Innen

○

○

○

○

n ∈ N G = (V, E) 2n + 1
v ∈ V d(v) ≥ n G
d(v) ≥ n − 1
G G
G G

k k ≥ 2 k + 1
f1
f1 ≥ c + 2
n
n
n
G 2k
k
k
• k

k
• k
k+1
K
K
n ≥ 3 G n
G n
2 G
G
G G
n
n
(Z, ◦), a ◦ b = (a + b)/2 (a, b ∈ Z)

(Q, ◦), a ◦ b = (a + b)/2 (a, b ∈ Q)
(A, ◦), A [0, 1]
(f ◦ g)(x) = (f(x), g(x))
R,
(R \ {0},
n n
n n
m
m
A = {0, 5, 10, 15, 20, 25, 30}
A = {0, 5, 10, 15, 20, 25, 30}

A \ {0}
B = {5, 10, 15, 20}
n n
n n
(G, ·) u ∈ G G
◦ a ◦ b := a · u · b (G, ◦)?
(G, ·) a, b
(a · b) 2 = a 2 · b 2 ,

m m
a ◦ b = a + b + 1
Dn n
ϕ 2π
n τ
Dn
{e, ϕ, ϕ 2 , . . . , ϕ n−1 , τ, τ · ϕ, τ · ϕ 2 . . . , τ · ϕ n−1 }
ϕ n = τ 2 = e ϕ · τ = τ · ϕ n−1
Dn
π
6
t
= 60◦

n = 2
D2 = {e, a, b, c},
∗ e a b c
e e a b c
a a e c b
b b c e a
c c b a e
G
G
G
G
ϕ 2π
3 =
120 ◦ τ
G
{ϕ}, {τ}, {ϕ, τ}
G
D4
(G, ·) a a −1
(G, ·) a b −1 ·a·b
(G, ·) G
(G, ·) a ∈ G
a |G| = e,
e
c c 100 = e c 1999 = e

c
(G, ·)
G
H τ G H
H
G
K ϕ G K
K
G
C ∗ ◦
a ∗ b = a + b + 1, a ◦ b = a + b + i.
(C, ∗) (C, ◦)
ϕ : a ↦→ ai (C, ∗)
(C, ◦)
1 c
0 1
a a
a a
c ∈ R
a = 0, a ∈ R
(Q, +).

D4
a + b √ 2 | a, b ∈ Z
{a + bi | a, b ∈ Z}
(Zm, +, ·)
(R, +, ·)
(R, +)
(R, ·)
(R, +, ·)
(R, +, ·)
mod 2m
0, 2, 4, . . . , 2m − 2 ,
2m = 10
2m = 20
(T, +, ·)

(R, +, ·) 0
e a ∈ R a n = 0
n ∈ N a e − a
(R, +, ·) a
a
R a a 2 = a
R
Z10
a Zm
a a
Z10
Z10 5
Z10 5
L := a + bi √ 5 | a, b ∈ Z

L 1 + i √ 5, 1 − i √ 5, 2, 3
(L, +, ·)
(Z4, +, ·)
R I R R = I
I R
(T, +, ·) T
(P )
Z/P
R =
a b
c d
| a, b, c, d ∈ Z I =
a b
c d
| a, b, c, d ∈ 2Z
I R
R/I
N R 0
N
N
N R/N
a b
M =
| a, b ∈ Z (M, +, ·)
2b a
E = ( a + b √ 2 | a, b ∈ Z , +, ·)
G = ( a + b √ 2 | a, b ∈ Z , +, ·) K = ( a + b √ 3 | a, b ∈ Z , +, ·)

L 1 + i √ 5, 1 − i √ 5, 2, 3
(L, +, ·)
(Z4, +, ·)
R I R R = I
I R
(T, +, ·) T
(P )
Z/P
R =
a b
c d
| a, b, c, d ∈ Z I =
a b
c d
| a, b, c, d ∈ 2Z
I R
R/I
N R 0
N
N
N R/N
a b
M =
| a, b ∈ Z (M, +, ·)
2b a
E = ( a + b √ 2 | a, b ∈ Z , +, ·)
G = ( a + b √ 2 | a, b ∈ Z , +, ·) K = ( a + b √ 3 | a, b ∈ Z , +, ·)

15 − 2 · 3 = 9

n ∈ N G = (V, E)
2n + 1 v ∈ V d(v) ≥ n
G
d(v) ≥ n − 1
2n + 1
n
n − 1
n
n − 1
n n + 1
G G
G G
G
G G

G
G1 = (V1, E1), G2 = (V2, E2), . . .
v1 v2 G
G G
v1 v ′ 1
G
v2 v1, v2, v ′ 1
G
v1
v ′ 1
G1
G2
v2
k k ≥ 2
k + 1
v1, v2, . . . , vs
vs s−1
vt vt, vt+1, . . . , vs, vt
vt
vs k k + 1
vs

v1, v2, . . . , vs vs
k = 2
G n
d
G n G
G d n − d − 1
G
G
d1 ≤ d2 ≤ . . . ≤ dn,
n − dn − 1 ≤ n − dn−1 − 1 ≤ . . . ≤ n − d1 − 1,
di + dn+1−i = n − 1
n − 1
di ≥ 1 di ≤ n − 2

n = 4 1 2
1, 1, 2, 2
n = 4
v1
v2
G
v4
v3
v1 ↦→ v2 v2 ↦→ v4 v3 ↦→ v1
v4 ↦→ v3
n = 5
v5
v1
v4
G
v2
v3
v1 ↦→ v1 v2 ↦→ v3 v3 ↦→ v5
v4 ↦→ v2 v5 ↦→ v4
v5
v1
v2
v1
G
v4
G
v2
v4
v3
v3

