QUESTION 6 Consider the groups U(8) and ZA (1) Determine the identity element in the group U(8) × Z4 (ii) Determine all the elements of order 4 in the group U(8) X Z.. (iii) Determine the subgroup of U(8) × Z4 generated by the element (7,1). Attach File

QUESTION 6 Consider the groups U(8) and ZA (1) Determine the identity element in the group U(8) × Z4 (ii) Determine all the elements of order 4 in the group U(8) X Z.. (iii) Determine the subgroup of U(8) × Z4 generated by the element (7,1). Attach File

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 18E: Prove that SL(2,R)={ [ abcd ]|adbc=1 } is a subgroup of GL(2,R), the general linear group of order 2...

See similar textbooks

Related questions

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

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

Q: 8. Find a non-trivial normal subgroup of the octic group. Demonstrate that this subgroup is normal.

A: According to the given information, it is required to find a non-trivial normal subgroup of the…

Q: Theory 4. Prove that every group of order (5)(7)(47) is abelian and cyclic. DuOuo thot no groun of…

A: We have to be solve given problem:

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

Q: Theorem(6.16): Every cyclic group or order n is isomorphic to (Zn,+n) and every infinite cycle group…

A: See the attachment

Q: Question 1. Let G is a finite group and ø : G → G' be a surjective homomorphism (that is, ø is a…

A: Multiple : Let G be finite group. Consider the map , ϕ : G → G' Here , ϕ is a homomorphism and ϕ…

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

Q: prove that a group of order 45 is abelian.

Q: 1- The index 2- infinite group 3- permute elements B prove that (cent G,*) is normal subgroup of the…

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: 3. Find all Abelian groups (up to isomorphism) of order 100. From these isomorphism classes…

Q: Give an example of a cyclic group of smallest order that containsboth a subgroup isomorphic to Z12…

Q: Classify groups of order 2p as best as you can. Give a proof for your assertions and, when…

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

Q: Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to Z12…

A: Let D2n = {e, y, y2 …… yn+1}{s, sr,…. Srn} .y = rotation .s = reflection Then {e, y, y2, …… yn+1} is…

Q: KE Syl-(G). Prove that (a). HG and KG. (U). G has a cyclic subgroup of order 77. Syl(G),

Q: Let Z denote the group of integers under addition. Is every subgroupof Z cyclic? Why? Describe all…

Q: Theorem 2. Let G, and G, be groups, then @ Gx G,= G, × G, (6) If H = {(a, e,)| a e G} and H, = {(e,…

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

Q: create the addition and multiplication table in Z5 a) show that (Z5, +) is an abelian group b) show…

A: We know in Z5 addition is defined as, a+b = (a+b) (mod 5), for all a,b in Z5 In Z5\{0}…

Q: se A cyclic group has a unique generator. f G and G' are both groups then G' nG is a group. A cyclic…

A: 1.False Thus a cyclic group may have more than one generator. However, not all elements of G need be…

Q: A group G has order 4n. where n is odd. Show that G has no subgroup of order 8.

Q: 5. Prove that no group of order 96 is simple. 6. Prove that no group of order 160 is simple. 7. Show…

Q: Theorem 5 (Converse of Lagrange's theorem for abelian groups): Let G be an abelian group of order n…

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

Q: QUESTION 9 Find up to isomorphism all Abelian groups of order 36.

A: We have to find the all abelian groups of order 36 up to isomorphism.

Q: Prove that a simple group cannot have a subgroup of index 4.

A: We will prove this by method of contradiction. Let's assume that there exists a simple group G that…

Q: Prove that a cyclic group with even number of elements contains ex- actly one element of order 2.

A: The solution is given as

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: 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: iv Sketch the Caley Graph of the additive Group of direct product Z3× Z4 with respect to the…

A: Consider the conditions given in the question. Clearly from the hint a torus is involved in the…

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: {a3 }, {a2 }, {a5 }, {a4 } Which among is not a subgroup of a cyclic group of order 12?

