4. Let G be a finite group. Recall from Homework 9, Textbook §2.8 Exercise 25 thatthe normalizer of any nonempty subset S C G is defined to beN(S) = {g G | gSg-¹ = S}.€To simplify notations, the normalizer of a singleton subset {a} in G is denoted byN(a) instead of N({a}). It consists of all elements in G that commute with a:N (a) = {g G | gag¯¹ = a} = {g € G | ga = ag}.Use the Orbit-Stabilizer Theorem for suitable actions to prove the following state-ments.1(a) class(x)| = |G: N(x)| for all x € G.(b) If H is a subgroup of G, then the number of conjugates of H in G is |G: N(H)|.

Given Information: is a finite group.Normalizer: , where .To Prove:a) .b) Number of conjugates of…

Answered: (a) class(x)| = |G: N(x)| for all x €…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(a) class(x)| = |G: N(x)| for all x € G. (b) If H is a subgroup of G, then the number of conjugates of H in G is |G: N(H)|.