Feuille d`exercices 1

RE
E=RR={(x, y),|x|=|y|}
E=ZR={(x, y), x =y= 0 xy > 0}
E=ZR={(x, y), x y }
E=RR={(x, y), x −y∈Z}
E=N∗R={(x, y), x y }
EP(E)
∆
A∆B= (A\B)∪(B\A)
G E f :E→G
E f
E=ZG= (Z,+) f(x) = x+ 1
G g ∈G g G n ≥1
gn= 1 +∞
−1 (Q,+) (Q∗,×)
g∈GZ→G m 7→ gm
g
k, n ≥1kZ/nZ
k n
G G
g7→ g−1g7→ g2
g7→ g3
G g ∈G cg:G→G x 7→ gxg−1

cgG
c:G→Aut(G)
cAut(G)c
G
G
G
2F2
G' {1}G'Z/2Z
G x ∈G ω(x)x
(x, y)∈G2m=ω(x)n=ω(y)
m n ω(xy) = mn
m n ω(xy) = (m, n)
(m, n)∈(N?)2(m0, n0)∈(N?)2
m0|m, n0|n, (m0, n0) = 1 (m, n) = m0n0
Gexp G
z∈Gexp G=ω(z)
G x, y ∈G[x, y] = xyx−1y−1x y
G D(G)G
(x, y)G[x, y] = e D(G)
G
D(G)G
G G G/D(G)
HCG G/H
D(G)⊂H
M
f:G−→ M f G/D(G)
D(Sn)An

f:G→H
H f G f
G f H f
ZG={g∈G;∀x∈G, xg =gx}G ZHH
f f(ZG)⊂ZHf
N G π :G→G/N
π s:G/N →G π ◦s= Id
H⊂G H ∩N= 1 HN =G
π
G N H
H G/N G ∼ −→ N×H
θ:H→Aut(N)H N
G N H θ
H N G
H
θ
θ
H p G p
G H G
ρ:G−→ SG/H K
G|p!K
K⊂H