G G
G G
G G
v5
v1
v4
G
v2
v3
v1 ↦→ v5 v2 ↦→ v3 v3 ↦→ v1
v4 ↦→ v4 v5 ↦→ v2
G
G
v
v5
v1
v4
G
v2
v3

G G
G v
v
v
n
f1
f1 ≥ c + 2
n
n − 1
k fk
d n
2(n−1)

d
(fi · i) = f1 + 2f2 + . . . + dfd = 2(n − 1).
i=1
n
d
fi = f1 + f2 + . . . + fd = n.
i=1
d
(2 − i)fi = f1 − f3 − 2f4 − . . . − (d − 2)fd = 2
i=1
f1 = 2 + f3 + 2f4 + . . . ≥ 2 + f3 + f4 + . . . = c + 2,
c
n
n = 2 2 ≥ 2 n > 2
n − 1
n−1 G ′ f ′ 1 ≥ c′ +2
• G ′ f1 = f ′ 1 c = c ′
• G ′ f1 = f ′ 1 + 1 c = c′ + 1
• G ′ f1 = f ′ 1
+ 1 c = c′
f1 = c + 2

s t
m
s t u
s t
s t u
⌈ m
2
⌉ + ⌈ m
2
⌉ + 1 s t
m = 6
s
u
t

s1 v1, v2, . . . , vm
w = v ⌊ m
2 ⌋
m
w
s2 u1, u2, . . . , um w k
uk s1 l
uk ul
w
• ul s1 w ul = vj j < ⌊ m
2 ⌋
ul m
u1 u2
ul(= vj) vj+1 vm l > ⌊ m
2 ⌋ um um−1 ul(= vj)
vj+1 vm
• uk = vj j > ⌊ m
2 ⌋ m
u1 u2 uk(= vj)
vj−1 v1 k > ⌊ m
2 ⌋ um um−1 uk(= vj) vj−1 v1
w
m
n
n
n

n
n − 1
n − 1
n
n − 1
n − 1
n
n − 2
n
•
•

v
v v
v

G 2k
k
k
k k
k
k k

n
n = 1 n n + 1
v
n
v1, v2, . . . , vn
(n + 1)
• v v1 v1 v, v1, v2, . . . , vn
• v vn v v1, v2, . . . , vn, v
• i
v vi vi
vn v v1, v2, . . . , vi−1, v, vi, vi+1, . . . , vn

v
✣✍ ✻
✲ ✲ ✲◆
✲ ✲
v1 vi−1 vi vn
k
• k
k
• k
k + 1
k
G = (V, E) W ⊆ V E ′ ⊆ E V
W l
(V, E ′ ) W
l
V − W
k l
k k + 1
k
K

K
K
K
K v1, v2, . . . , vn K
n
K
K
K
v1 K w w, v1, v2, . . . , vn n + 1
vn
w v1
v2
n ≥ 3 G n
G n
2
G
v1, v2, . . . , vm
m K

K
K
v1 vm v1
vi1 , vi2 , . . . , vid vm vj1 , vj2 , . . . , vje d, e ≥ n/2 i1 −
1, i2 −1, . . . , id −1 j1, j2, . . . , je m−1
d + e ≥ n ≥ m x y
ix − 1 = jy v1
vm v1 vk+1
vm vk
v1
vk
vk+1
v1, v2, . . . , vk, vm, vm−1, . . . , vk+1, v1 m
G
G G
vm

n
n n 2
n
3n − 6
2(3n − 6) ≥ n
2
12n − 24 ≥ n 2 − n
0 ≥ n 2 − 13n + 24 = 1 2
(2n − 13) − 73
4
n ≥ 11 n 2 − 13n + 24
n1,2 = 13 ± √ 13 2 − 4 · 24
2
= 13 ± √ 73
2
n1 ≈ 10,77 n2 ≈ 2,23
n ≥ 11
a
c
d
b
b ′
d ′
a ′
c ′
n
c ′
a
d ′
b
c
a ′
d
b ′

t
e n
n + t = e + 2.
t e
e = 3
2
t
e = 3n − 6
t + n = 2
e + n = e + 2.
3
e = 4
2t = 2t
e = 2n − 4
t + n = 1
e + n = e + 2,
2
6 = 3·4−6 12 = 3·6−6
30 = 3·12−6 12 = 2·8−4
n

e t
t = 2t
e ≥ 4
2
e + 2 = n + t ≤ n + 1
2 e.
e ≤ 2n − 4 n
n−2
2n − 4
w1 w2
n − 2 v1 v2 vn−2
w1
v1 v2 vn−2
w2

