Theorem 24.1 Number of Conjugates of a Let G be a finite group and let a be an element of G. Then, Icl(a)l = IG:C(a)l. PROOF Consider the function T that sends the coset xC(a) to the conjugate xax-' of a. A routine calculation shows that T is well-defined, is one-to- one, and maps the set of left cosets onto the conjugacy class of a. Thus, the number of conjugates of a is the index of the centralizer of a.

Theorem 24.1 Number of Conjugates of a Let G be a finite group and let a be an element of G. Then, Icl(a)l = IG:C(a)l. PROOF Consider the function T that sends the coset xC(a) to the conjugate xax-' of a. A routine calculation shows that T is well-defined, is one-to- one, and maps the set of left cosets onto the conjugacy class of a. Thus, the number of conjugates of a is the index of the centralizer of a.

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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Question

Show that the function T defined in the proof of Theorem 24.1 is
well-defined, is one-to-one, and maps the set of left cosets onto the
conjugacy class of a.

Theorem 24.1 Number of Conjugates of a
Let G be a finite group and let a be an element of G. Then,
Icl(a)l = IG:C(a)l.
PROOF Consider the function T that sends the coset xC(a) to the conjugate
xax-' of a. A routine calculation shows that T is well-defined, is one-to-
one, and maps the set of left cosets onto the conjugacy class of a. Thus, the
number of conjugates of a is the index of the centralizer of a.

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