Abstract algebra Suppose that f: G→ G such that f(x) = axa. Then f is a group homomorphism ifand only ifO a = eO a^4 = ea^2 = eO a^3 = eIf H and K are subgroups of G. IH|= 20 and IKl-32 then a possible value of IHNK| is2.8.

Answered: Image /qna-images/answer/595bfac1-9a4f-4535-a10c-84eaced0c2b3.jpg

Answered: Suppose that f: G G such that f(x) =…

Suppose that f: G G such that f(x) = axa*. Then f is a group homomorphism if %3D and only if a = e O a^4 = e O a^2 = e a^3 = e

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.7: Direct Sums (optional)

Problem 11E: 11. Assume that are subgroups of the abelian group such that the sum is direct. If is a subgroup...

See similar textbooks

Related questions

Q: Let G :- [0, 1) be the set of real numbers x with 0<x< 1. Define an operation + on G by X* y:= {x+y…

A: Here we check associativity property.

Q: Dn Prove that is isomorphic to a subgroup of Sn

Q: Show that SL(n, R) is a normal subgroup of GL(n, R). Further, by apply- ing Fundamental Theorem of…

A: Suppose, ϕ:GLn,R→R\0 such that ϕA=A for all A∈GLn,R Now, sinceA∈GLn,R if and only if A≠0 Now, we see…

Q: TRUE or FALSE: Let G be a group. Let æ, y, z E G. If ryz = e then yzx = e.

A: The solution to the given question is explained below.

Q: 8. Show that (Z,,x) is a monoid. Is (Z,,x4) an abelian group? Justify your answer

Q: Prove that GL(2,R) is not an abelian group

A: Solution is given below:

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

Q: 9. Describe the group of the polynomial (x* – 1) e Q[x] over Q.

Q: Let G be a group containing elements a and b. Express (ab)^2 without parentheses. Do not assume that…

A: Given that G is a group and let a,b belongs to G. The given expression is

Q: 33. If a, bɛR with a # 0, let T:R→ R be the function given by T(x) = ax + b. Prove that the set G =…

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: Suppose that f (x) is a fifth-degree polynomial that is irreducible overZ2. Prove that x is a…

Q: The center of agrouf G is denoted by ZEGJ, one of the foltowing statemnts is fatse! O ZEfn) =fe} for…

A: To Determine :- The center of group is denoted as Z G. (a) Z Sn = e for n > 2.

Q: 7. If x and g are elements of group G, prove that x=g 'xg. Warning: You may not assume that G is…

Q: nilpotent

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

Q: Define the mapping a: R² →R by 7((x,y))=x. (Note that IR is a group under addition with identity 0).…

Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

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

Q: implies y = z. Prove that G is Abelian. Prove that a group G is Abelian if and only if (gh)-1 = g¬…

Q: G group is called meta- Abelian if it has an Abelian subgroup N which is normal in G G and is…

A: (Solving the first question as per our guidelines) Question: Show that S3 is meta-Abelian.

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

Q: %3D Let x belong to a group. If x² +e while x° = e, prove that about the order of r?

A: Given that x2≠e and x6=e To prove that x4≠e and x5≠e Suppose that x4=e also x6=e therefore…

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

Q: Sz,0) be a permutation group. Then all elements in One to one, onto function. Onto function.

Q: (1) Z/12Z

Q: True or false? The group S3 under function composition ◦ is not a cyclic group

Q: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

Q: State the first isomorphism theorem for groups and use it to show that the groups/mz and Zm are…

Q: Suppose that f: G → G such that f(x) and only if = axa. Then f is a group homomorphism if a = e a^4…

Q: (G, .> be a group such that a.a = e for all a E G. Show that G is an abelian grou 2. Let

A: We have to solve given problem:

Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y, (x*y) = (y*x) now associative…

Q: (2) (Z x Z)/(6Z × 14Z)

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: f:H1 x H2 x ... x Hn ---> H1 + H2 + ... + Hn via h1h2...hn ---> (h1, h2,...,hn

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

Q: If G is a non-abelian group of order 8 with Z(G) {e}, prove that |Z(G)| = 2.

Q: 4) If center of G equal G (ie cent(G) = G) then G is not necessary abelian group. true O false

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

Q: Suppose that f:G → G such that f(x) = axa. Then f is a group homomorphism if and only if a^4 = e %3D…

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: Decide if the abelian group Z/2 × Z/2 is cyclic or not. Prove your answer

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

Q: Let GL(2,R) be the general linear group of 2 by 2 matrices. 1 x is a) Show that o: R GL (2,R) where…

A: We first prove this is a homomorphism.

Q: Let x belong to a group. If x2e while x : x + e and x + e. What can we say about the order of x? =…

Q: Let (G,*) and (H,*) be finite abelian groups. If G x G = H x H then G=H. Show that they are…

A: Given that, G×G=H×H⇒G=H Since G,* and H,* are both finite abelian groups we get,…

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: Suppose that fG G such that f(x) = axa. Then fis a group homomorphism if and only if O a^3 = e a^2 e…

Question

Abstract algebra

Suppose that f: G
→ G such that f(x) = axa. Then f is a group homomorphism if
and only if
O a = e
O a^4 = e
a^2 = e
O a^3 = e
If H and K are subgroups of G. IH|= 20 and IKl-32 then a possible value of IHNK| is
2.
8.

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

$Factorization$

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

Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
14. Let be an abelian group of order where and are relatively prime. If and , prove that .
Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.
15. Prove that if for all in the group , then is abelian.
Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.
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.
let Un be the group of units as described in Exercise16. Prove that [ a ]Un if and only if a and n are relatively prime. Exercise16 For an integer n1, let G=Un, the group of units in n that is, the set of all [ a ] in n that have multiplicative inverses. Prove that Un is a group with respect to multiplication.
2. Show that is a normal subgroup of the multiplicative group of invertible matrices in .
32. Let be a fixed element of the group . According to Exercise 20 of section 3.5, the mapping defined by is an automorphism of . Each of these automorphism is called an inner automorphism of . Prove that the set forms a normal subgroup of the group of all automorphism of . Exercise 20 of Section 3.5 20. For each in the group , define a mapping by . Prove that is an automorphism of .
Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
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 ] }