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By Antoine Chambert-Loir

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Extra info for Algèbre [Lecture notes]]

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De groupes) ; en effet, pour x, y ∈ A, on a Int(a)(x y) = a(x y)a−1 = (axa −1 )(aya−1 ) = Int(a)(x) Int(a)(y). On a Int(a) = idA si et seulement si ax = xa pour tout x ∈ A, c’est-à-dire si se seulement si a appartient au centre de A. En outre, si b est un autre élément inversible de A, on a Int(a) ○ Int(b) = Int(ab) puisque Int(a)(Int(b)(x)) = a(bxb−1 )a−1 = (ab)x(b−1 a−1 ) = (ab)x(ab)−1 = Int(ab)(x) pour tout x ∈ A. Il s’ensuit que Int(a) est un automorphisme de A et que son automorphisme réciproque est l’automorphisme Int(a−1 ).

Soit f ′ ∶ S ∪ S′ → A l’application telle que f ′ (s) = f (s) et f ′ (s′ ) = f (s)−1 pour tout s ∈ S. Il existe un unique morphisme de monoïdes φ′ ∶ M′ (S) → A tel que φ′ ○ j′ = f ′ . La relation d’équivalence définie par φ′ (pour laquelle m ≡ m′ si et seulement si φ′ (m) = φ′ (m′ )) est compatible avec la structure de monoïde de M′ (S) ; comme φ′ (ss′ ) = f ′ (s) f ′ (s′ ) = f (s) f (s)−1 = e et φ′ (s′ s) = f ′ (s′ ) f ′ (s) = f (s)−1 f (s) = e, on a ss ′ ≡ e et s′ s ≡ e. Par suite, cette relation d’équivalence est moins fine que la relation ∼ et il existe une application φ ∶ F(S) → A telle que φ(p(m)) = φ′ (m) pour tout m ∈ M′ (S).

5). — Soit A un monoïde (resp. un groupe) opérant dans un ensemble X. Par restriction, tout sous-monoïde (resp. sous-groupe) de A opère dans X. Plus généralement, soit B un monoïde et soit u ∶ B → A un morphisme de monoïdes. Alors B opère dans X par la formule b ⋅ x = u(b) ⋅ x pour b ∈ B et x ∈ X. On dit qu’une partie Y de X est stable pour l’opération de A si l’on a a ⋅ y ∈ Y pour tout y ∈ Y et tout a ∈ A. Dans ce cas, l’opération de A dans X se restreint en une opération de A dans Y. 6). — Soit A un monoïde.

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Algèbre [Lecture notes]] by Antoine Chambert-Loir

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