∗ ∗ n
6.
∗ ∗ n
6
X a, b ∈ X
∗ a, b, X
∗
a∗x =b X e ∈ X
∗ y ∈ X, y ∗ e = e ∗ y = y ya
a
e
a
∗ a ∗ ya = ya ∗ a = e e
ya
a
ya
1
a
−
1
−
a∗x = b a 1 ∗ (a ∗ x) = a −
1
−
∗b
∗ ∗
α, β, γ ∈ X α ∗ (β ∗ γ ) = (α ∗ β ) ∗ γ
a 1 ∗ (a ∗ x) (a 1 ∗ a) ∗ x e∗x x −
−
= = = =
1
−
a a a a
∗b ∗b 1 ∗b 1 ∗b 1
−
−
−
a∗x = b a
b
X
∗
∗
X
∗
X
a
1
∗
−
a ∈ X X
∗ X
∗
G
∗
∗
a, b ∈ G a∗b= b∗a
G e e,
a,
∗
∗
a∗b a∗b= e a∗b= a a∗b= b a∗b =a a, b, c.
b = e, a∗b = b a∗b = e
∗
∗
Z2 , Z3 Z4
a∗a = b∗b = c∗c = e V
Z4 O O
< b >= O O {0, 2}
{e}
i i i i i i i i i i i i i i i i i i i i
4 4
< a >= {e, a j j}
j j V O O
U U U U U U U U U U U U U U U U U U U U
{e, b} < b >= O O
U U U U U U U U U U U U U U U U
< c >= {e, c} 44
i i i i i i i i i i i i i i i i
< e >= {e}
G 2
n−1
{e,a,a , · · · , a
G =
n
}
G = {e,a,a2 , · · · , an 1 } b = as ∈ G
n
−
(a1 , a2 , a3 , . . . , an )
H
n mdc(n, s)
G
S 1 , S 2 , S 3 , . . . , Sn ai ∈ S i i = 1, 2, 3, . . . , n
n
S 1 × S 2 × S 3 × . . . × S n
S i
i=1
n
i=1
G1 , G2 , . . . , Gn
(a1 a2 , a3 , . . . , an )
(a1 a2 , a3 , . . . , an )·(b1 , b2 , b3 , . . . , bn )
Gi n
Gi
(a1 b1 , a2 b2 , a3 b3 , . . . , a3 bn ) Gi
i=1
ai , bi ∈ Gi n
Gi Gi
i=1
(b1 , b2 , b3 , . . . , bn )
ai · bi ∈ Gi
ei
n
(e1 , e2, ..., en )
Gi
Gi
i=1
(a1 a2 , a3 , . . . , an )
1
−
1
−
a1 , a2 , a3 , . . . , an
Zm
m
1
−
1
−
n
Gi
i=1
× Zn
Zmn
mdc (m, n) = 1
n n
Zmi
i=1
mi
Zm1 m2 m3 , ...mn
i = 1, 2, 3, . . . , n
Z( p1 )r1
× Z( p2 )r2 × . . . × Z( pn)rn × Z × Z × . . . ×Z
pi
ri ( pi )ri
Z
Z pi
× Z pj < pr
max(i, j )
p
p i+j = r
Z pr
Z pr
× Z( p2 )r2 × . . . × Z( pn )rn ( p1 )r1 · ( p2 )r2 · . . . · ( pn )rn ( p1 )s1 · ( p2 )s2 · . . . · ( pn )sn 0 ≤ si ≤ ri ( pi )ri si Z( pi )ri ( pi )ri ( pi )ri ( pi )ri si mdc(( pi )ri , ( pi )ri si ) = ( pi )ri si ( pi )ri = ( pi )si ( p1 )r1 s1 × ( p2 )r2 s2 × . . . × ( pn )rn sn ( pi )ri si Z( p1 )r1
pi m ( pi )ri
−
si
−
−
−
−
−
Z( pi )ri
−
−
−
m m G
m r1
r2
Z( p1 )r1
rn
m = ( p1 ) · ( p2 ) · . . . · ( pn ) ri = 1 pi Z p1 p2 , ...pn G
G G
m
× Z( p2)r2 × . . . × Z( pn)rn
m
i
N
N
f (x, y) = f (x)f (y), ∀ x, y ∈ G N = {x ∈ G; f (x) = e} e
G
G/N
f : G → G/N
G
e¯
G/N
f : G → G
h : G → G/K g : G/K → f (G) f (x) = g(h(x))
x∈G
(HN )/N ∼ = H/(H ∩ N )
G
N
K
y = g 1 xg −
x, y ∈ G, x ∼ y ⇐⇒ ∃ g ∈ G x ∼ x, ∀ x ∈ G x = e 1 xe, ∀ x ∈ G x∼y y∼x 1 u=g x = u 1 yu y∼x x ∼ y y ∼ z x ∼ z 1 z = u xu u = gh
G
H
G/H ∼ = (G/K )(H/K ).
