Example

Tag 0A5L, permanent. Cite it as [Bourbaki, Tag 0A5L].

Lie Groups and Lie Algebras, chapter II, § 5. Magnus algebras

Suppose that X is a set with two elements x, y and let H_1 = \{x, y\}, H_2 = \{xy\}. Every element w of F(X) can therefore be written

w \equiv x^a y^b (x, y)^c \mod. C^3 \quad \text{with } a, b, c \text{ in } \mathbf{Z}.

For w = (xy)^n, a = b = n and c = n(1 - n)/2 (cf. Exercise 9), whence

(xy)^n \equiv x^n y^n (x, y)^{n(1-n)/2} \mod. C^3.

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

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