Page 480 - Advanced Linear Algebra
P. 480
464 Advanced Linear Algebra
%~ %
% c ~ % c
c
% c ~ %
c
* . ²%³
where
* ²%³ ~¸ . % ~% ¹
.
is the centralizer of . But c * ² % ³ . if and only if and are in the same
%
coset of *²%³ . Thus, there is a one-to-one correspondence between the
.
conjugates of and the cosets of * . ² % ³ . Hence,
%
b . b¸%¹ ~ ². ¢ * ²%³³
.
Since the distinct conjugacy classes form a partition of (because conjugacy is
.
an equivalence relation), we have
(( ~ . b¸%¹ ~ b ². ¢ * ²%³³
.
.
%: %:
where is a set consisting of exactly one element from each conjugacy class
:
.
.
¸%¹ . Note that a conjugacy class ¸%¹ has size if and only if % c ~ % for
all . , that is, % ~ % for all . and these are precisely the elements in
.
the center A².³ of . Hence, the previous equation can be written in the form
((.~ (A².³ b ². ¢ * ²%³³
(
.
%: Z
where : Z is a set consisting of exactly one element from each conjugacy class
¸%¹ of size greater than . This is the class equation for ..
.
The Complex Roots of Unity
If is a positive integer, then the complex th roots of unity are the complex
solutions to the equation
%c ~
The set < of complex th roots of unity is a cyclic group of order . To see
this, note first that < is an abelian group since Á < implies that <
and c < . Also, since % c has no multiple roots, < has order .
Now, in any finite abelian group , if is the maximum order of all elements
.
.
of , then ~ for all . . Thus, if no element of < has order , then
and every . satisfies the equation % c ~ , which has fewer
than solutions. This contradiction implies that some element of < must have
order and so < is cyclic.
