Answered: 5. Let G be a group and let a € G. An…

5. Let G be a group and let a € G. An element b E G is called a conjugate of a if there exists an element x E G such that b = xax-1. Show that any conjugate of a has the same order as a.

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: 1. If (G,*) a group such that a² = e for all a € G. Prove that G is abelian. 2. Define an operation…

Q: (c) Let G be a group and a, b € G be elements of G each of order 3 such that a² = 6² and prove tha a…

A: Given: A group G, and a,b∈G order of a and b is 3 and a2=b2. To Prove: a=b.

Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

A: We have given a map , ϕ : A → B , where A = ℝ , + , B = ℝ* , · such that , ϕx = 2x We know that…

Q: (h, k) E H x K. What is the order Let H and K be finite groups and let G = H x K. Let g of g?

A: To calculate the order of the element (h,k) in the direct product of groups

Q: 4.1. Let a and b be elements of a group G. Assume that a has order 7 and that a³b = ba³. Prove that…

Q: Show if the given set is a group defined by the given operation. Check all properties. If so, check…

Q: Suppose T: G-> H is group isomorphism and S:H-> K is also a group isomorphism. SHow the…

Q: Consider the group G=A4. List the elements of A4 in cycle notation, and determin the order of A4.

A: Consider the group G=A4. We have to list the elements of A4 in cycle notation, and we have to…

Q: Let G be the set consisting of the following matrices: (6 9) (6 ) ( ) -1/2 v3/2 V3/2 1/2 -1/2 V3/2)…

Q: 4. Let G be a group and let a and b be elements of G. Prove that -1 (a*b)¹ -1 =b¹ a

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

Q: . Let G be a group and a e G. An element be G is called a conjugate of a if there exists an element…

Q: Let G be a group. Assume a and b are elements of G and set H₁ = and H₂ =. Assume |H₁| = 21 and |H₂|…

Q: Q5. Let A and B be two groups. Let 0: A x B → B defined by 0(a, b) = b Is 0 isomorphism? Find…

A: To check whether a function θ is isomorphism, it is required to check θ is homorphism θ is one-one…

Q: 16. Let G be a group and let g and h be in G. Show that . - as defined in #15.

A: Given: G is a group, g,h∈G, ϕg:G→G is a function defined as ϕg(x)=gxg-1 To prove: ϕg∘ϕh=ϕgh Let x∈G…

Q: Let a and b be elements of a group. If |a| and |b| are relatively prime, show that intersects =…

A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…

Q: Let G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar groupis G isomorphic?

A: Here given that G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar group is G isomorphic…

Q: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…

Q: Let G be a group and let a e G. The set C(a) = {x € G | xa = that commute with a is called the…

Q: Let G, H, and K be groups. Show that (G x H) × K = G × (H × K). Hint: An element of (G × H) × K will…

Q: Help: Let G and G' be groups with identity elements e and e' respectively and let f:G goes to G' be…

A: Let G and G' be groups with identity elements e and e' respectively and let f:G goes to G' be a…

Q: 3. Let {: ) G = | a,be Q, a² +b² # 0 26 a Determine if G is a group with respect to matrix…

A: Satisfy all four properties for proving G be a group.

Q: 4.3. Let a and b be elements of a group G. Prove that ab and ba have the same order.

Q: Let a be an element of a group. Suppose abs(a5) = 12. Show that abs(a) = 12 or abs(a) = 60.

Q: Solve the separable differential equation dy 1la – 6yVz² +1 0. dz Subject to the initial condition:…

Q: 5. Let G be a group, and let HCG be a subgroup. Show that if HG, then there exist elements a, b, a',…

A: According to the question,

Q: 5. Let M, and M, in Prob. 4 be closed subspaces. Show that then T(M,) T*(M,*).

A: Orthogonal complement of a subspace: Let V be an inner product space and M be a subspace then M⊥ is…

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: Given that A and B is a group. Find out if þ: A→B is a homomorphism. If it is a homomorphism, also…

A: Group homomorphism is nothing but a function defined between two groups. The function must be closed…

Q: 1. Let (G, *) and (H, *) be groups. The direct product of G and H is the set G x H equipped with…

Q: 2.10 Let G be the set of all 2×2 matrices ( .), where a, b ER and a² + b² +0. Show that G forms a…

Q: Let G be a group with operation * and e be the identity element of G. Show that for any element a in…

Q: et G = {fab | a, b € R, a 0}, where f(x) = ax + b for all real x. Consider the group G under the…

A: Given that G is a group defined as G = { fa,b | a, b ∈ R , a≠ 0} . where fa,b(x)=ax+b…

Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

Q: Exercise 14. Let G₁ and G₂ be a groups. Then the set G₁ X G₂ is a group with the following binary…

Q: Find order of all the elements of the Group = {1,-1, i, -i}. The binary operator * is defined as a *…

A: 1 is identity . so order of 1 is equal to 1.

Question

100%

5. Let G be a group and let a € G. An element b E G is called a conjugate of a if there exists an element x E G
such that b = xax-1. Show that any conjugate of a has the same order as a.

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

3. Let G be a group with identity e. Show that if a ★ a=e for all a in G, then G is Abelian.
Let G {0, 1, 2, 3, 4, 5, 6, 7} and assume that G is a group with the operation satisfying the following conditions: a. a = 0 for all a E G, and a b ≤a+b, for every pair a and b in G. Write down the multiplication table (Cayley table) of G. = Hint: First establish that 0 a = a for every a E G. Then think of writing down the multiplication table as similar to playing Sudoku.
Direct product AxB is isomorphic CxD.
Let G be a group and a € G be such that o(r) = 20. Find the orders of a, at, 10, r14 and 30
3. Let G be a group, HAG, KAG, Hn K = {e}. Show that G G/H x G/K.
he following is a multiplication table for a group G. The product a * b is given in the ble where a is from the leftmost column and b from the top row. Let H e V W X Y 2 = e e V W X W V Y Z X W X Y e W X Z V SOSN e What are the orders of the elements? X 3 Y 2 Y 2 Z X V Y e e X W V W e V Y 3 NS Y ১৯৯ {e, z). Show that this is a subgroup of G. Calculate the right cosets of H. Without calculating the left cosets of H, explain why H is a normal subgroup of G. Construct a multiplication table for the quotient group G/H. Is G/H cyclic? Give reasons. Describe the subgroup Z ≤ G/H generated by the element Hx = G/H. What is the order of (G/H)/Z?
Formulate a Cayleyan (multiplication) table and state the reason if why it is a group or not.
Determine whether U(12) with multiplication mod 12 is a cyclic group or not.
plz 2Q and with details.
Define * on Q* by a * b = - ab Show that (Q*,*) is a group. 1.1 1.2 Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba. 1.3 Let G be a group and Z(G) = {a € G:ax = xa for all x e G}. Show that Z(G)is a normal subgroup of G. Let G be a group and let p: G → G be the map p(x) = x-1. (a) Prove that o is bijective. (b) Prove that p is an automorphism if G is abelian. Let a, ßE S, (Symmetric group), where a = (1,2)(4,5) and ß = (1,6,5,3,2). Verify that (aß)-1 = B-'a-1 . 1.4 1.5
1. Let G and H be groups, and let a: G H be a homomorphism. Show that if a is bijective, then a¹: HG is also a homomorphism.