Article GROUP_3, MML version 4.99.1005
:: GROUP_3:th 1
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
(b2 * b3) * (b3 ") = b2 & (b2 * (b3 ")) * b3 = b2 & (b3 " * b3) * b2 = b2 & (b3 * (b3 ")) * b2 = b2 & b2 * (b3 * (b3 ")) = b2 & b2 * (b3 " * b3) = b2 & b3 " * (b3 * b2) = b2 & b3 * (b3 " * b2) = b2;
:: GROUP_3:th 2
theorem
for b1 being non empty Group-like associative multMagma holds
b1 is non empty Group-like associative commutative multMagma
iff
the multF of b1 is commutative(the carrier of b1);
:: GROUP_3:th 3
theorem
for b1 being non empty Group-like associative multMagma holds
(1). b1 is commutative;
:: GROUP_3:th 4
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4, b5 being Element of bool the carrier of b1
st b2 c= b3 & b4 c= b5
holds b2 * b4 c= b3 * b5;
:: GROUP_3:th 5
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1
st b3 c= b4
holds b2 * b3 c= b2 * b4 & b3 * b2 c= b4 * b2;
:: GROUP_3:th 6
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3, b4 being Subgroup of b1
st b3 is Subgroup of b4
holds b2 * b3 c= b2 * b4 & b3 * b2 c= b4 * b2;
:: GROUP_3:th 7
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Subgroup of b1 holds
b2 * b3 = {b2} * b3;
:: GROUP_3:th 8
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Subgroup of b1 holds
b3 * b2 = b3 * {b2};
:: GROUP_3:th 10
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being Subgroup of b1 holds
(b3 * b2) * b4 = b3 * (b2 * b4);
:: GROUP_3:th 11
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being Subgroup of b1 holds
(b2 * b4) * b3 = b2 * (b4 * b3);
:: GROUP_3:th 12
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being Subgroup of b1 holds
(b3 * b4) * b2 = b3 * (b4 * b2);
:: GROUP_3:th 13
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1
for b4 being Subgroup of b1 holds
(b4 * b2) * b3 = b4 * (b2 * b3);
:: GROUP_3:th 15
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3, b4 being Subgroup of b1 holds
(b3 * b2) * b4 = b3 * (b2 * b4);
:: GROUP_3:funcnot 1 => GROUP_3:func 1
definition
let a1 be non empty Group-like associative multMagma;
func Subgroups A1 -> set means
for b1 being set holds
b1 in it
iff
b1 is strict Subgroup of a1;
end;
:: GROUP_3:def 1
theorem
for b1 being non empty Group-like associative multMagma
for b2 being set holds
b2 = Subgroups b1
iff
for b3 being set holds
b3 in b2
iff
b3 is strict Subgroup of b1;
:: GROUP_3:funcreg 1
registration
let a1 be non empty Group-like associative multMagma;
cluster Subgroups a1 -> non empty;
end;
:: GROUP_3:th 18
theorem
for b1 being non empty strict Group-like associative multMagma holds
b1 in Subgroups b1;
:: GROUP_3:th 19
theorem
for b1 being non empty Group-like associative multMagma
st b1 is finite
holds Subgroups b1 is finite;
:: GROUP_3:funcnot 2 => GROUP_3:func 2
definition
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Element of the carrier of a1;
func A2 |^ A3 -> Element of the carrier of a1 equals
(a3 " * a2) * a3;
end;
:: GROUP_3:def 2
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
b2 |^ b3 = (b3 " * b2) * b3;
:: GROUP_3:th 21
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4 being Element of the carrier of b1
st b2 |^ b3 = b4 |^ b3
holds b2 = b4;
:: GROUP_3:th 22
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
(1_ b1) |^ b2 = 1_ b1;
:: GROUP_3:th 23
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
st b2 |^ b3 = 1_ b1
holds b2 = 1_ b1;
:: GROUP_3:th 24
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
b2 |^ 1_ b1 = b2;
:: GROUP_3:th 25
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
b2 |^ b2 = b2;
:: GROUP_3:th 26
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
b2 |^ (b2 ") = b2 & b2 " |^ b2 = b2 ";
:: GROUP_3:th 27
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
b2 |^ b3 = b2
iff
b2 * b3 = b3 * b2;
:: GROUP_3:th 28
