3. Sous-groupes

G H ⊂G
G H
H < G
{e}< G
G < G
2Z<Z
G H G
∀a, b ∈H, ab−1∈H
H G
G H G
∀a, b ∈H, ab ∈H
∀a∈H, a−1∈H
H G
G H ={x:x2=e}
G
G
H={x2:x∈G}
G

H G
H G
∀a, b ∈H, ab ∈H.
GL(n, R)
SL(n, R)SL(n, Z)
O(n)SO(n) = SL(n, R)∪O(n)
G G
G
Z(G) = {g∈G:∀h∈G, gh =hg}.
Z(G)G
ZD4
a G
hai:= {an:n∈Z}
G a
a a
06=n∈Z
α∈[0,1[ R∈(R2)
απ R α
G¯ G φ :G→¯ G
∀a, b ∈G, φ(ab) = φ(a)φ(b).

φ:Z→2Z
φ(n)=2n
G=R¯ G=R>0
φ:G→¯ G
φ(x) = exp(x)
G g ∈G
φg:G→G
φg(x) = gxg−1
g
G
(G)
g∈G Tg:G→
G
Tg(x) = gx.
Tg
Tg(x) = Tg(y)gx =gy
g−1x=y
x∈G Tg(g−1x) = x.
Tg(G)
H={Tg:g∈G} ⊂ (G).
H(g)
=Te∈H H
Tg, Th∈H Tg◦Th=Tgh ∈H
Tg∈H Tg−1∈H Tg
Tg−1◦Tg=Tg−1g=Te=,
Tg◦Tg−1=Tgg−1=Te=.

φ:G→H
φ(g) = Tg.
φ H
φ(g) = φ(h)Tg=Th
g=ge =Tg(e) = Th(e) = he =h.
φ(gh) = Tgh =Tg◦Th=φ(g)φ(h)
x∈G
Tghx= (gh)x=g(Thx) = Tg◦Th(x).
φ
D3S6={1,2,3,4,5,6}