- #1
catcherintherye
- 48
- 0
G is a finite group, |G| =p^n, p prime
*:GxX -> X is group action. X is a finite set,
I am required to prove the following [tex] |X|\equiv |X^G|modp [/tex]
Now we start by asserting that [tex] x_1, x_2, ...,x_m [/tex]
is the set of m orbit representatives. That orbit x [tex] <x_i> = {x_i} \\
iff x_i [/tex] is a fixed point.
we arrange the x_i's so that fixed points precede the non-fixed points.
[tex] {x_1,x_2,...x_a}, |X^G|=a, x_a+1,...x_m [/tex] are the remaining orbit reps.
numerical form of class eqn says
[tex]
|X| = \sum_{i=1}^a \frac{|G|}{|G_x_i|} + \sum_{i=a+1}^m \frac{|G|}{|G_X_i}
[/tex]
since [tex] x_1, x_2,...,x_a fixed G_x_i = G [/tex] for 1<=i<=a
|G|/|G_x_i| =1 for 1<=i<=a
[tex]
|X| = a+ \sum_{i=a+1}^m \frac{|G|}{|G_X_i}
[/tex]
for i=a+1,...,m
[tex] x_i not fixed, G_x_i not equal G [/tex]
but [tex] |G| = p^n so |G_x_i| = p^e_i [/tex]
where e_i < n but where does this fact come from?? I don't see how it follows that order of the stability subgroup must be a power of a prime??
*:GxX -> X is group action. X is a finite set,
I am required to prove the following [tex] |X|\equiv |X^G|modp [/tex]
Now we start by asserting that [tex] x_1, x_2, ...,x_m [/tex]
is the set of m orbit representatives. That orbit x [tex] <x_i> = {x_i} \\
iff x_i [/tex] is a fixed point.
we arrange the x_i's so that fixed points precede the non-fixed points.
[tex] {x_1,x_2,...x_a}, |X^G|=a, x_a+1,...x_m [/tex] are the remaining orbit reps.
numerical form of class eqn says
[tex]
|X| = \sum_{i=1}^a \frac{|G|}{|G_x_i|} + \sum_{i=a+1}^m \frac{|G|}{|G_X_i}
[/tex]
since [tex] x_1, x_2,...,x_a fixed G_x_i = G [/tex] for 1<=i<=a
|G|/|G_x_i| =1 for 1<=i<=a
[tex]
|X| = a+ \sum_{i=a+1}^m \frac{|G|}{|G_X_i}
[/tex]
for i=a+1,...,m
[tex] x_i not fixed, G_x_i not equal G [/tex]
but [tex] |G| = p^n so |G_x_i| = p^e_i [/tex]
where e_i < n but where does this fact come from?? I don't see how it follows that order of the stability subgroup must be a power of a prime??