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 * b3) |^ b4 = (b2 |^ b4) * (b3 |^ b4);
:: GROUP_3:th 29
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4 being Element of the carrier of b1 holds
(b2 |^ b3) |^ b4 = b2 |^ (b3 * b4);
:: GROUP_3:th 30
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
(b2 |^ b3) |^ (b3 ") = b2 & (b2 |^ (b3 ")) |^ b3 = b2;
:: GROUP_3:th 32
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
b2 " |^ b3 = (b2 |^ b3) ";
:: GROUP_3:th 33
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
for b4 being natural set holds
(b2 |^ b4) |^ b3 = (b2 |^ b3) |^ b4;
:: GROUP_3:th 34
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
for b4 being integer set holds
(b2 |^ b4) |^ b3 = (b2 |^ b3) |^ b4;
:: GROUP_3:th 35
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
st b1 is non empty Group-like associative commutative multMagma
holds b2 |^ b3 = b2;
:: GROUP_3:th 36
theorem
for b1 being non empty Group-like associative multMagma
st for b2, b3 being Element of the carrier of b1 holds
b2 |^ b3 = b2
holds b1 is commutative;
:: GROUP_3:funcnot 3 => GROUP_3:func 3
definition
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Element of bool the carrier of a1;
func A2 |^ A3 -> Element of bool the carrier of a1 equals
{b1 |^ b2 where b1 is Element of the carrier of a1, b2 is Element of the carrier of a1: b1 in a2 & b2 in a3};
end;
:: GROUP_3:def 3
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1 holds
b2 |^ b3 = {b4 |^ b5 where b4 is Element of the carrier of b1, b5 is Element of the carrier of b1: b4 in b2 & b5 in b3};
:: GROUP_3:th 38
theorem
for b1 being set
for b2 being non empty Group-like associative multMagma
for b3, b4 being Element of bool the carrier of b2 holds
b1 in b3 |^ b4
iff
ex b5, b6 being Element of the carrier of b2 st
b1 = b5 |^ b6 & b5 in b3 & b6 in b4;
:: GROUP_3:th 39
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1 holds
b2 |^ b3 <> {}
iff
b2 <> {} & b3 <> {};
:: GROUP_3:th 40
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1 holds
b2 |^ b3 c= (b3 " * b2) * b3;
:: GROUP_3:th 41
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4 being Element of bool the carrier of b1 holds
(b2 * b3) |^ b4 c= (b2 |^ b4) * (b3 |^ b4);
:: GROUP_3:th 42
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4 being Element of bool the carrier of b1 holds
(b2 |^ b3) |^ b4 = b2 |^ (b3 * b4);
:: GROUP_3:th 43
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1 holds
b2 " |^ b3 = (b2 |^ b3) ";
:: GROUP_3:th 44
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
{b2} |^ {b3} = {b2 |^ b3};
:: GROUP_3:th 45
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4 being Element of the carrier of b1 holds
{b2} |^ {b3,b4} = {b2 |^ b3,b2 |^ b4};
:: GROUP_3:th 46
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4 being Element of the carrier of b1 holds
{b2,b3} |^ {b4} = {b2 |^ b4,b3 |^ b4};
:: GROUP_3:th 47
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4, b5 being Element of the carrier of b1 holds
{b2,b3} |^ {b4,b5} = {b2 |^ b4,b2 |^ b5,b3 |^ b4,b3 |^ b5};
:: GROUP_3:funcnot 4 => GROUP_3:func 4
definition
let a1 be non empty Group-like associative multMagma;
let a2 be Element of bool the carrier of a1;
let a3 be Element of the carrier of a1;
func A2 |^ A3 -> Element of bool the carrier of a1 equals
a2 |^ {a3};
end;
:: GROUP_3:def 4
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b2 |^ b3 = b2 |^ {b3};
:: GROUP_3:funcnot 5 => GROUP_3:func 5
definition
let a1 be non empty Group-like associative multMagma;
let a2 be Element of bool the carrier of a1;
let a3 be Element of the carrier of a1;
func A3 |^ A2 -> Element of bool the carrier of a1 equals
{a3} |^ a2;
end;
:: GROUP_3:def 5
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b3 |^ b2 = {b3} |^ b2;
:: GROUP_3:th 50
theorem
for b1 being set
