3 group (the direct product of symmetric group S3) nilpotent?

Q: find Aut(Z30). Use the FUndamental Theorem of Abelian Groups to express this group as an external…

Q: 6. Embed the group Qs into the SU(2).

A: Given: Q0=e,i,j,k e-2=e, i2=j2=k2=ijk=e, Where, e is the identity element and e commutes with the…

Q: 64. Express Ug(72)and U4(300)as an external direct product of cyclic groups of the form Zp

A: see my attachments

Q: 1- Prove that if (Q - {0},) is a group, and H = 1+2n 1+2m 9 n, m e Z} is a subset of Q-{0}, then…

A: We need to prove that H=1+2n1+2m∋n,m∈Z is a subgroup of Q-0 A subset W of a group V is said to be a…

Q: 1+2n Prove that if (Q-(0},) is a group, and H = a n, m e Z} 1+2m is a subset of Q-{0}, then prove…

Q: (1) Z/12Z

Q: 1. Show that G is closed under x. 2. Show that (G. x) in a cyclic grouP generated by t.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Use the Cayley table of the dihedral group D3 to determine the left AND right cosets of H={R0,F}.…

Q: Consider the group G={x € R such that x#0} under the binary operation *: Th identity element of G is…

A: Solution: Since for any x,y∈G, the operation * is defined as x*y=-2xy The identity element is e=-12…

Q: Determine the order of (Z ⨁ Z)/<(2, 2)>. Is the group cyclic?

A: Given, the group We have to find the order of the group and also check, This is a…

Q: Let G be an Abelian group and H 5 {x ∊ G | |x| is 1 or even}. Givean example to show that H need not…

Q: Q3: Describe the quotient group of a- (²/z, ·+) b- (2/z,+)

Q: Prove that group A4 has no subgroups of order

A: Topic- sets

Q: Find all the producers and subgroups of the (Z10, +) group.

A: NOTE: A group has subgroups but not producers. Given group is ℤ10 , ⊕10 because binary operation in…

Q: Q2 : Find the left regular representation of the group Z5 and express the group element in the…

Q: Prove that the alternating group is a group with respect to the composition of functions?

A: Sn is the set of all permutations of elements from 1,2,.....,n which is known as the symmetric group…

Q: (a) Compute the list of subgroups of the group Z/45Z and draw the lattice of subgroups. (prove that…

A: In the given question we have to write all the subgroup of the group ℤ45ℤ and also draw the the…

Q: Find the order of the element (2, 3) in the direct product group Z4 × 28. Compute the exponent and…

Q: This is abstract algebra: Prove that if "a" is the only elemnt of order 2 in a group, then "a"…

Q: Show that Z12 is not isomorphic to Z2 ⊕ Z6. ℤn denotes the abelian cyclic group of order n. Justify…

A: To show : ℤ12 is not isomorphic to ℤ2⊕ℤ6 Pre-requisite : P1. A group G is said to be cyclic if there…

Q: Find a non-trivial, proper normal subgroup of the dihedral group Dn-

Q: Let G be a group of order 25. Prove G is cyclic or g^5=e for all g in G. Generalize to any group of…

A: The Result to be proved is: If G is a group of order p2, where p is a prime, then either G is cyclic…

Q: 5. Prove that the cyclic group Z/15Z is isomorphic to the product group Z/3Z × Z/5Z.

A: Definitions: Isomorphism: A mapping between two sets is called an isomorphism if it is one-to-one,…

Q: Let Hand K be subgroups of an Abelian group. If |H| that HN Kis cyclic. Does your proof generalize…

A: This question is related to group theory. Solution is given as

Q: 2- Let (C,) be the group of non-zero -complex number and let H = {1, –1, i, -1}. Show that (H,;) is…

Q: 20. Consider the group U9 of all units in Z9. Given that U9 is a cyclic group under multiplication,…

Q: In the group (Z, +), find (-1), the cyclic subgroup generated by -1. Let G be an abelian group, and…

Q: (d) Prove that 1+ N is a group with respect to multiplication. (e) Verify that 1+N(Z27) is a cyclic…

A: d) To prove that 1+N is a group. Let a=1+n, b=1+n'∈1+N where n,n'∈N=Nℤn. Then,…

Q: Every element of a cyclic group generates the group. True or False then why

A: False Every element of cyclic group do not generate the group.

Q: 7. You have previously proved that the intersection of two subgroups of a group G is always a sub-…

Q: Find the Galois group of the polynomial r-1.

Q: 2. (a) Let be a subgroup of the center of G. Show that if G/N is a cyclic group, then G must be…

Q: The group (Z, t6) contains only 4 subgroups

Question

The group U(15) is an internal direct product of the cyclic subgroups generated by 7 and
by 11,
U(15) = (7) × (11). True or false?
O True
O False

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

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

9. Suppose that and are subgroups of the abelian group such that . Prove that .
Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.
5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:
11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup of for prove that is a direct sum.
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.
2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .
Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
Prove that each of the following subsets H of GL(2,C) is subgroup of the group GL(2,C), the general linear group of order 2 over C a. H={ [ 1001 ],[ 1001 ],[ 1001 ],[ 1001 ] } b. H={ [ 1001 ],[ i00i ],[ i00i ],[ 1001 ] }
Consider the group U9 of all units in 9. Given that U9 is a cyclic group under multiplication, find all subgroups of U9.
Find two groups of order 6 that are not isomorphic.
Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.
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 .