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

Q: Theorem group tren G'45. Eremple Treorem is i0 9 N a group then Example abeliar.

Q: Prove that group A4 has no subgroups of order

A: Topic- sets

Q: Explain why a non-Abelian group of order 8 cannot be the internaldirect product of proper subgroups

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

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

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

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

Q: a) Clearly state all the subgroups of the additive group Explain why/how you have found all the…

A: We have to write subgroup of z_9.

Q: 15*. Find an explicit epimorphism from Z24 onto a group of order 6. (In your work, identify the…

A: To construct a homomorphism from Z24 , which is onto a group of order 6.

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: 1- group Fen 9 Example Theorem is group then

Question

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

1.Prove part of Theorem . Theorem 3.4: Properties of Group Elements Let be a group with respect to a binary operation that is written as multiplication. The identity element in is unique. For each, the inverse in is unique. For each . Reverse order law: For any and in ,. Cancellation laws: If and are in , then either of the equations or implies that .
Prove part e of Theorem 3.4. Theorem 3.4: Properties of Group Elements Let G be a group with respect to a binary operation that is written as multiplication. The identity element e in G is unique. For each xG, the inverse x1 in G is unique. For each xG,(x1)1=x. Reverse order law: For any x and y in G, (xy)1=y1x1. Cancellation laws: If a,x, and y are in G, then either of the equations ax=ay or xa=ya implies that x=y.
Exercises In Section 3.3, the centralizer of an element a in the group G was shown to be the subgroup given by Ca=xGax=xa. Use the multiplication table constructed in Exercise 20 to find the centralizer Ca for each element a of the octic group D4. Construct a multiplication table for the octic group D4 described in Example 12 of this section.
31. a. Prove Theorem : The center of a group is an abelian subgroup of. b. Prove Theorem : Let be an element of a group .the centralizer of in is subgroup of.
Exercises 1. List all cyclic subgroups of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .
Consider the additive group of real numbers. Prove or disprove that each of the following mappings : is an automorphism. Equality and addition are defined on in Exercise 52 of section 3.1. a. (x,y)=(y,x) b. (x,y)=(x,y) Sec. 3.1,52 Let G1 and G2 be groups with respect to addition. Define equality and addition in the Cartesian product by G1G2 (a,b)=(a,b) if and only if a=a and b=a (a,b)+(c,d)=(ac,bd) Where indicates the addition in G1 and indicates the addition in G2. Prove that G1G2 is a group with respect to addition. Prove that G1G2 is abelian if both G1 and G2 are abelian. For notational simplicity, write (a,b)+(c,d)=(a+c,b+d) As long as it is understood that the additions in G1 and G2 may not be the same binary operations.
3. Consider the group under addition. List all the elements of the subgroup, and state its 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 .
Lagranges Theorem states that the order of a subgroup of a finite group must divide the order of the group. Prove or disprove its converse: if k divides the order of a finite group G, then there must exist a subgroup of G having order k.
Exercises 3. Find the order of each element of the group in Example of section. Example 3. We shall take and obtain an explicit example of . In order to define an element of , we need to specify , , and . There are three possible choices for . Since is to be bijective, there are two choices for after has been designated, and then only one choice for . Hence there are different mappings in .
Let H be a subgroup of a group G. Prove that gHg1 is a subgroup of G for any gG.We say that gHg1 is a conjugate of H and that H and gHg1 are conjugate subgroups. Prove that H is abelian, then gHg1 is abelian. Prove that if H is cyclic, then gHg1 is cyclic. Prove that H and gHg1 are isomorphic.
In Example 3, the group S(A) is nonabelian where A={ 1,2,3 }. Exhibit a set A such that S(A) is abelian. Example 3. We shall take A={ 1,2,3 } and obtain an explicit example of S(A). In order to define an element f of S(A), we need to specify f(1), f(2), and f(3). There are three possible choices for f(1). Since f is to be bijective, there are two choices for f(2) after f(1) has been designated, and then only once choice for f(3). Hence there are 3!=321 different mappings f in S(A).