for b2 being non empty Group-like associative multMagma
for b3 being Element of the carrier of b2
for b4 being Element of bool the carrier of b2 holds
b1 in b4 |^ b3
iff
ex b5 being Element of the carrier of b2 st
b1 = b5 |^ b3 & b5 in b4;
:: GROUP_3:th 51
theorem
for b1 being set
for b2 being non empty Group-like associative multMagma
for b3 being Element of the carrier of b2
for b4 being Element of bool the carrier of b2 holds
b1 in b3 |^ b4
iff
ex b5 being Element of the carrier of b2 st
b1 = b3 |^ b5 & b5 in b4;
:: GROUP_3:th 52
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b2 |^ b3 c= (b3 " * b2) * b3;
:: GROUP_3:th 53
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1 holds
(b3 |^ b4) |^ b2 = b3 |^ (b4 * b2);
:: GROUP_3:th 54
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1 holds
(b3 |^ b2) |^ b4 = b3 |^ (b2 * b4);
:: GROUP_3:th 55
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1 holds
(b2 |^ b3) |^ b4 = b2 |^ (b3 * b4);
:: GROUP_3:th 56
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
(b4 |^ b2) |^ b3 = b4 |^ (b2 * b3);
:: GROUP_3:th 57
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
(b2 |^ b4) |^ b3 = b2 |^ (b4 * b3);
:: GROUP_3:th 58
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
for b4 being Element of bool the carrier of b1 holds
(b2 |^ b3) |^ b4 = b2 |^ (b3 * b4);
:: GROUP_3:th 59
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 |^ b2 = (b2 " * b3) * b2;
:: GROUP_3:th 60
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3, b4 being Element of bool the carrier of b1 holds
(b3 * b4) |^ b2 c= (b3 |^ b2) * (b4 |^ b2);
:: GROUP_3:th 61
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1 holds
b2 |^ 1_ b1 = b2;
:: GROUP_3:th 62
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
st b2 <> {}
holds (1_ b1) |^ b2 = {1_ b1};
:: GROUP_3:th 63
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
(b3 |^ b2) |^ (b2 ") = b3 & (b3 |^ (b2 ")) |^ b2 = b3;
:: GROUP_3:th 65
theorem
for b1 being non empty Group-like associative multMagma holds
b1 is non empty Group-like associative commutative multMagma
iff
for b2, b3 being Element of bool the carrier of b1
st b3 <> {}
holds b2 |^ b3 = b2;
:: GROUP_3:th 66
theorem
for b1 being non empty Group-like associative multMagma holds
b1 is non empty Group-like associative commutative multMagma
iff
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
b2 |^ b3 = b2;
:: GROUP_3:th 67
theorem
for b1 being non empty Group-like associative multMagma holds
b1 is non empty Group-like associative commutative multMagma
iff
for b2 being Element of bool the carrier of b1
for b3 being Element of the carrier of b1
st b2 <> {}
holds b3 |^ b2 = {b3};
:: GROUP_3:funcnot 6 => GROUP_3:func 6
definition
let a1 be non empty Group-like associative multMagma;
let a2 be Subgroup of a1;
let a3 be Element of the carrier of a1;
func A2 |^ A3 -> strict Subgroup of a1 means
the carrier of it = (carr a2) |^ a3;
end;
:: GROUP_3:def 6
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being Element of the carrier of b1
for b4 being strict Subgroup of b1 holds
b4 = b2 |^ b3
iff
the carrier of b4 = (carr b2) |^ b3;
:: GROUP_3:th 70
theorem
for b1 being set
for b2 being non empty Group-like associative multMagma
for b3 being Element of the carrier of b2
for b4 being Subgroup of b2 holds
b1 in b4 |^ b3
iff
ex b5 being Element of the carrier of b2 st
b1 = b5 |^ b3 & b5 in b4;
:: GROUP_3:th 71
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Subgroup of b1 holds
the carrier of b3 |^ b2 = (b2 " * b3) * b2;
:: GROUP_3:th 72
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
for b4 being Subgroup of b1 holds
(b4 |^ b2) |^ b3 = b4 |^ (b2 * b3);
:: GROUP_3:th 73
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
b2 |^ 1_ b1 = b2;
:: GROUP_3:th 74
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being strict Subgroup of b1 holds
