Consider the group G = ℤ9 ⨉ ℤ27 where the operation is done componentwise, that is (a, b) * (c, d) = (a +9 c, b +27 d), for all (a, b), (c, d) in G. (a) Find the elements of the subgroup <(1, 3)>. (b) Compute [G : <(1, 3)>]. (c) Find the elements of the left coset (1, 6) + <(1, 3)>.

Consider the group G = ℤ9 ⨉ ℤ27 where the operation is done componentwise, that is (a, b) * (c, d) = (a +9 c, b +27 d), for all (a, b), (c, d) in G. (a) Find the elements of the subgroup <(1, 3)>. (b) Compute [G : <(1, 3)>]. (c) Find the elements of the left coset (1, 6) + <(1, 3)>.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Let G = Z9 x Z12 × Z16. (a) Find all subgroups of G of order 144. (b) Let H = ((3,3, 6)), the cyclic…

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

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: (b) Is Z7 Z10 a cyclic group? Explain. (c) Find the order of the element (2, 2) in Z7 Z10.

A: Cyclic groups and order of an element

Q: Consider the groups U(8) and Z4 . (i) Determine the identity element in the group U(8)xZ4 . (ii)…

Q: O In the group G=Z%6 , let H (9) and K = (18) %3D List the elements inG / H , G/K and in H/K ,…

A: For G/H : H=0,9,18,271+H ={1,10,19,28}..........8+H={8,17,26,35} For G/K:…

Q: (3) Write down all proper subgroups in the group Z3 × Z3.

Q: (b) Let H= ((3,3, 6)), the cyclic subgroup of G generated by (3,3,6). Determine |G/H.

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

A: Given: Group of order 4m where m is odd.We need to show that :a) Given group must have a subgroup…

Q: d n=0

Q: Calculate the cyclic subgroup (15) < (Z24, +21)

Q: Let G be a group of order p", where p is a prime and n is a positive integer. Show that G contains a…

A: Given :- Let G be a group of order pn , where p is a prime and n is a positive integer. To…

Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

Q: If G is a finite solvable group, show that there exist subgroups of G {e} = H, C H, C H, C · · CH, =…

A: We need to prove that there exist subgroups of G, e=H0⊂H1⊂H2⊂...⊂Hn=G such that Hi+1Hi has prime…

Q: Consider the groups U(8) and Z4: (1) Determine the identity element in the group U(8) ×Za: (ii)…

Q: Let P₂(x) = {ax² + b | a, b e R and a #0}. If p(x) = ax² +b and g(x) = cx² + d are elements of…

A: In the given problem we have to show the the set p2(x) is a group with respect to the binary…

Q: To which group is Z12/ ker f isomorphic to? (a) Z₁ (b) V4 (c) D4 (d) A₁

Q: G = (R, +), H = {a+bv2: a,b € Z}

A: Given G = (R, +), H = {a+b√2 : a, b∈Z}. We check whether H is a subgroup of G.

Q: the group (a) find all possible orders of elements in G; (b) for each possible order of an element…

A: (ℤ/364ℤ,+) is isomorphic to (ℤ364,+). All possible orders of (ℤ364,+) are all possible divisors of…

Q: Explain why the group (Z4, +) is NOT isomorphic to the group (LI, ∘) Group (LI ∘): The symmetries…

A: If two groups are isomorphic then they should have same number of elements

Q: 10 2) (i) Prove that H = (0, 2, 4, 6, 8) is a subgroup of Z (ii) Prove that H is a normal subgroup…

A: Answer

Q: Combine the regularization method with any other method to solve the Fredholm integral equations of…

Q: Consider the subgroup H = (i) of the the group (C\{0},.). LIT TT TT?

