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Class notes algebre (smc)

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Class notes algebre (smc)

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  • March 30, 2023
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Université M. Ismail  Faculté des Sciences  Département de Mathématiques

Module d'Algèbre 6 - SMA -S4 / Série 1 2020-2021




Exercice 1. Soit E un ensemble quelconque muni d'une relation d'équivalence R. Sur une
partie A non vide de E , on dé
ni la relation binaire RA par :

∀x, y ∈ E, x RA y ⇔ x ∈ A, y ∈ A et xRy.

1. Montrer que RA est une relation d'équivalence sur A. Quelles sont ses classes d'équiva-
lence.
2. En notant respectivement par p : E → E/R la projection canonique induite par R,
pA : A → E/R sa restriction sur A et q : A → A/RA la projection canonique induite par
RA , montrer qi'il existe une application jA : A/RA → E/R telle que pA = jA ◦ q .
3. Montrer que jA est en fait injective et canonique (jA est appelée injection canonique de
A/RA dans E/R).

Exercice 2. Soit G un ensemble muni d'une loi de composition interne (l.c.i.) T supposée
associative. Montrer alors que

(i) ∃e ∈ G / ∀x ∈ G eT x = x
G est un groupe ⇔
(ii) ∀x ∈ G ∃x0 ∈ G / xT x0 = e


Exercice 3.

1. Caractériser les groupes (G, .) pour lesquels l'application :

ϕ: G → G
x 7→ x−1

est un automorphisme.
2. Soit (G, .) un groupe et a ∈ G. Montrer que l'application

f: Z → G
k 7→ ak

est un homomorphisme de groupes.

Exercice 4. Soit K un sous-ensemble non vide d'un groupe
ni G. Soit

NG (K) = {g ∈ G / g −! Kg = K} et CG (K) = {g ∈ G / ∀x ∈ K, g −! xg = x}

respectivement appelés le normalisateur et le centralisateur de K dans G.
1. Montrer que NG (K) et CG (K) sont des sous-groupes de G et que CG (K) ⊆ NG (K).
2. Montrer que NG (K) = G si et seulement si K = g∈G g −! Kg.
S



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