(Z, ◦), a ◦ b = (a + b)/2 (a, b ∈ Z)
(Q, ◦), a ◦ b = (a + b)/2 (a, b ∈ Q)
(A, ◦), A [0, 1]
(f ◦ g)(x) = (f(x), g(x))
R,
(R \ {0},
(Z, ◦), a◦b = (a+b)/2 (a, b ∈ Z) ◦
3 ◦ 2 = 5
2
∈ Z. (Z, ◦)
(Q, ◦), a ◦ b = (a + b)/2 (a, b ∈ Q) ◦
(a ◦ b) ◦ c =
a ◦ (b ◦ c) =
a+b
2
+ c
2
a + b+c
2
2
= a + b + 2c
= 2a + b + c
,
4
.
4
a = b = 1, c = 0.
(a ◦ b) ◦ c = 1 3
= = a ◦ (b ◦ c).
2 4
(A, ◦), A [0, 1]
(f ◦ g)(x) = (f(x), g(x)) ◦

(f ◦ g) ◦ h = f ◦ (g ◦ h) f, g, h ∈ A
R,
(R \ {0},
(a/b)/c =
a
b
c
a/(b/c) = a
b
c
= a
b · c ,
= a · c
b .
a = 1, b = 2, c = 2. 1
4 ,
a
b b
a
εk = cos k 2π
8
2π
+ i sin k , 0 ≤ k ≤ 7 (∗)
8

ε3 = cos 3π
4
ε4 = cos π + i sin π
ε5 = cos 5π
4
+ i sin 3π
4
✛
+ i sin 5π
4
■
✻
❄
✒
✠ ❘
ε2 = cos π π
2 + i sin 2
ε6 = cos 3π
2
ε1 = cos π π
4 + i sin 4
✲
ε0 = cos 0 + i sin 0
ε7 = cos 7π
4
+ i sin 3π
2
+ i sin 7π
4
εk · εm = εk+m, k + m
a, b, c
(a · b) · c = a · (b · c).
εk · 1 = 1 · εk = εk.
εk · x =
1 x = 1
εk = ε−k. −k
x · εk = 1
x
n
n

a ∈ C n a n = 1. n
εk = cos k 2π
n
2π
+ i sin k , 0 ≤ k ≤ n − 1 (∗)
n
a b
n a n = 1 b n = 1. (a · b) n = a n · b n = 1,
a · b n
a, b, c
(a · b) · c = a · (b · c)
C · C
n a
1 · a = a · 1 = a.
a n
εk · x = 1 x = 1
εk = ε−k = εn−k, εn−k
x · εk = 1 x
n n
a n
b k a n = 1 b k = 1. (a · b) n·k =
(a n ) k · (b k ) n = 1, a · b n · k
a, b, c
(a · b) · c = a · (b · c)
C · C

n n
a n 1 · a = a · 1 = a.
a n
{0, 1, 2, 3, 4 (mod 5)}, (∗)
0
a · b = a · b, (∗∗)
· (mod 5) 0 1 2 3 4
0 0 0 0 0 0
1 0 1 2 3 4
2 0 2 4 1 3
3 0 3 1 4 2
4 0 4 3 2 1
3 · 4 = 2,
3 · 4 ≡ 12 ≡ 2 (mod 5).

(a · b) · c = (a · b) · c
a · (b · c) = a · (b · c),
1 1
3
2 3 · 2 = 1.
2 · 3 = 1, 3 2.
0 0, 1,
{1, 2, 3, 4 (mod 5)}, (∗ ∗ ∗)
· (mod 5) 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1
1

1
{0, 1, 2, 3, 4, 5, 6, 7 (mod 8)} (∗)
· (mod 8) 0 1 2 3 4 5 6 7
0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7
2 0 2 4 6 0 2 4 6
3 0 3 6 1 4 7 2 5
4 0 4 0 4 0 4 0 4
5 0 5 2 7 4 1 6 3
6 0 6 4 2 0 6 4 2
7 0 7 6 5 4 3 2 1
6 · 5 = 6,
6 · 5 ≡ 30 ≡ 6 (mod 8).

1 1
0 0, 1,
{1, 2, 3, 4, 5, 6, 7 (mod 8)} (∗∗)
· (mod 8) 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7
2 2 4 6 0 2 4 6
3 3 6 1 4 7 2 5
4 4 0 4 0 4 0 4
5 5 2 7 4 1 6 3
6 6 4 2 0 6 4 2
7 7 6 5 4 3 2 1
0
4 · 2 = 0.
{1, 3, 5, 7 (mod 8)} (∗ ∗ ∗)

· (mod 8) 1 3 5 7
1 1 3 5 7
3 3 1 7 5
5 5 7 1 3
7 7 5 3 1
1
1
m
m
1
0 0, 1,

m
n n
1 1
a · x = 1 (∗ ∗ ∗∗)
a · x ≡ 1 (mod n)
(a, n) = 1 | 1.
x = a ϕ(n)−1
(mod n)
a ϕ(n)−1
n
n
A = {0, 5, 10, 15, 20, 25, 30}
A = {0, 5, 10, 15, 20, 25, 30}

A \ {0}
B = {5, 10, 15, 20}
+ (mod 35) 0 5 10 15 20 25 30
0 0 5 10 15 20 25 30
5 5 10 15 20 25 30 0
10 10 15 20 25 30 0 5
15 15 20 25 30 0 5 10
20 20 25 30 0 5 10 15
25 25 30 0 5 10 15 20
30 30 0 5 10 15 20 25
A
a + b = a + b
0.
20 15.
A

