Consider the groups U(8) and Z4. (i) Determine the identity element in the group U(8)xZ4 . (ii) Determine all the elements of order 4 in the group U(8)xZ4 . (iii) Determine the subgroup of U(8)xZ4 generated by the element(7,1) .

Consider the groups U(8) and Z4. (i) Determine the identity element in the group U(8)xZ4 . (ii) Determine all the elements of order 4 in the group U(8)xZ4 . (iii) Determine the subgroup of U(8)xZ4 generated by the element(7,1) .

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 27E: 27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. ...

See similar textbooks

Related questions

Q: In the group Z24, let H =(4) and N= (6). (a) State the Second Isomorphism Theorem. (b) List the…

A: As per our guidelines only first three subquestions are solved. To get solution of remaining…

Q: Describe all the elements in the cyclic subgroup of generated by the 2×2 matrix [1 1 0 1]

A: Describe the elements in the subgroup of the General linear group GLn, ℝ generated by the 2×2…

Q: (1) Z/12Z (2) (Zx 끄)/(6Zx 14Z) (3) (Z4 × Z4)/((2, 3)) (4) (Z4 x Z10)/((2, 4))

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: Show that the center of a group of order 60 cannot have order 4.

Q: Give an example, with justification, of an abelian group of rank 7 and with torsion group being…

A: consider the equation

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: does the set of polynomials with real coefficients of degree 5 specify a group under the addition of…

Q: Find Aut(Z15) . Use the Fundamental Theorem of Abelian Groups to express this group as an external…

A: We know the following theorem. Theorem: Aut(Zn) is isomorphic to the group of multiplicative units…

Q: There exists a quartic polynomial (i.e. a polynomial of degree 4) over Q whose Galois group is…

Q: List the six elements of GL(2, Z2). Show that this group is non-Abelian by finding two elements that…

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

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: QUESTION 6 Consider the groups U(8) and Z4 (1) Determine the identity element in the group U(8) x Z.…

Q: (1) Z/12Z

Q: QUESTION 6 Consider the groups U(8) and Za (i) Determine the identity element in the group U(8) × Z.…

Q: Show that every abelian group of order 255 (3)(5)(17) is isomorphic to Z55 and hence cyclic. [Ilint:…

A: We have to solve given problem:

Q: Find the order of each of the elements of the group ((Z/8Z*, * ). Is this group cyclic? Do the same…

A: To investigate the orders of the elements in the given groups

Q: ake the minimum three groups along with subgroups (Mathematically) and verify the langrage theorem

A: Lagrangian theorem says that, If H is a subgroup of a group G then H\G. Now take, Z3={0,1,2} The…

Q: 2. Consider the group (R, +) and the subgroup H = (2nx |n€ Z}. %3! [cos(0) - sin(@) sin(0) cos(0)…

A: Kindly keep a note in step 1 stating that as per our guidelines we are supposed to answer only one…

Q: 2. Prove that a free group of rank > 1 has trivial center.

A: Given:Prove that a free group of rank>1 has trivial center

Q: QUESTION 6 Consider the groups U(8) and Z () Determine the identity element in the group U(8) x ZA…

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: QUESTION 6 Consider the groups U(8) and Z4: (1) Determine the identity element in the group U(8) ×…

Q: (2) (Z x Z)/(6Z × 14Z)

Q: Find Aut(Z20). Use the fundamental theorem of Abelian groups to express this group as an external…

A: Find Aut(Z20) by using the fundamental theorem of Abelian groups

Q: ii) Find the structure of its Galois group, G.

A: To Determine :- The structure of its Galois group, G.

Q: (7) Define GL2 (R) to be the group of invertible 2 x 2 matric manifold, cc this group has the…

A: Define GL2R to be the group of invertible 2×2 matrices. To prove that this group has the structure…

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: In the group Z24, let H =(4) and N=(6). (a) State the Second Isomorphism Theorem. (b) List the…

A: Given that, Group is Z24, let H=4 and N=6. Let (G,∗) be a group, H a subgroup of G, and g∈G. The…

Q: The group (Z6,6) contains only 4 subgroups

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

Q: 4. Construct a 2-dimensional CW-complex whose fundamental group is Z x Z/2 (and prove it).

