Corollary

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

Let G be a p-group. If G is not reduced to e, its centre is not reduced to e.

Let G operate on itself by inner automorphisms. The set of fixed points is the centre Z of G. By Proposition 11,

\operatorname{Card}(Z) \equiv \operatorname{Card}(G) \equiv 0 \pmod{p},

whence $\operatorname{Card}(Z) \neq 1$ and $Z \neq \{e\}$ .

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

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