· (mod 35) 0 5 10 15 20 25 30
0 0 0 0 0 0 0 0
5 0 25 15 5 30 20 10
10 0 15 30 10 25 5 20
15 0 5 10 15 20 25 30
20 0 30 25 20 15 10 5
25 0 20 5 25 10 30 15
30 0 10 20 30 5 15 25
A
A
15
0
· (mod 35) 5 10 15 20 25 30
5 25 15 5 30 20 10
10 15 30 10 25 5 20
15 5 10 15 20 25 30
20 30 25 20 15 10 5
25 20 5 25 10 30 15
30 10 20 30 5 15 25
A \ {0}
A \ {0}
15
15

B 5 · 20 = 0.
n
n
n n
(G, ·) u ∈ G G
◦ a ◦ b := a · u · b (G, ◦)?
◦ G G G

(a ◦ b) ◦ c = (a · u · b) · u · c =
(G, ·) · G
= a · u · (b · u · c) = a ◦ (b ◦ c)
(G, ·) e (G, ◦) ε
a ◦ ε = a
a · u · ε = a
a · u · ε = a · e
G
a −1
a −1 · a · u · ε = a −1 · a · e
e · u · ε = e · e
u · ε = e
u −1
ε = u −1 · e
ε = u −1
u −1
ε ◦ a = a
ε · u · a = a
ε · u · a = e · a
ε · u = e
ε = e · u −1
ε = u −1
u −1

a
a ◦ x = ε
a · u · x = u −1
u · x = a −1 · u −1
x = u −1 · a −1 · u −1
a ◦ x = ε x = u −1 · a −1 · u −1
x ◦ a = ε a (G, ◦)
u −1 · a −1 · u −1 .
(G, ◦)
a · b = b · a
a b
a b
a · b = (a · b) −1
(a · b) −1 = b −1 · a −1
b −1 · a −1 = b · a
a · b = b · a

a b
(a · b) · (a · b) = e
a · (b · a) · b = e
a
a · a · (b · a) · b = a
b
a · a · (b · a)b · b = a · b
a · a = e b · b = e
b · a = a · b
(G, ·) a, b
(a · b) 2 = a 2 · b 2 ,
a b
(a · b) 2 = a 2 · b 2
(a · b) 2 = a · b · a · b
a 2 · b 2 = a · a · b · b
a · b · a · b = a · a · b · b
a −1 b −1
b · a = a · b

εk = cos k 2π
8
2π
+ i sin k , 0 ≤ k ≤ 7 (∗)
8
ε1
n ε n 1
1 ε1
2 ε2 1
3 ε3 1
4 ε4 1
5 ε5 1
6 ε6 1
7 ε7 1
8 ε8 1
π π
= (cos 4 + i sin
π π
= (cos 4 + i sin
π π
= (cos 4 + i sin
π π
= (cos 4 + i sin
π π
= (cos 4 + i sin
π π
= (cos + i sin
4
= (cos π
4
4 )2 = cos π π
2 + i sin 2 = ε2
4 )3 = cos 3π 3π
4 + i sin 4 = ε3
4 )4 = cos π + i sin π = ε4
4 )5 = cos 5π 5π
4 + i sin 4 = ε5
4 )6 = cos 3π 3π
2 + i sin 2 = ε6
4 )7 = cos 7π 7π
4 + i sin 4 = ε7
π + i sin 4 )8 = cos 2π + i sin 2π = ε0
|ε1| = 8, ε 8 1 = 1 ε1
|ε2| = 4, ε 4 2 = 1 ε1

|ε3| = 8, |ε4| = 2, |ε5| = 8, |ε6| = 4, |ε7| = 8 |ε0| = 1.
ε0 1 {ε0}
ε1 8 {ε0, ε1, ε2, ε3, ε4, ε5, ε6, ε7}
ε2 4 {ε0, ε2, ε4, ε6}
ε3 8 {ε0, ε1, ε2, ε3, ε4, ε5, ε6, ε7}
ε4 2 {ε0, ε4}
ε5 8 {ε0, ε1, ε2, ε3, ε4, ε5, ε6, ε7}
ε6 4 {ε0, ε2, ε4, ε6}
ε7 8 {ε0, ε1, ε2, ε3, ε4, ε5, ε6, ε7}
ε1, ε3, ε5 ε7
ε2 ε6 {ε2, ε4, ε6, ε0} . ε2
ε4 {ε4, ε0} .
ε0 {ε0} .
ε1
m m
a ◦ b = a + b + 1

m
m a
−a,
m
◦ a, b ∈ Z, a ◦ b = a + b + 1 ∈ Z
(a ◦ b) ◦ c = (a + b + 1) ◦ c = (a + b + 1) + c + 1 = a + b + c + 2,
a ◦ (b ◦ c) = a ◦ (b + c + 1) = a + (b + c + 1) + 1 = a + b + c + 2,
a ◦ e = a + e + 1 = a
e = −1.
a◦a −1 = a+a −1 +1 = −1 a −1 = −a−2.
0 1 = 0 0 2 = 1 0 3 = 2 . . . 0 k = k − 1 . . .
a −1 =
−a − 2
0 −1 = −2 0 −2 = −3 0 −3 = −4 . . . 0 −k = −(k − 1) − 2 = −k − 1 . . .
0 0 = e = −1
Dn n
ϕ 2π
n