(b3 |^ b2) |^ (b2 ") = b3 & (b3 |^ (b2 ")) |^ b2 = b3;
:: GROUP_3:th 76
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3, b4 being Subgroup of b1 holds
(b3 /\ b4) |^ b2 = (b3 |^ b2) /\ (b4 |^ b2);
:: GROUP_3:th 77
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Subgroup of b1 holds
Ord b3 = Ord (b3 |^ b2);
:: GROUP_3:th 78
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Subgroup of b1 holds
b3 is finite
iff
b3 |^ b2 is finite;
:: GROUP_3:funcreg 2
registration
let a1 be non empty Group-like associative multMagma;
let a2 be Element of the carrier of a1;
let a3 be finite Subgroup of a1;
cluster a3 |^ a2 -> finite strict;
end;
:: GROUP_3:th 79
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being finite Subgroup of b1 holds
ord b3 = ord (b3 |^ b2);
:: GROUP_3:th 80
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
((1). b1) |^ b2 = (1). b1;
:: GROUP_3:th 81
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being strict Subgroup of b1
st b3 |^ b2 = (1). b1
holds b3 = (1). b1;
:: GROUP_3:th 82
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
((Omega). b1) |^ b2 = (Omega). b1;
:: GROUP_3:th 83
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being strict Subgroup of b1
st b3 |^ b2 = b1
holds b3 = b1;
:: GROUP_3:th 84
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Subgroup of b1 holds
Index b3 = Index (b3 |^ b2);
:: GROUP_3:th 85
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Subgroup of b1
st Left_Cosets b3 is finite
holds index b3 = index (b3 |^ b2);
:: GROUP_3:th 86
theorem
for b1 being non empty Group-like associative multMagma
st b1 is non empty Group-like associative commutative multMagma
for b2 being strict Subgroup of b1
for b3 being Element of the carrier of b1 holds
b2 |^ b3 = b2;
:: GROUP_3:prednot 1 => GROUP_3:pred 1
definition
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Element of the carrier of a1;
pred A2,A3 are_conjugated means
ex b1 being Element of the carrier of a1 st
a2 = a3 |^ b1;
end;
:: GROUP_3:dfs 7
definiens
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Element of the carrier of a1;
To prove
a2,a3 are_conjugated
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
a2 = a3 |^ b1;
:: GROUP_3:def 7
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
b2,b3 are_conjugated
iff
ex b4 being Element of the carrier of b1 st
b2 = b3 |^ b4;
:: GROUP_3:prednot 2 => not GROUP_3:pred 1
notation
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Element of the carrier of a1;
antonym a2,a3 are_not_conjugated for a2,a3 are_conjugated;
end;
:: GROUP_3:th 88
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
b2,b3 are_conjugated
iff
ex b4 being Element of the carrier of b1 st
b3 = b2 |^ b4;
:: GROUP_3:th 89
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
b2,b2 are_conjugated;
:: GROUP_3:th 90
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
st b2,b3 are_conjugated
holds b3,b2 are_conjugated;
:: GROUP_3:prednot 3 => GROUP_3:pred 2
definition
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Element of the carrier of a1;
redefine pred a2,a3 are_conjugated;
symmetry;
:: for a1 being non empty Group-like associative multMagma
:: for a2, a3 being Element of the carrier of a1
:: st a2,a3 are_conjugated
:: holds a3,a2 are_conjugated;
reflexivity;
:: for a1 being non empty Group-like associative multMagma
:: for a2 being Element of the carrier of a1 holds
:: a2,a2 are_conjugated;
end;
:: GROUP_3:th 91
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4 being Element of the carrier of b1
st b2,b3 are_conjugated & b3,b4 are_conjugated
holds b2,b4 are_conjugated;
:: GROUP_3:th 92
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
st (b2,1_ b1 are_conjugated or 1_ b1,b2 are_conjugated)
holds b2 = 1_ b1;
:: GROUP_3:th 93
