Article AUTGROUP, MML version 4.99.1005
:: AUTGROUP:th 1
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Subgroup of b1 holds
for b3, b4 being Element of the carrier of b1
st b4 is Element of the carrier of b2
holds b4 |^ b3 in b2
iff
b2 is normal(b1);
:: AUTGROUP:funcnot 1 => AUTGROUP:func 1
definition
let a1 be non empty strict Group-like associative multMagma;
func Aut A1 -> FUNCTION_DOMAIN of the carrier of a1,the carrier of a1 means
(for b1 being Element of it holds
b1 is Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a1) &
(for b1 being Function-like quasi_total multiplicative Relation of the carrier of a1,the carrier of a1 holds
b1 in it
iff
b1 is one-to-one & b1 is being_epimorphism(a1, a1));
end;
:: AUTGROUP:def 1
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being FUNCTION_DOMAIN of the carrier of b1,the carrier of b1 holds
b2 = Aut b1
iff
(for b3 being Element of b2 holds
b3 is Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b1) &
(for b3 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b1 holds
b3 in b2
iff
b3 is one-to-one & b3 is being_epimorphism(b1, b1));
:: AUTGROUP:th 3
theorem
for b1 being non empty strict Group-like associative multMagma holds
Aut b1 c= Funcs(the carrier of b1,the carrier of b1);
:: AUTGROUP:th 4
theorem
for b1 being non empty strict Group-like associative multMagma holds
id the carrier of b1 is Element of Aut b1;
:: AUTGROUP:th 5
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b1 holds
b2 in Aut b1
iff
b2 is being_isomorphism(b1, b1);
:: AUTGROUP:th 6
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of Aut b1 holds
b2 " is Function-like quasi_total multiplicative Relation of the carrier of b1,the carrier of b1;
:: AUTGROUP:th 7
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of Aut b1 holds
b2 " is Element of Aut b1;
:: AUTGROUP:th 8
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being Element of Aut b1 holds
b2 * b3 is Element of Aut b1;
:: AUTGROUP:funcnot 2 => AUTGROUP:func 2
definition
let a1 be non empty strict Group-like associative multMagma;
func AutComp A1 -> Function-like quasi_total Relation of [:Aut a1,Aut a1:],Aut a1 means
for b1, b2 being Element of Aut a1 holds
it .(b1,b2) = b1 * b2;
end;
:: AUTGROUP:def 2
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Function-like quasi_total Relation of [:Aut b1,Aut b1:],Aut b1 holds
b2 = AutComp b1
iff
for b3, b4 being Element of Aut b1 holds
b2 .(b3,b4) = b3 * b4;
:: AUTGROUP:funcnot 3 => AUTGROUP:func 3
definition
let a1 be non empty strict Group-like associative multMagma;
func AutGroup A1 -> non empty strict Group-like associative multMagma equals
multMagma(#Aut a1,AutComp a1#);
end;
:: AUTGROUP:def 3
theorem
for b1 being non empty strict Group-like associative multMagma holds
AutGroup b1 = multMagma(#Aut b1,AutComp b1#);
:: AUTGROUP:th 9
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being Element of the carrier of AutGroup b1
for b4, b5 being Element of Aut b1
st b2 = b4 & b3 = b5
holds b2 * b3 = b4 * b5;
:: AUTGROUP:th 10
theorem
for b1 being non empty strict Group-like associative multMagma holds
id the carrier of b1 = 1_ AutGroup b1;
:: AUTGROUP:th 11
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of Aut b1
for b3 being Element of the carrier of AutGroup b1
st b2 = b3
holds b2 " = b3 ";
:: AUTGROUP:funcnot 4 => AUTGROUP:func 4
definition
let a1 be non empty strict Group-like associative multMagma;
func InnAut A1 -> FUNCTION_DOMAIN of the carrier of a1,the carrier of a1 means
for b1 being Element of Funcs(the carrier of a1,the carrier of a1) holds
b1 in it
iff
ex b2 being Element of the carrier of a1 st
for b3 being Element of the carrier of a1 holds
b1 . b3 = b3 |^ b2;
end;
:: AUTGROUP:def 4
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being FUNCTION_DOMAIN of the carrier of b1,the carrier of b1 holds