τ Dn
{e, ϕ, ϕ 2 , . . . , ϕ n−1 , τ, τ · ϕ, τ · ϕ 2 . . . , τ · ϕ n−1 }
ϕ n = τ 2 = e ϕ · τ = τ · ϕ n−1
Dn
π
6 = 60◦ t
n = 2
D2 = {e, a, b, c},
∗ e a b c
e e a b c
a a e c b
b b c e a
c c b a e
G
G
G
G

ϕ
2π
3 = 120◦ τ
G
{ϕ}, {τ}, {ϕ, τ}
G
G
G |G| = 3! = 6,
ϕ 2π
3 τ
G
e
ϕ
ϕ 2
ϕ · τ =
= τ · ϕ 2
D3
ϕ 2 · τ =
= τ · ϕ
e
2π
3
ϕ
τ
τ

2π
3
ϕ 2
ϕ, τ ϕ · τ
τ ϕ 2 ϕ·τ = τ ·ϕ 2 . ϕ 2 ·τ
τ · ϕ
τ · ϕ = ϕ · τ,
e 1 e
τ τ 2 = e 2 τ −1 = τ
ϕ ϕ 3 = e 3 ϕ −1 = ϕ 2
ϕ 2 (ϕ 2 ) 3 = (ϕ 3 ) 2 = e 3 (ϕ 2 ) −1 = ϕ
τ · ϕ = ϕ 2 · τ (τ · ϕ) 2 = τ · ϕ · τ · ϕ = τ · ϕ · ϕ 2 · τ = e 2 (τ · ϕ) −1 = τ · ϕ
ϕ · τ = τ · ϕ 2 (ϕ · τ) 2 = ϕ · τ · ϕ · τ = ϕ · τ · τ · ϕ 2 = e 2 (ϕ · τ) −1 = ϕ · τ
< ϕ >= {e, ϕ, ϕ 2 } =< ϕ 2 >, < τ >= {e, τ} < ϕ, τ >= G
< τ·ϕ >= {e, τ·ϕ} < ϕ·τ >= {e, ϕ·τ} < ϕ, ϕ·τ >= G =< ϕ·τ, τ·ϕ >
τ · ϕ = ϕ · τ
D3
G

{e} 1
{e, τ} 2
{e, ϕ · τ} 2
{e, τ · ϕ} 2
{e, ϕ, ϕ 2 } 3
{e, ϕ, ϕ 2 , τ, τ · ϕ, ϕ · τ} 6
D4
ϕ π
2
τ
G = {e, ϕ, ϕ 2 , ϕ 3 , τ, τ · ϕ, τ · ϕ 2 , τ · ϕ 3 }
ϕ 4 = τ 2 = e ϕ · τ = τ · ϕ 3
(G, ·) a a −1
G e
a ∈ G |a| = n,
a n = e, (∗)
a n e
a −1 n
a n
a −1 · a −1 · . . . · a −1 · a · a · . . . · a · a
e, a −1 · a = e, n
(a −1 ) n · a n = e (∗∗)

a n = e, (a −1 ) n = e |a −1 |
n. |a −1 | ≤ |a|.
a −1 a
|a| ≤ |a −1 |.
a a −1
|a −1 | = |a|.
a a −1
(G, ·) a b −1 ·a·b
G e
a ∈ G |a| = n,
a n = e, (∗)
a n e
b −1 · a · b n
(b −1 · a · b) · (b −1 · a · b) · . . . · (b −1 · a · b) =
= b −1 · a · b · b −1 · a · b · . . . · b −1 · a · b =
= b −1 · a n · b, (∗∗)
b b −1 e. a n = e,
b −1 · e · b = b −1 · b = e. |b −1 · a · b|
n. |b −1 · a · b| ≤ |a|.
b −1 · a · b |b −1 · a · b| = n,
(b −1 · a · b) n = e, (∗ ∗ ∗)
b −1 · a · b n e
(b −1 · a · b) n = b −1 · a n · b.
e.
b −1 · a n · b = e.

−1
a n = b · e · b −1
a n = e
|a| n. |a| ≤ |b −1 · a · b|,
a b −1 ·a·b
|a| = |b −1 · a · b|.
a b −1 · a · b
(G, ·)
G
(G, ·)
a ∈ G
a |G| = e,
e
|a| = n. |a| |G|, |G| = n · s
s
a |G| = a n·s = (a n ) s = e s = e
c c 100 = e c 1999 = e
c

c 100 = e, (∗)
c 1999 = e. (∗∗)
c 1999 = c 100·19+99 = (c 100 ) 19 · c 99 = c 99 .
e
c 100 = c 99 · c = e · c = c
c = e, c
c 99 = e. (∗ ∗ ∗)
a a n = e,
|a| n. |c| 100, |c| 1999,
|c| (100, 1999) = 1, c
(G, ·)
|a| = n. (∗)
a n = e, n
a
n p
n = p · k 1 < k < n
a n = a k·p = (a k ) p = e
a k p
a k p
a k

