Abstract Algebra: Theory and Applications · Problem 14.5.24

The Sylow Theorems — Thomas Judson

Mathematics — Abstract Algebra · p-group-fixed-points-congruence

Problem

Let

G

be a group with order

p^n

where

p

is prime and

X

a finite

G

-set. If

X_G = \{x \in X : gx = x \text{ for all } g \in G\}

is the set of elements in

X

fixed by the group action, then prove that

|X| \equiv |X_G| \pmod{p}