FEUILLE N 2 : GROUPES, SOUS

◦
(G, ·)e H ⊂G
G
(i)e∈H
(ii)∀x, y ∈H xy ∈H
(iii)∀x∈H x−1∈H
H≤G
Z,R(C,+)
n∈N∗Gln(R)n×n
A∈Gln(R)A−1∈Gln(R)
AA−1=A−1A=In=
1 0 · · · 0
0
0
0. . . 0 1
.
n= 2 Gln(R)
n≥2Gln
G:= R×R∗? G
(x, y)?(x0, y0)=(x0+xy0, yy0).
(1) (G, ?)
(2) H(R∗,×)R×H G
(3) GGl2(R)
G= 1b
0c:b, c ∈R
Gl2(R) (G, ?)
aRx, y ∈R
x?y:= x+y−axy.
(1) ?
(2) ?
(3) R\ { 1
a}?

◦
E R∗×R∗×R
(a, b, c)?(α, β, γ)=(aα, bβ, aγ +cβ)
E?
(G, ·)G
(G, ·)H, K G
(H∪K G)⇐⇒ (H⊂K K ⊂H).
G g ∈G
φg:G→G
x7→ gxg−1
(1) g∈G φgG
(2) (G)G
◦(φ◦ψ)(x) = φ(ψ(x))
(3)
Φ : G→(G)
g7→ φg
(i) Φ
(ii) Φ = Z(G)Z(G) := {x∈G:xy =yx, ∀y∈G}
G
G Z(G) = G
G n ≥1x7→ xnG
∀x∈G xn−1∈Z(G)
G f G G f(x) = x−1f
G
G H G H2:= {xy :x, y ∈H}H−1:=
{x−1:x∈H}HH−1:= {xy−1:x, y ∈H}
(1) H2=H H−1=H HH−1=H
(2) H, K G HK := {xy :x∈H, y ∈K}KH := {yx :y∈
K, x ∈H}. HK G HK =KH
(Z,+)
(1) H(Z,+) G={0}a∈N∗H=aZ
(2) H:= {8b+12c:b, c ∈Z}HZa H =aZ
(3) H:= 8Z∩12ZHZa H =aZ
(4) x, y ∈ZxZ∩yZ Z x y
x y