G = (2, ") "
a " b = ab - 2a - 2b + 6, a, b " G.
G
"
ab - 2a - 2b + 6 = (a - 2) (b - 2) + 2.
a > 2 b > 2 (a - 2) (b - 2) > 0
a " b = ab - 2a - 2b + 6 = (a - 2) (b - 2) + 2 > 2.
a, b " G a " b " G "
G.
" . a, b, c
G.
(a " b) " c = (ab - 2a - 2b + 6) " c = (ab - 2a - 2b + 6) c - 2 (ab - 2a - 2b + 6) - 2c + 6 =
= abc - 2ab - 2ac - 2bc + 4a + 4b + 4c - 6,
a " (b " c) = a " (bc - 2b - 2c + 6) = a (bc - 2b - 2c + 6) - 2a - 2 (bc - 2b - 2c + 6) + 6 =
= abc - 2ab - 2ac - 2bc + 4a + 4b + 4c - 6.
(a " b) " c = a " (b " c) "
a, b
G.
a " b = ab - 2a - 2b + 6 = ba - 2b - 2a + 6 = b " a.
"
" . "
e
a " e = a, a " G.
ae - 2a - 2e + 6 = a.
 e = 3. 3 G,
e = 3 " .
G. a " G " b
a " b = 3
ab - 2a - 2b + 6 = 3.
2a - 3
b = .
a - 2
2a - 3 2a - 3
G, > 2. a > 2, a - 2 > 0.
a - 2 a - 2
2a-3 > 2 (a - 2) . -3 > -4 a
2a - 3
a-1 = .
a - 2
G "
R+.
Õ
Õ (x) = x - 2, x " (2, ")
Õ (a " b) = (ab - 2a - 2b + 6) - 2 = ab - 2a - 2b + 4 = (a - 2) (b - 2) = Õ (a) Õ (b) .
Õ Õ (a) = Õ (b) , a - 2 = b - 2,
a = b.
Õ (2, ") R+.
c
c a " (2, ") Õ (a) = c. a = c + 2. Õ (a) =
Õ (c + 2) = (c + 2) - 2 = c. c " R+ a " (2, ")
Õ (a) = c.
R+.
 "
A = a + b 2 : a, b " Q\ {0}
"
" x, y " A x y-1 " A. x = a + b 2
y = c + d 2, a, b, c, d " a, b " Q\ {0} .
" "
"
" " -1
a + b 2 c - d 2
a + b 2
x y-1 = a + b 2 c + d 2 = " = =
c2 - 2d2
c + d 2
"
"
ac - 2bd + (-ad + bc) 2 ac - 2bd -ad + bc
= = + 2.
c2 - 2d2 c2 - 2d2 c2 - 2d2
"
c2 - 2d2 = 0 c2 - 2d2 = 0, c2 = 2d2 c = Ä…d 2

c
"
ac - 2bd -ad + bc
x y-1 = + 2
c2 - 2d2 c2 - 2d2
A A
A
G = {0, 1, 2, ..., n - 1} +n n
rn (a + b) = rn (rna + b) = rn (a + rnb) , a, b " Z. "
rna a
n
n a +n b = rn (a + b) , a, b " Z.
+n G
n G
+n . a, b, c
G. (")
(a +n b) +n c = rn (a + b) +n c = rn (rn (a + b) + c) = rn (a + b + c) ,
a +n (b +n c) = a +n rn (b + c) = rn (a + rn (b + c)) = rn (a + b + c) .
 (a +n b) +n c = a +n (b +n c) +n
+n
a, b G.
a +n b = rn (a + b) = rn (b + a) = b +n a.
+n
0
a +n e = a e
+n a +n e = rn (a + e) = a a + e
{0, 1, 2, ..., n - 1, n, ..., 2n - 2, } .
o
a + e " {0, 1, 2, ..., n - 1} rn (a + e) = a + e = a e = 0
o
a + e " {n, ..., 2n - 2, } rn (a + e) = a + e - n = a e = n.
n " G. e = 0.
/
a " G.
a
a +n b = e b a +n b = rn (a + b) = e
a + b {0, 1, 2, ..., n - 1, n, ..., 2n - 2, } .
o
a + b " {0, 1, 2, ..., n - 1} rn (a + b) = a + b = 0 a = -b
o
a + b " {n, ..., 2n - 2, } rn (a + e) = a + b - n = 0
b = n - a.
G a = -b 1o
0. a = b = 0 0
o
0. a = 0 b

b = n - a