G
H τ G
H
H G
K ϕ G
K
K G
G = {e, ϕ, ϕ 2 , τ, τ ·ϕ = ϕ 2 ·τ, ϕ·τ = τ ·ϕ 2 }, H = {e, τ}
x · H, H · x
e e · H = {e, τ} H · e = {e, τ}
τ τ · H = {τ, e} H · τ = {τ, e}
ϕ ϕ · H = {ϕ, ϕ · τ} H · ϕ = {ϕ, τ · ϕ}
ϕ 2 ϕ 2 · H = {ϕ 2 , τ · ϕ} H · ϕ 2 = {ϕ 2 , ϕ · τ}
ϕ · τ ϕ · τ · H = {ϕ · τ, ϕ} H · ϕ · τ = {ϕ · τ, ϕ 2 }
τ · ϕ τ · ϕ · H = {τ · ϕ, ϕ 2 } H · τ · ϕ = {τ · ϕ, ϕ}
e τ
ϕ τ · ϕ
ϕ · τ ϕ 2
G H

H G
K =< ϕ >= {e, ϕ, ϕ 2 } G
x · K, K · x
e e · K = {e, ϕ, ϕ 2 } K · e = {e, ϕ, ϕ 2 }
τ τ · K = {τ, τ · ϕ, ϕ · τ} K · τ = {τ, ϕ · τ, τ · ϕ}
ϕ ϕ · K = {ϕ, ϕ 2 , e} K · ϕ = {ϕ, ϕ 2 , e}
ϕ 2 ϕ 2 · K = {ϕ 2 , e, ϕ} K · ϕ 2 = {ϕ 2 , e, ϕ}
ϕ · τ ϕ · τ · K = {ϕ · τ, τ, τ · ϕ} K · ϕ · τ = {ϕ · τ, τ · ϕ, τ}
τ · ϕ τ · ϕ · K = {τ · ϕ, ϕ · τ, τ} K · τ · ϕ = {τ · ϕ, τ, ϕ · τ, }
e ϕ ϕ 2
τ τ · ϕ ϕ · τ
G K
K G
C ∗ ◦
a ∗ b = a + b + 1, a ◦ b = a + b + i.
(C, ∗) (C, ◦)

ϕ : a ↦→ ai
(C, ∗) (C, ◦)
(C, ∗) ∗ a b
a + b + 1
−1 a −a − 2
(C, ◦) ◦ a b
a + b + i
−i a −a − 2i
a ↦→ ai
ai = bi a = b z + iw
x + iy z + iw
ϕ(a ∗ b) = ϕ(a + b + 1) = ai + bi + i (∗)
ϕ(a) ◦ ϕ(b) = (ai) ◦ (bi) = ai + bi + i (∗∗)
1 c
0 1
a a
a a
c ∈ R
a = 0, a ∈ R
(Q, +).

(R ∗ , ·)
(R, +)
c ↦→
x ↦→ e x
1 c
0 1
1 c
0 1
a ↦→
1 1
2a 2a 1 1
2a 2a (Q, +)
(R + , ·)
c ∈ R , ·
a a
a a
a = 0, a ∈ R , ·
c ↦→
ϕ(c1) · ϕ(c2) =
ϕ (c1 + c2) =
1 c1
0 1
1 c
0 1
1 c1 + c2
0 1
1 c2
·
0 1
=
1 c1 + c2
0 1
x ↦→ e x
ϕ(x + y) = e x+y = e x · e y = ϕ(x) · ϕ(y)

2.5
2
1.5
1
0.5
–4 –3 –2 –1 1
x
e x
x ↦→ ln(x)
2
0
–2
–4
–6
x
5 10 15 20 25 30
ln(x)

|a| = n, a n = e, ϕ(a n ) =
(ϕ(a)) n = e ′ , e ′
|a| = |ϕ(a)|.
−1
a a
a a
a a
a a
x x
x x
b b
·
b b
x x
·
x x
a a
·
a a
=
=
=
2ab 2ab
2ab 2ab
a a
a a
a a
a a
x x
x x
=
1
2
1
2
a ↦→
1 1
2a 2a 1 1
2a 2a
ϕ(a) · ϕ(b) =
1 1
2a 2a 1 1
2a 2a 1
·
2 b 1
2 b
1
2b 1
2b 1
2
1
2
=
1
2ab 1
2ab 1
2ab 1
2ab
= ϕ(ab)

D4
ϕ(15) = 8.
1 2 4 7 8 11 13 14
1 4 2 4 4 2 4 2
ϕ(24) = 8.
1 5 7 11 13 17 19 23
1 2 2 2 2 2 2 2
1 ε1 ε2 ε3 ε4 ε5 ε6 ε7
1 8 4 8 2 8 4 8
D4
e ϕ ϕ 2 ϕ 3 τ τ · ϕ τ · ϕ 2 τ · ϕ 3
1 4 2 4 2 2 2 2

a + b √ 2 | a, b ∈ Z
{a + bi | a, b ∈ Z}
(Zm, +, ·)
n, m n + x = m
n, m
n + x = m
H H
(H, +)
(R, +) · (a+b √ 2)−(c+d √ 2) = (a−c)+(b−c) √ 2
H − H ⊆ H H ⊆ R R
(a + b √ 2)(c + d √ 2) = (ac +
2bd)+(ad+cb) √ 2 H ·H ⊆ H
R
i · i = −1
Zm
a + x ≡ b

