5. Let G be a finite group, and let class (₁), ..., class(xn) be all of the distinct non- singleton conjugacy classes in G. Show that |G| = |Z(G)| + n | class(x₁)| i=1 n = |Z(G)| + Σ|G : N(xi)|. i=1 This is called the class equation. (Hint: Let G act on X = G by conjugation, and use the Orbit Decomposition Theorem.)
5. Let G be a finite group, and let class (₁), ..., class(xn) be all of the distinct non- singleton conjugacy classes in G. Show that |G| = |Z(G)| + n | class(x₁)| i=1 n = |Z(G)| + Σ|G : N(xi)|. i=1 This is called the class equation. (Hint: Let G act on X = G by conjugation, and use the Orbit Decomposition Theorem.)
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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Transcribed Image Text:5. Let G be a finite group, and let class(x₁),..., class(xn) be all of the distinct non-
singleton conjugacy classes in G. Show that
n
|G| = |Z(G)| + Σ | class(x₁)|
i=1
n
= |Z(G)| +
i=1
|G : N(xi)|.
This is called the class equation. (Hint: Let G act on X = G by conjugation, and
use the Orbit Decomposition Theorem.)
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