K ≤ H
G
H
−
−
x∼y
y = g 1 xg
g∈G
−
−
−
y = g 1 xg x ∼ z −
z = h 1 yh −
g, h ∈ G
x ∼ y 1
g
g xg = x −
xe = x, ∀ x ∈ G y = xg ⇒ x = yg 1 , ∀ x ∈ G (xg )h = x(gh) , ∀ x, y, g ∈ G −
X
G
G
X
∗ : G × X → X
G
ex = e x ∈ X (g1 g2 )(x) = g1 (g2 x) x ∈ X G − set X x ∈ X g ∈ G
g1 , g2 ∈ G gx = x
X
X g = {x ∈ X/gx = x} X
Gx = {g ∈ G/gx = x}
G − set x
X
x ∈ G, G x
G − set X
x
G
x ∈ X
x
Gx.
X |G|
|Gx|
G
G − set
|Gx| = (G : Gx ).
x ∈ X
pn
G
X
|G|
G − set
|X G | (modp) C x = {y : x ∼ y } = {xg : g ∈ G} G G x1 , x2 , . . . , xn
x∈G
n
G = C x1 ∪ C x2 ∪ . . . ∪ C xn
|G| = |C x1 | + |C x2 | + . . . + |C xn | . C x = {x}
G
|G| = |Z (G)| +
x ∈ Z (G)
|C xi |
xi ∈ / Z (G)
p G
∃a ∈ G
a
p G
|G| = 1
|G| = 1
|X | ≡
1 ≤ |L| ≤ |G|
L G G = x p r p ·m r ≥ 1 a = x p r
−1
·
m
O (x) =
|G| =e a = e, a p
G =e x ∈ G, x |x| ∃ a ∈ x O (a) = p |N | p N = x L = G/N |L| < |G| N G p =e g =e g g p = e g∈ / N, g p ∈ N ∃g ∈ L (g p )n = g pn |N | = n |g| ∃ a ∈ g O (a) = p p
|G|
= G Z = Z (G) [G : C G (xi )]
G
∃ xi ∈ / Z (G) = G. H = C G (xi ) O (a) = p
|N | |H | ∃ a ∈ H
|H | < |G|
H
G
(N [H ] : H ) =
(G : H ) (modp)
|G| = mpn n≥1 pi pi pi+1
1≤i≤n
1≤i≤n
pi i+1
p
i
pi (N [H ] : H ) N [H ] /H N [H ] /H
(G : H ) N [H ] ,
i < n
|N [H ] /H | γ : N [H ] → N [H ] /H γ [K ] = {x ∈ N [H ] /γ (x) ∈ K } N [H ] i+1 p . H < γ 1 [K ] ≤ N [H ] N [H ] 1
−
−
|γ 1 [K ] = pi+1 | . −
|G| = mpn pn gP g
1
−
P 1 P 2 P 1 P 2 y(xP 1 ) = (yx)P 1 L |L| = (G : P 1 ) P P 2 2 = 0 xP 1 ∈ LP 2 yxP 1 = xP 1 LP 2 1 1 y ∈ P 2 x yxP 1 = P 1 y ∈ P 2 x yx ∈ P 1 y ∈ P 2 1 1 x P 2 x ≤ P 1 P 1 = x P 2 x P 1 P 2 |P 1 | = |P 2 | , P 1
L y ∈ P 2
P 2 L L ≡ |L| (modp)
−
−
−
−
|G| |G| S S |S| ≡ |S P | (modp) x ∈ P P ≤ N [T ] T ≤ N [T ] N [T ] N [T ] S P = {P } |S| ≡ |S P | = 1(modp)
x ∈ P T ∈ S S P T ∈ S P
xT x = T
1
−
xT x
1
−
N [T ] N [T ]
T = P
S S (G : GP ) |G| ,
GP
|G|
P ∈ S (G : GP )
|S| =
th