A: Please check the detailed sol" in next step

Q: The groups Z/6Z, S3, GL(2,2), and De (the symmetries of an equilateral triangle) are all groups of…

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

A: We will be using definition of subgroup and verify that H indeed satisfy the definition.

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

Q: Find the center and the commutator subgroup of S3 x Z12-

A: Solution

Q: The group (Z6,+6) contains only 4 subgroups

Q: 3. Show that Q has no subgroup isomorphic to Z2 × Z2.

A: The objective is to show that ℚ has no subgroup isomorphic to ℤ2×ℤ2

Q: If R2 is the plane considered as an (additive) abelian group, show that any line L through the L in…

Q: t subgroups and quotient groups of a solvable group are solvable.

Q: Write out the Cayley table for the group {1, -1, i, -i} under multiplication where i = sqrt(-1). (In…

Q: Find the order of each element of the group Z/12Z under addition

Q: under the addition operation. a) Identify all subgroups of order 9. Explain how these groups are…

A: We shall answer first three subparts only you have asked more than 3. For others kindly post again…

Q: How do you interprete the main theorem of Galois Thoery in terms of subgroup and subfield diagrams?

Q: 1- Let (C,) be a group of non-zero complex number and let H = {x + iy}| x² + y² = 2}. Then (H,) is a…

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

Q: Consider the groups U(8) and Z4 (i) Determine the identity element in the group U(8) × Z4" (ii)…

Q: Use the fundamental theorem of Abelian groups to express Z20 as an external direct product of cyclic…

Q: QUESTION 6 Consider the groups U(8) and ZA (1) Determine the identity element in the group U(8) × Z4…

Question

Consider the groups U(8) and Z₄.

(i) Determine the identity element in the group U(8)xZ₄ .

(ii) Determine all the elements of order 4 in the group U(8)xZ₄ .

(iii) Determine the subgroup of U(8)xZ₄ generated by the element(7,1) .

Expert Solution

This question has been solved!

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

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

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, algebra and related others by exploring similar questions and additional content below.

Similar questions

27. a. Show that a cyclic group of order has a cyclic group of order as a homomorphic image. b. Show that a cyclic group of order has a cyclic group of order as a homomorphic image.
The Galois group of x4 +x2 +1 is the same as that of x6 -1 and is of order 2.
Explain why a group of order 4m where m is odd must have a subgroupisomorphic to Z4 or Z2 ⨁ Z2 but cannot have both a subgroupisomorphic to Z4 and a subgroup isomorphic to Z2 ⨁ Z2. Show thatS4 has a subgroup isomorphic to Z4 and a subgroup isomorphic toZ2 ⨁ Z2.
Prove that a simple group cannot have a subgroup of index 4.29. Prove that there is no simple group of order p2q, where p and
An element x of a group satisfies x2 = e precisely when x = x-1. Use this observation to show how that a group of even order must contain an odd numder of elements of order 2.
Prove that a simple group of order 60 has a subgroup of order 6 anda subgroup of order 10.
List the six elements of GL(2, Z2). Show that this group is non-Abelian by finding two elements that do not commute
Determine the order of (Z ⨁ Z)/<(2, 2)>. Is the group cyclic?
Prove that order of the subgroup divides the order of the group by the index of the subgroup (Lagrange theorem).Specify all steps.
Q2 : Find the left regular representation of the group Z5 andexpress the group element in the regular representation in disjointcycles.For a Abelian group, its left regular representation= right regular representation
does the set of polynomials with real coefficients of degree 5 specify a group under the addition of polynomials?
Find Aut(Z20). Use the fundamental theorem of Abelian groups to express this group as an external direct product of cyclic groups of prime power order.