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
b2 |^ carr (Omega). b1 = {b3 where b3 is Element of the carrier of b1: b2,b3 are_conjugated};
:: GROUP_3:funcnot 7 => GROUP_3:func 7
definition
let a1 be non empty Group-like associative multMagma;
let a2 be Element of the carrier of a1;
func con_class A2 -> Element of bool the carrier of a1 equals
a2 |^ carr (Omega). a1;
end;
:: GROUP_3:def 8
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
con_class b2 = b2 |^ carr (Omega). b1;
:: GROUP_3:th 95
theorem
for b1 being set
for b2 being non empty Group-like associative multMagma
for b3 being Element of the carrier of b2 holds
b1 in con_class b3
iff
ex b4 being Element of the carrier of b2 st
b4 = b1 & b3,b4 are_conjugated;
:: GROUP_3:th 96
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
b2 in con_class b3
iff
b2,b3 are_conjugated;
:: GROUP_3:th 97
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
b2 |^ b3 in con_class b2;
:: GROUP_3:th 98
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
b2 in con_class b2;
:: GROUP_3:th 99
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
st b2 in con_class b3
holds b3 in con_class b2;
:: GROUP_3:th 100
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
con_class b2 = con_class b3
iff
con_class b2 meets con_class b3;
:: GROUP_3:th 101
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
con_class b2 = {1_ b1}
iff
b2 = 1_ b1;
:: GROUP_3:th 102
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
(con_class b2) * b3 = b3 * con_class b2;
:: GROUP_3:prednot 4 => GROUP_3:pred 3
definition
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Element of bool the carrier of a1;
pred A2,A3 are_conjugated means
ex b1 being Element of the carrier of a1 st
a2 = a3 |^ b1;
end;
:: GROUP_3:dfs 9
definiens
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Element of bool the carrier of a1;
To prove
a2,a3 are_conjugated
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
a2 = a3 |^ b1;
:: GROUP_3:def 9
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1 holds
b2,b3 are_conjugated
iff
ex b4 being Element of the carrier of b1 st
b2 = b3 |^ b4;
:: GROUP_3:prednot 5 => not GROUP_3:pred 3
notation
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Element of bool the carrier of a1;
antonym a2,a3 are_not_conjugated for a2,a3 are_conjugated;
end;
:: GROUP_3:th 104
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1 holds
b2,b3 are_conjugated
iff
ex b4 being Element of the carrier of b1 st
b3 = b2 |^ b4;
:: GROUP_3:th 105
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1 holds
b2,b2 are_conjugated;
:: GROUP_3:th 106
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1
st b2,b3 are_conjugated
holds b3,b2 are_conjugated;
:: GROUP_3:prednot 6 => GROUP_3:pred 4
definition
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Element of bool the carrier of a1;
redefine pred a2,a3 are_conjugated;
symmetry;
:: for a1 being non empty Group-like associative multMagma
:: for a2, a3 being Element of bool the carrier of a1
:: st a2,a3 are_conjugated
:: holds a3,a2 are_conjugated;
reflexivity;
:: for a1 being non empty Group-like associative multMagma
:: for a2 being Element of bool the carrier of a1 holds
:: a2,a2 are_conjugated;
end;
:: GROUP_3:th 107
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4 being Element of bool the carrier of b1
st b2,b3 are_conjugated & b3,b4 are_conjugated
holds b2,b4 are_conjugated;
:: GROUP_3:th 108
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
{b2},{b3} are_conjugated
iff
b2,b3 are_conjugated;
:: GROUP_3:th 109
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
for b3 being Subgroup of b1
st b2,carr b3 are_conjugated
holds ex b4 being strict Subgroup of b1 st
the carrier of b4 = b2;
:: GROUP_3:funcnot 8 => GROUP_3:func 8
definition
