Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
4. Z(G) = N Co(a). aEG

4. Z(G) = N Co(a). aEG

Advanced Engineering Mathematics
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
icon
Related questions
Question

[Groups and Symmetries] How do you solve this question. Definitions are provided 

Center/Centralizer/Normalizer Let G be a group. • The center of G, denoted by Z(G), is the set of all elements in G that commute with every element of G; Z(G) = {g E G : gx = xg Vx € G}. • Given an element a E G, the centralizer of a E G, denoted by CG(@), is the set of elements in G that commute with a; Cg(a) = {g € G : ga = ag}. • If H 3 G, the centralizer of H in G, denoted by CG(H), is the set of elements in G that commute with every element in H; CG(H)= {g € G : gh = hg Vh E H}. • If H < G, the normalizer of H in G is the set NG(H)= {gE G : gHg = H}. It follows by definition that G is Abelian if and only if Z(G) = G. Moreover, CG(G) = Z(G). Theorem 2.27. Let G be a group. Then, 1. Z(G) < Cg(a) < G for all a E G. 2. Z(G) 3 CG(H)< Ng(H)<G for all H 3 G. 3. G is Abelian if and only if Z(G) = Cg(a) for all a E G. 4. Z(G) = N Cala). aEG
Transcribed Image Text:Center/Centralizer/Normalizer Let G be a group. • The center of G, denoted by Z(G), is the set of all elements in G that commute with every element of G; Z(G) = {g E G : gx = xg Vx € G}. • Given an element a E G, the centralizer of a E G, denoted by CG(@), is the set of elements in G that commute with a; Cg(a) = {g € G : ga = ag}. • If H 3 G, the centralizer of H in G, denoted by CG(H), is the set of elements in G that commute with every element in H; CG(H)= {g € G : gh = hg Vh E H}. • If H < G, the normalizer of H in G is the set NG(H)= {gE G : gHg = H}. It follows by definition that G is Abelian if and only if Z(G) = G. Moreover, CG(G) = Z(G). Theorem 2.27. Let G be a group. Then, 1. Z(G) < Cg(a) < G for all a E G. 2. Z(G) 3 CG(H)< Ng(H)<G for all H 3 G. 3. G is Abelian if and only if Z(G) = Cg(a) for all a E G. 4. Z(G) = N Cala). aEG
Question 3. Let b 0 1 0 0 1 a H = c) : a, b, c e Q 1 under matrix multiplication. Notice that H is a group (you do not have to prove this). (a) Find Z(H). (b) Prove that Z(H)=Q, Q is under addition.
Transcribed Image Text:Question 3. Let b 0 1 0 0 1 a H = c) : a, b, c e Q 1 under matrix multiplication. Notice that H is a group (you do not have to prove this). (a) Find Z(H). (b) Prove that Z(H)=Q, Q is under addition.
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS