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Additional resources for Algèbre [Lecture notes]]

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On note M(S) l’ensemble des suites finies (éventuellement vide) d’élements de S ; un élément (m1 , . . , m n ) de M(S) est appelé mot sur l’ensemble S ; l’entier n est appelé sa longueur. On munit l’ensemble M(S) de la loi de composition donnée par (s1 , . . , s m )(t1 , . . , t n ) = (s1 , . . , s m , t1 , . . , t n ) si m, n ∈ N et s1 , . . , s m , t1 , . . , t n sont des éléments de S. Cette loi est associative et possède la suite vide ε = () pour élément neutre. Ainsi, M(S) est un monoïde, qu’on appelle le monoïde libre sur l’ensemble S.

Plus généralement, si Y est une partie de X, on appelle fixateur de Y l’intersection des fixateurs des éléments de Y. C’est un sous-monoïde de A ; si A est un groupe, c’est un sous-groupe de A. 6. OPÉRATIONS D’UN GROUPE DANS UN ENSEMBLE 37 l’ensemble des éléments a ∈ A tels que a ⋅ Y = Y. C’est un sous-monoïde de A, et un sous-groupe de A si A est un groupe. 8. — Soit A un groupe opérant (à gauche) dans un ensemble X. Soit x un élément de X. L’application de A dans X a ↦ a ⋅ x est appelée application orbitale définie par x, et son image, égale à A ⋅ x est appelée l’orbite de x.

C) Supposons que A soit un groupe. Soit B un sous-groupe de A. On appelle normalisateur de B dans A l’ensemble NA (B) des éléments a ∈ A tels que Int(a)(B) = aBa−1 = B. (2) En effet, on a Int(e)(B) = B. Soit ensuite a, a ′ ∈ NA (B) ; alors Int(aa′ )(B) = Int(a)(Int(a′ )(B)) = Int(a)(B) = B, donc aa′ ∈ NA (B). Enfin, si a ∈ NA (B), on a Int(a−1 )(B) = Int(a−1 )(Int(a)(B)) = Int(a−1 a)(B) = Int(e)(B) = B, donc a−1 ∈ NA (B). Cela prouve que NA (B) est un sous-groupe de A. 38. 28 CHAPITRE 2. 3) (Sous-groupes de Z).

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Algèbre [Lecture notes]] by Antoine Chambert-Loir

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