let a1 be non empty Group-like associative multMagma;
let a2 be Element of bool the carrier of a1;
func con_class A2 -> Element of bool bool the carrier of a1 equals
{b1 where b1 is Element of bool the carrier of a1: a2,b1 are_conjugated};
end;
:: GROUP_3:def 10
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1 holds
con_class b2 = {b3 where b3 is Element of bool the carrier of b1: b2,b3 are_conjugated};
:: GROUP_3:th 111
theorem
for b1 being set
for b2 being non empty Group-like associative multMagma
for b3 being Element of bool the carrier of b2 holds
b1 in con_class b3
iff
ex b4 being Element of bool the carrier of b2 st
b1 = b4 & b3,b4 are_conjugated;
:: GROUP_3:th 113
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1 holds
b2 in con_class b3
iff
b2,b3 are_conjugated;
:: GROUP_3:th 114
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
b3 |^ b2 in con_class b3;
:: GROUP_3:th 115
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1 holds
b2 in con_class b2;
:: GROUP_3:th 116
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1
st b2 in con_class b3
holds b3 in con_class b2;
:: GROUP_3:th 117
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Element of bool the carrier of b1 holds
con_class b2 = con_class b3
iff
con_class b2 meets con_class b3;
:: GROUP_3:th 118
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
con_class {b2} = {{b3} where b3 is Element of the carrier of b1: b3 in con_class b2};
:: GROUP_3:th 119
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
st b1 is finite
holds con_class b2 is finite;
:: GROUP_3:prednot 7 => GROUP_3:pred 5
definition
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Subgroup of a1;
pred A2,A3 are_conjugated means
ex b1 being Element of the carrier of a1 st
multMagma(#the carrier of a2,the multF of a2#) = a3 |^ b1;
end;
:: GROUP_3:dfs 11
definiens
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Subgroup of a1;
To prove
a2,a3 are_conjugated
it is sufficient to prove
thus ex b1 being Element of the carrier of a1 st
multMagma(#the carrier of a2,the multF of a2#) = a3 |^ b1;
:: GROUP_3:def 11
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Subgroup of b1 holds
b2,b3 are_conjugated
iff
ex b4 being Element of the carrier of b1 st
multMagma(#the carrier of b2,the multF of b2#) = b3 |^ b4;
:: GROUP_3:prednot 8 => not GROUP_3:pred 5
notation
let a1 be non empty Group-like associative multMagma;
let a2, a3 be Subgroup of a1;
antonym a2,a3 are_not_conjugated for a2,a3 are_conjugated;
end;
:: GROUP_3:th 121
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict Subgroup of b1 holds
b2,b3 are_conjugated
iff
ex b4 being Element of the carrier of b1 st
b3 = b2 |^ b4;
:: GROUP_3:th 122
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
b2,b2 are_conjugated;
:: GROUP_3:th 123
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict Subgroup of b1
st b2,b3 are_conjugated
holds b3,b2 are_conjugated;
:: GROUP_3:prednot 9 => GROUP_3:pred 6
definition
let a1 be non empty Group-like associative multMagma;
let a2, a3 be strict Subgroup of a1;
redefine pred a2,a3 are_conjugated;
symmetry;
:: for a1 being non empty Group-like associative multMagma
:: for a2, a3 being strict Subgroup of a1
:: st a2,a3 are_conjugated
:: holds a3,a2 are_conjugated;
reflexivity;
:: for a1 being non empty Group-like associative multMagma
:: for a2 being strict Subgroup of a1 holds
:: a2,a2 are_conjugated;
end;
:: GROUP_3:th 124
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3, b4 being strict Subgroup of b1
st b3,b4 are_conjugated & b4,b2 are_conjugated
holds b3,b2 are_conjugated;
:: GROUP_3:funcnot 9 => GROUP_3:func 9
definition
let a1 be non empty Group-like associative multMagma;
let a2 be Subgroup of a1;
func con_class A2 -> Element of bool Subgroups a1 means
for b1 being set holds
b1 in it
iff
ex b2 being strict Subgroup of a1 st