Q: Prove that (Z × Z)/((0,1)) is an infinite cyclic group. Prove that (Z × Z)/((1,1)) is an infinite…

Q: 5. Consider the "clock arithmetic" group (Z15,0) a) Using Lagrange's Theorem, state all possible…

Q: Consider the group GL₂(R). Let A (a) Find all elements of H = (A). (b) What is A|? √2 2 √2 2 NANS √2

Q: Consider the group A₁. Let H = {e, (12)(34), (13)(24), (14)(23)}. (a) Show that H is a subgroup of…

Q: 1. Consider the group U(5). (a) What is |U(5)|? (b) For each a ∈ U(5), find |a|, , and | | 2.…

A: U(n) - Set of all positive integers less than n and relatively prime to n. Then U(n) is a group of…

Q: True or False: The group Dg has a subgroup isomorphic to D3. (a) True (b) False (c)

A: Answer :- A)False

Q: b. Find the center and the commutator subgroup of S2 × Z7.

Question

Consider the group G = ℤ₉ ⨉ ℤ₂₇ where the operation is done componentwise, that is
(a, b) * (c, d) = (a +₉ c, b +₂₇ d), for all (a, b), (c, d) in G.

(a) Find the elements of the subgroup <(1, 3)>.
(b) Compute [G : <(1, 3)>].
(c) Find the elements of the left coset (1, 6) + <(1, 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

Step 2

Step 3

Step 4

Trending now

This is a popular solution!

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

List the elements of the subgroups <1 >, <2>, <3>, <4>, <6> and <0> in Z12
Assignment: Q1: Let G be and acG 8.t a finite group la1=12.If H= (a> find all other generalors of H.
5. (a) Decide whether (Z, -) forms a group where : Z x Z Z is the usual operation of subtraction, i.e. (m, n) m - n. Justify your answer fully. X1 (b) Consider R? = R x R. Elements of R? have the form in which x1, x2 E R. Define the operation : R2 x R2 R? by ) - „). y2 \Y1x2+ y2, Show that (R?, *) does not form a group. Next, find a suitable subset of R? that forms a group with the operation * as given above. Finally, decide with proof whether this group is abelian. Show (R\{-1}, o) forms a group, where for any a, be R\{-1}, (c) a ob = (a+1)(b+1) – 1. Next, find a group element g such that 2og o3 = 5. Is this element unique? Justify your answer.
why is Q/Z(G) is a subgroup of G/Z(G)?
Let G = GL(2, R) and let g = (a) Letz 8 J Write down all the elements of the right coset Kr. 3 (b) Let x = a C d in terms of a, b, c, d. Let K(g) be the cyclic subgroup generated by g. b] be an element of GL(2, R). Write down all the elements of the right coset Ka
Find the order of the element 5(9)in the factor group U(16)/(9).
6. Additional Classifications: Classify the factor group according to the fundamental theorem of finitely generated abelian groups: a. (Z₁ × Z₁2)/((3,6)) b. (Z x Z)/((3,6))
Consider the groups U(8) and (i) Determine the identity element in the group U(8) × Z4. (ii) Determine all the elements of order 4 in the group U(8) × Z4. (ii) Determine the subgroup of U(8) × Z4 generated by the element (7.1).
If ring (R1, +, *) is isomorphic to ring (R2, +, *) with f being the isomorphism,then prove the following statements:a. f(e1) = e2, where e1 is the identity element for the group (R1, +) and e2 isthe identity element of the group (R2, +).b. f(-a) = -f(a), for each element ‘a’ in R1, where –a represent the additiveinverse of element ‘a’ in R1 and –f(a) represents the additive inverse of theelement f(a) in R2.c. If (R1, +, *) is a ring with (multiplicative) identity I1 and (R2, +, *) is a ringwith (multiplicative) identity I2, then f(I1) = I2.
2. Let G= Z₂x Z₁. (a) Find subgroups H and K of G such that H K but G/HG/K. (b) Find subgroups A and B of G such that G/A G/B but A & B.
√4. Let G be the multiplicative group of units Us = {[1], [2], [3], [4]} Zs.
(3) Find the splitting g and UIIC .4 (4) Find the Galois group of the polynomial r* r+1. (5) Find the roots a he polynomia