(mod m) a, b
Zm
(R, +, ·)
(R, +)
(R, ·)
(R, +, ·)
(R, +) a, b ∈ R
a + b = b + a (e + e)(a + b) e ∈ R
(R, ·) a, b ∈ R c = (e + e)
d = (a + b)
(e + e)(a + b) = c(a + b) = ca + cb = (e + e)a + (e + e)b = a + a + b + b,
(e + e)(a + b) = (e + e)d = ed + ed = e(a + b) + e(a + b) = a + b + a + b.
a + a + b + b = a + b + a + b.
a + b = b + a
(R, +, ·)
a · 0 = 0 a ∈ R 0 ∈ R
0
0 · 0 −1 = e e ∈ R 0 · 0 −1 = 0
e = 0 a ∈ R a = a · e = 0
mod 2m
0, 2, 4, . . . , 2m − 2

2m = 10
2m = 20
0, 2, 4, 6, 8
(Zm, +, ·)
(2a + l1 · 10) + (2b + l2 · 10) =
2(a+b)+(l1+l2)10 (a+b) = c+l·5 0 ≤ c ≤ 4 2(a+b) = 2c+l·10
2c {0, 2, 4, 6, 8}
2a + 2b = 2(a + b) 0 x
2m − x
(R ∗ , ·)
·
6 (6 · 2 = 2 ; 6 · 4 = 4 ; 6 · 6 =
6 ; 6 · 8 = 8)
8 8
6 2
8 · 2 = 6 8 2
0, 2, 4, 6, 8, 10, 12, 14, 16, 18
2 · 10 = 0

(T, +, ·)
(T ∗ , ·) T
a ∈ T ∗ a · x
T ∗ x T ∗
x1, x2 ∈ T ∗ x1 = x2 a · x1 = a · x2
a · x1 = a · x2
a · x1 a · (x1 − x2) = 0 x1 − x2 0
T T ∗ x T ∗ a·x
T ∗ a·x = b
a, b ∈ T ∗ (T ∗ , ·) (T, +, ·)
0
(2, 6) ; (3, 4) ; (6, 10) ; (6, 8) ; (4, 9) ; (6, 6) ; (3, 8) ; (9, 8) ; (4, 6)
(R, +, ·) 0
e a ∈ R
a n = 0 n ∈ N a e − a
e
(e − a)(e + a + · · · + a n−1 ) = e(e + a + · · · + a n−1 ) − a(e + a + · · · + a n−1 ) =
= e + a + · · · + a n−1 − a − a 2 − · · · − a n =
= e − a n = e.

e − a (e + a + · · · + a n−1 )
(R, +, ·) a
a
a
ab = 0 b = 0 a −1 (ab) = (a −1 a)b = b = 0
R a a 2 = a
R
(a + a) = (a + a) 2
(a + a) 2 = (a + a)a + (a + a)a = a 2 + a 2 + a 2 + a 2 .
a 2 + a 2 + a 2 + a 2 = a + a + a + a.
(a + a) = a + a + a + a a + a = 0
(R) = 2
(a + b) = (a + b) 2
a, b ∈ R
(a + b) = (a + b) 2 =
= (a + b)a + (a + b)b = a 2 + ba + ab + b 2 = a + ba + ab + b,
ba + ab = 0 (R) = 2 ab + ab = 0
ba = ab

Z10
a Zm
a a
Z10
·
0 1, 3, 7, 9
{1x : 1 ≤ x ≤ 10}
{3x : 1 ≤ x ≤ 10} {7x : 1 ≤ x ≤ 10} {9x : 1 ≤ x ≤ 10}
1, 3, 7, 9 2, 4, 6, 8
2∗2 ≡ 4, 4∗8 ≡ 2, 2∗8 ≡ 6, 6∗2 ≡ 2, 2∗4 ≡ 8, 8∗4 ≡ 2, 4∗4 ≡

6, 6 ∗ 4 ≡ 4, 4 ∗ 2 ≡ 8, 8 ∗ 8 ≡ 4, 6 ∗ 8 ≡ 8, 8 ∗ 2 ≡ 6 (mod 10)
5 0
5 1, 3, 5, 7, 9
1, 3, 7, 9
2, 4, 6, 8, 5
a m {ax : 1 ≤ x ≤ m}
m b x ax = b
a a m b
x ax = b ax ≡ b (mod m)
(a, m)|b a
(a, m) = 1
Z10 5
Z10 5
5 · 5 = 5 5 ≡ 5 2 (mod 10) 5
Z10
5 Z10 5|a · b 5|a 5|b
(T, +, ·) (T ∗ , ·) T ∗
ax = b a, b ∈ T ∗
T ∗

p
p|p · p p p|p ɛ
ɛp = p a
aɛp = ap (aɛ − a)p = 0 aɛ − a = 0
aɛ = a a
ɛ
L := a + bi √ 5 | a, b ∈ Z
L 1+i √ 5, 1−i √ 5, 2, 3
(L, +, ·)
(1 + i √ 5)(1 − i √ 5) = 1 + 5 = 6 = 2 · 3 1 ± i √ 5|2 · 3
(1 ± i √ 5)(a + bi √ 5) 2, 3 a, b
1 ± i √ 5 |2 1 ± i √ 5 |3
2 3
1 + i √ 5 = αβ (1 + i √ 5)(1 − i √ 5) = αβαβ = αβαβ
(1 + i √ 5)(1 − i √ 5) = 6 αα ββ ααββ = 1 · 6 2 · 3
αα = (a + bi √ 5)(a − bi √ 5) = a 2 + 5b 2 = 2 a, b
ααββ = 2 · 3 α β
α = a + bi √ 5 1 = αα = a 2 + 5b 2
a = ±1 α 1+i √ 5 = αβ
α β
L
(Z4, +, ·)