b1 = b2 & a2,b2 are_conjugated;
end;
:: GROUP_3:def 12
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being Element of bool Subgroups b1 holds
b3 = con_class b2
iff
for b4 being set holds
b4 in b3
iff
ex b5 being strict Subgroup of b1 st
b4 = b5 & b2,b5 are_conjugated;
:: GROUP_3:th 127
theorem
for b1 being set
for b2 being non empty Group-like associative multMagma
for b3 being Subgroup of b2
st b1 in con_class b3
holds b1 is strict Subgroup of b2;
:: GROUP_3:th 128
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict Subgroup of b1 holds
b2 in con_class b3
iff
b2,b3 are_conjugated;
:: GROUP_3:th 129
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being strict Subgroup of b1 holds
b3 |^ b2 in con_class b3;
:: GROUP_3:th 130
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
b2 in con_class b2;
:: GROUP_3:th 131
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict Subgroup of b1
st b2 in con_class b3
holds b3 in con_class b2;
:: GROUP_3:th 132
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict Subgroup of b1 holds
con_class b2 = con_class b3
iff
con_class b2 meets con_class b3;
:: GROUP_3:th 133
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
st b1 is finite
holds con_class b2 is finite;
:: GROUP_3:th 134
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being strict Subgroup of b1 holds
b3,b2 are_conjugated
iff
carr b3,carr b2 are_conjugated;
:: GROUP_3:attrnot 1 => GROUP_3:attr 1
definition
let a1 be non empty Group-like associative multMagma;
let a2 be Subgroup of a1;
attr a2 is normal means
for b1 being Element of the carrier of a1 holds
a2 |^ b1 = multMagma(#the carrier of a2,the multF of a2#);
end;
:: GROUP_3:dfs 13
definiens
let a1 be non empty Group-like associative multMagma;
let a2 be Subgroup of a1;
To prove
a2 is normal
it is sufficient to prove
thus for b1 being Element of the carrier of a1 holds
a2 |^ b1 = multMagma(#the carrier of a2,the multF of a2#);
:: GROUP_3:def 13
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1 holds
b2 is normal(b1)
iff
for b3 being Element of the carrier of b1 holds
b2 |^ b3 = multMagma(#the carrier of b2,the multF of b2#);
:: GROUP_3:exreg 1
registration
let a1 be non empty Group-like associative multMagma;
cluster non empty strict unital Group-like associative normal Subgroup of a1;
end;
:: GROUP_3:th 137
theorem
for b1 being non empty Group-like associative multMagma holds
(1). b1 is normal(b1) & (Omega). b1 is normal(b1);
:: GROUP_3:th 138
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict normal Subgroup of b1 holds
b2 /\ b3 is normal(b1);
:: GROUP_3:th 139
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1
st b1 is non empty Group-like associative commutative multMagma
holds b2 is normal(b1);
:: GROUP_3:th 140
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1 holds
b2 is normal Subgroup of b1
iff
for b3 being Element of the carrier of b1 holds
b3 * b2 = b2 * b3;
:: GROUP_3:th 141
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1 holds
b2 is normal Subgroup of b1
iff
for b3 being Element of the carrier of b1 holds
b3 * b2 c= b2 * b3;
:: GROUP_3:th 142
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1 holds
b2 is normal Subgroup of b1
iff
for b3 being Element of the carrier of b1 holds
b2 * b3 c= b3 * b2;
:: GROUP_3:th 143
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1 holds
b2 is normal Subgroup of b1
iff
for b3 being Element of bool the carrier of b1 holds
b3 * b2 = b2 * b3;
:: GROUP_3:th 144
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
b2 is normal Subgroup of b1
iff
for b3 being Element of the carrier of b1 holds
b2 is Subgroup of b2 |^ b3;
:: GROUP_3:th 145
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
b2 is normal Subgroup of b1
iff
for b3 being Element of the carrier of b1 holds
