Proposition 11

Tag 05DS, permanent. Cite it as [Bourbaki, Tag 05DS].

Algebra, chapter I, § 6. Extensions, solvable groups, nilpotent groups

Let $E$ be a finite set and $G$ a $p$ -group operating on $E$ . Let $E^G$ denote the set of $x \in E$ such that $gx = x$ for all $g \in G$ (the fixed points). Then

\operatorname{Card}(E^G) \equiv \operatorname{Card}(E) \pmod{p}.

E — E^G is a disjoint union of orbits not reduced to a point. The cardinal of such an orbit is a power of $p$ distinct from $p^0 = 1$ and hence a multiple of $p$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary, § 6
Lemma, § 6
Theorem 3, § 6
Exercise 10, § 2
Exercise 6, § 4

Cites

Nothing.