{0} I − I ⊆ I I · I ⊆ I
0 − 0 = 0 r · 0 = 0 {0, 2} I − I ⊆ I I · I ⊆ I
0 − 2 = 2 r · 2 = 0 2 r
{0, 1, 2, 3}
{0, 2, 3} 2 + 3 = 1
1
R I R R = I
I R
i ∈ I ri
r R ri ∈ I r ∈ R
I R R = I
r1, r2 ∈ R r1 = r2 r1i = r2i
(r1 − r2)i = 0 i R
(T, +, ·) T
{0} T T
{0} = I ⊆ T 0 = t ∈ I e = t −1 t ∈ I
T = T {e} ⊆ T I ⊆ I I = T
(P )
Z/P
P − P ⊆ P
ZP ⊆ P P Z ⊆
P

Z P 0 ∈ Z
0 + P
0 1
1 + P 1
Z/P 0 1
R =
a b
| a, b, c, d ∈ Z I =
c d
a b
| a, b, c, d ∈ 2Z
c d
I R
R/I
I
I−I ⊆ I RI ⊆ I
S M ∈ R, S ∈ I
m11 M =
m12
m21 m22
m11s11 + m12s21 m11s12 + m12s22
MS =
m21s11 + m22s21 m21s12 + m22s22
s11 =
s12
s21 s22
MS S MS ∈ I
RI ⊆ I
s11m11 + s12m21 s11m12 + s12m22
SM =
,
s21m11 + s22m21 s21m12 + s22m22
IR ⊆ I I R
0 ∈ R
0
0 0+I

M ∈ R
M
4
1
4 2
4 4 4 4
1 + 2 + 3 + 4 = 24
24
N R
0
N
N
N R/N
Z6 2, 3 ∈ N 2 + 3 /∈ N
n ∈ N \{0} m ∈ M \{0}
∀r ∈ R \ {0} (rn)m = r(nm) = 0 rn ∈ R
0 RN ⊆ N
R/N N N
a1, a2 ∈ R\N a1a2 /∈ N (a1+N)(a2+N) = (a1a2+N)
N a1 + N a2 + N

a b
M =
| a, b ∈ Z
2b a
(M, +, ·) E = ( a + b √ 2 | a, b ∈ Z , +, ·)
a, b ∈ Z ϕ : M → E ϕ
a
b
2b a
a + b √ 2 a + b √ 2
M
ϕ
ϕ
a1 b1
2b1 a1
+
a2 b2
2b2 a2
a1 + a2 b1 + b2
= ϕ
=
2(b1 + b2) a1 + a2
= (a1 + a2) + (b1 + b2) √ 2 =
√ √
= a1 + b1 2 + a2 + b2 2 = ϕ
a1 b1
2b1 a1
·
a2 b2
2b2 a2
a1 b1
2b1 a1
+ ϕ
a2 b2
2b2 a2
a1a2 + 2b1b2 a1b2 + b1a2
= ϕ
=
2(a1b2 + b1a2) a1a2 + 2b1b2
= (a1a2 + 2b1b2) + (a1b2 + b1a2) √ 2 =
√ √
= (a1 + b1 2)(a2 + b2 2) = ϕ
a1 b1
2b1 a1
· ϕ
a2 b2
2b2 a2
G = ( a + b √ 2 | a, b ∈ Z , +, ·) K = ( a + b √ 3 | a, b ∈ Z , +, ·)
ϕ
G K
ϕ(u) = ϕ(1 · u) = ϕ(1)ϕ(u) u ∈ G ϕ(1) = 1 ϕ(2) =
ϕ(1 + 1) = ϕ(1) + ϕ(1) = 1 + 1 = 2 ϕ(m) = m
=

m ∈ Z G x 2 = 2 ϕ(x 2 ) =
ϕ(2) = 2 K
ϕ(x) = a + b √ 3
(a + b √ 3) 2 = 2
ϕ(x 2 ) = (ϕ(x)) 2 = 2
a 2 + 2ab √ 3 + 3b 2 = 2
a 2 + 3b 2 = 2
2ab = 0
b 2 = 0 a 2 = 2 a
K G K

m ∈ Z G x 2 = 2 ϕ(x 2 ) =
ϕ(2) = 2 K
ϕ(x) = a + b √ 3
(a + b √ 3) 2 = 2
ϕ(x 2 ) = (ϕ(x)) 2 = 2
a 2 + 2ab √ 3 + 3b 2 = 2
a 2 + 3b 2 = 2
2ab = 0
b 2 = 0 a 2 = 2 a
K G K