b2 |^ b3 is Subgroup of b2;
:: GROUP_3:th 146
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
b2 is normal Subgroup of b1
iff
con_class b2 = {b2};
:: GROUP_3:th 147
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
b2 is normal Subgroup of b1
iff
for b3 being Element of the carrier of b1
st b3 in b2
holds con_class b3 c= carr b2;
:: GROUP_3:th 148
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict normal Subgroup of b1 holds
(carr b2) * carr b3 = (carr b3) * carr b2;
:: GROUP_3:th 149
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being strict normal Subgroup of b1 holds
ex b4 being strict normal Subgroup of b1 st
the carrier of b4 = (carr b2) * carr b3;
:: GROUP_3:th 150
theorem
for b1 being non empty Group-like associative multMagma
for b2 being normal Subgroup of b1 holds
Left_Cosets b2 = Right_Cosets b2;
:: GROUP_3:th 151
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
st Left_Cosets b2 is finite & index b2 = 2
holds b2 is normal Subgroup of b1;
:: GROUP_3:funcnot 10 => GROUP_3:func 10
definition
let a1 be non empty Group-like associative multMagma;
let a2 be Element of bool the carrier of a1;
func Normalizator A2 -> strict Subgroup of a1 means
the carrier of it = {b1 where b1 is Element of the carrier of a1: a2 |^ b1 = a2};
end;
:: GROUP_3:def 14
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
for b3 being strict Subgroup of b1 holds
b3 = Normalizator b2
iff
the carrier of b3 = {b4 where b4 is Element of the carrier of b1: b2 |^ b4 = b2};
:: GROUP_3:th 154
theorem
for b1 being set
for b2 being non empty Group-like associative multMagma
for b3 being Element of bool the carrier of b2 holds
b1 in Normalizator b3
iff
ex b4 being Element of the carrier of b2 st
b1 = b4 & b3 |^ b4 = b3;
:: GROUP_3:th 155
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1 holds
Card con_class b2 = Index Normalizator b2;
:: GROUP_3:th 156
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of bool the carrier of b1
st (con_class b2 is finite or Left_Cosets Normalizator b2 is finite)
holds ex b3 being finite set st
b3 = con_class b2 & card b3 = index Normalizator b2;
:: GROUP_3:th 157
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
Card con_class b2 = Index Normalizator {b2};
:: GROUP_3:th 158
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
st (con_class b2 is finite or Left_Cosets Normalizator {b2} is finite)
holds ex b3 being finite set st
b3 = con_class b2 &
card b3 = index Normalizator {b2};
:: GROUP_3:funcnot 11 => GROUP_3:func 11
definition
let a1 be non empty Group-like associative multMagma;
let a2 be Subgroup of a1;
func Normalizator A2 -> strict Subgroup of a1 equals
Normalizator carr a2;
end;
:: GROUP_3:def 15
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1 holds
Normalizator b2 = Normalizator carr b2;
:: GROUP_3:th 160
theorem
for b1 being set
for b2 being non empty Group-like associative multMagma
for b3 being strict Subgroup of b2 holds
b1 in Normalizator b3
iff
ex b4 being Element of the carrier of b2 st
b1 = b4 & b3 |^ b4 = b3;
:: GROUP_3:th 161
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
Card con_class b2 = Index Normalizator b2;
:: GROUP_3:th 162
theorem
for b1 being non empty Group-like associative multMagma
for b2 being strict Subgroup of b1
st (con_class b2 is finite or Left_Cosets Normalizator b2 is finite)
holds ex b3 being finite set st
b3 = con_class b2 & card b3 = index Normalizator b2;
:: GROUP_3:th 163
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being strict Subgroup of b1 holds
b2 is normal Subgroup of b1
iff
Normalizator b2 = b1;
:: GROUP_3:th 164
theorem
for b1 being non empty strict Group-like associative multMagma holds
Normalizator (1). b1 = b1;
:: GROUP_3:th 165
theorem
for b1 being non empty strict Group-like associative multMagma holds
Normalizator (Omega). b1 = b1;