Article GROUP_8, MML version 4.99.1005
:: GROUP_8:th 1
theorem
for b1 being Element of NAT
for b2 being non empty finite strict Group-like associative multMagma
st b1 is prime & ord b2 = b1
holds ex b3 being Element of the carrier of b2 st
ord b3 = b1;
:: GROUP_8:th 2
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3, b4 being Element of the carrier of b2
for b5, b6 being Element of the carrier of b1
st b3 = b5 & b4 = b6
holds b3 * b4 = b5 * b6;
:: GROUP_8:th 3
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being Element of the carrier of b2
for b4 being Element of the carrier of b1
st b3 = b4
for b5 being Element of NAT holds
b3 |^ b5 = b4 |^ b5;
:: GROUP_8:th 4
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being Element of the carrier of b2
for b4 being Element of the carrier of b1
st b3 = b4
for b5 being integer set holds
b3 |^ b5 = b4 |^ b5;
:: GROUP_8:th 5
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Subgroup of b1
for b3 being Element of the carrier of b2
for b4 being Element of the carrier of b1
st b3 = b4 & b1 is finite
holds ord b3 = ord b4;
:: GROUP_8:th 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
st b3 in b2
holds b2 * b3 c= the carrier of b2;
:: GROUP_8:th 7
theorem
for b1 being non empty Group-like associative multMagma
for b2 being Element of the carrier of b1
st b2 <> 1_ b1
holds gr {b2} <> (1). b1;
:: GROUP_8:th 8
theorem
for b1 being non empty Group-like associative multMagma
for b2 being integer set holds
(1_ b1) |^ b2 = 1_ b1;
:: GROUP_8:th 9
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being integer set holds
b2 |^ (b3 * ord b2) = 1_ b1;
:: GROUP_8:th 10
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1
st b2 is not being_of_order_0(b1)
for b3 being integer set holds
b2 |^ b3 = b2 |^ (b3 mod ord b2);
:: GROUP_8:th 11
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1
st b2 is not being_of_order_0(b1)
holds gr {b2} is finite;
:: GROUP_8:th 12
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1
st b2 is being_of_order_0(b1)
holds b2 " is being_of_order_0(b1);
:: GROUP_8:th 13
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
b2 is being_of_order_0(b1)
iff
for b3 being integer set
st b2 |^ b3 = 1_ b1
holds b3 = 0;
:: GROUP_8:th 14
theorem
for b1 being non empty strict Group-like associative multMagma
st ex b2 being Element of the carrier of b1 st
b2 <> 1_ b1
holds for b2 being strict Subgroup of b1
st b2 <> b1
holds b2 = (1). b1
iff
b1 is cyclic &
b1 is finite &
(ex b2 being Element of NAT st
Ord b1 = b2 & b2 is prime);
:: GROUP_8:th 15
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3, b4 being Element of the carrier of b1
for b5 being Element of bool the carrier of b1 holds
b4 in (b2 * b5) * b3
iff
ex b6 being Element of the carrier of b1 st
b4 = (b2 * b6) * b3 & b6 in b5;
:: GROUP_8:th 16
theorem
for b1 being non empty Group-like associative multMagma
for b2 being non empty Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
Card b2 = Card ((b3 " * b2) * b3);
:: GROUP_8:funcnot 1 => GROUP_8:func 1
definition
let a1 be non empty strict Group-like associative multMagma;
let a2, a3 be strict Subgroup of a1;
func Double_Cosets(A2,A3) -> Element of bool bool the carrier of a1 means
for b1 being Element of bool the carrier of a1 holds
b1 in it
iff
ex b2 being Element of the carrier of a1 st
b1 = (a2 * b2) * a3;
end;
:: GROUP_8:def 1
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being strict Subgroup of b1
for b4 being Element of bool bool the carrier of b1 holds
b4 = Double_Cosets(b2,b3)
iff
for b5 being Element of bool the carrier of b1 holds
b5 in b4
iff
ex b6 being Element of the carrier of b1 st
b5 = (b2 * b6) * b3;
:: GROUP_8:th 17
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
for b4, b5 being strict Subgroup of b1 holds
b2 in (b4 * b3) * b5
iff
ex b6, b7 being Element of the carrier of b1 st
b2 = (b6 * b3) * b7 & b6 in b4 & b7 in b5;
:: GROUP_8:th 18
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being Element of the carrier of b1
for b4, b5 being strict Subgroup of b1
st (b4 * b2) * b5 <> (b4 * b3) * b5
for b6 being Element of the carrier of b1
st b6 in (b4 * b2) * b5
holds not b6 in (b4 * b3) * b5;
:: GROUP_8:funcreg 1
registration
let a1 be non empty Group-like associative multMagma;
let a2 be Subgroup of a1;
cluster Left_Cosets a2 -> non empty;
end;
:: GROUP_8:funcnot 2 => GROUP_2:func 17
notation
let a1 be non empty Group-like associative multMagma;
let a2 be Subgroup of a1;
synonym index(a1,a2) for index a2;
end;
:: GROUP_8:th 19
theorem
for b1 being non empty Group-like associative multMagma
for b2, b3 being Subgroup of b1
for b4 being Subgroup of b2
st b1 = b2 "\/" b3 & b4 = b2 /\ b3 & b1 is finite
holds index b4 <= index b3;
:: GROUP_8:th 20
theorem
for b1 being non empty finite Group-like associative multMagma
for b2 being Subgroup of b1 holds
0 < index b2;
:: GROUP_8:th 21
theorem
for b1 being non empty Group-like associative multMagma
st b1 is finite
for b2 being Subgroup of b1
for b3, b4 being Subgroup of b2
st b2 = b3 "\/" b4
for b5 being Subgroup of b3
st b5 = b3 /\ b4
for b6 being Subgroup of b4
st b6 = b3 /\ b4
for b7 being Subgroup of b2
st b7 = b3 /\ b4 & Left_Cosets b4 is finite & Left_Cosets b3 is finite & index b3,index b4 are_relative_prime
holds index b4 = index b5 & index b3 = index b6;
:: GROUP_8:th 22
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
st b3 in b2
for b4 being integer set holds
b3 |^ b4 in b2;
:: GROUP_8:th 23
theorem
for b1 being non empty strict Group-like associative multMagma
st b1 <> (1). b1
holds ex b2 being Element of the carrier of b1 st
b2 <> 1_ b1;
:: GROUP_8:th 24
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1
st b1 = gr {b2}
for b3 being strict Subgroup of b1
st b3 <> (1). b1
holds ex b4 being Element of NAT st
0 < b4 & b2 |^ b4 in b3;
:: GROUP_8:th 25
theorem
for b1 being non empty strict Group-like associative cyclic multMagma
for b2 being strict Subgroup of b1 holds
b2 is non empty Group-like associative cyclic multMagma;