b2 = InnAut b1
iff
for b3 being Element of Funcs(the carrier of b1,the carrier of b1) holds
b3 in b2
iff
ex b4 being Element of the carrier of b1 st
for b5 being Element of the carrier of b1 holds
b3 . b5 = b5 |^ b4;
:: AUTGROUP:th 12
theorem
for b1 being non empty strict Group-like associative multMagma holds
InnAut b1 c= Funcs(the carrier of b1,the carrier of b1);
:: AUTGROUP:th 13
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of InnAut b1 holds
b2 is Element of Aut b1;
:: AUTGROUP:th 14
theorem
for b1 being non empty strict Group-like associative multMagma holds
InnAut b1 c= Aut b1;
:: AUTGROUP:th 15
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being Element of InnAut b1 holds
(AutComp b1) .(b2,b3) = b2 * b3;
:: AUTGROUP:th 16
theorem
for b1 being non empty strict Group-like associative multMagma holds
id the carrier of b1 is Element of InnAut b1;
:: AUTGROUP:th 17
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of InnAut b1 holds
b2 " is Element of InnAut b1;
:: AUTGROUP:th 18
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being Element of InnAut b1 holds
b2 * b3 is Element of InnAut b1;
:: AUTGROUP:funcnot 5 => AUTGROUP:func 5
definition
let a1 be non empty strict Group-like associative multMagma;
func InnAutGroup A1 -> strict normal Subgroup of AutGroup a1 means
the carrier of it = InnAut a1;
end;
:: AUTGROUP:def 5
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being strict normal Subgroup of AutGroup b1 holds
b2 = InnAutGroup b1
iff
the carrier of b2 = InnAut b1;
:: AUTGROUP:th 20
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being Element of the carrier of InnAutGroup b1
for b4, b5 being Element of InnAut b1
st b2 = b4 & b3 = b5
holds b2 * b3 = b4 * b5;
:: AUTGROUP:th 21
theorem
for b1 being non empty strict Group-like associative multMagma holds
id the carrier of b1 = 1_ InnAutGroup b1;
:: AUTGROUP:th 22
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of InnAut b1
for b3 being Element of the carrier of InnAutGroup b1
st b2 = b3
holds b2 " = b3 ";
:: AUTGROUP:funcnot 6 => AUTGROUP:func 6
definition
let a1 be non empty strict Group-like associative multMagma;
let a2 be Element of the carrier of a1;
func Conjugate A2 -> Element of InnAut a1 means
for b1 being Element of the carrier of a1 holds
it . b1 = b1 |^ a2;
end;
:: AUTGROUP:def 6
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1
for b3 being Element of InnAut b1 holds
b3 = Conjugate b2
iff
for b4 being Element of the carrier of b1 holds
b3 . b4 = b4 |^ b2;
:: AUTGROUP:th 23
theorem
for b1 being non empty strict Group-like associative multMagma
for b2, b3 being Element of the carrier of b1 holds
Conjugate (b2 * b3) = (Conjugate b3) * Conjugate b2;
:: AUTGROUP:th 24
theorem
for b1 being non empty strict Group-like associative multMagma holds
Conjugate 1_ b1 = id the carrier of b1;
:: AUTGROUP:th 25
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
(Conjugate 1_ b1) . b2 = b2;
:: AUTGROUP:th 26
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
(Conjugate b2) * Conjugate (b2 ") = Conjugate 1_ b1;
:: AUTGROUP:th 27
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
(Conjugate (b2 ")) * Conjugate b2 = Conjugate 1_ b1;
:: AUTGROUP:th 28
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
Conjugate (b2 ") = (Conjugate b2) ";
:: AUTGROUP:th 29
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of the carrier of b1 holds
(Conjugate b2) * Conjugate 1_ b1 = Conjugate b2 &
(Conjugate 1_ b1) * Conjugate b2 = Conjugate b2;
:: AUTGROUP:th 30
theorem
for b1 being non empty strict Group-like associative multMagma
for b2 being Element of InnAut b1 holds
b2 * Conjugate 1_ b1 = b2 & (Conjugate 1_ b1) * b2 = b2;
:: AUTGROUP:th 31
theorem
for b1 being non empty strict Group-like associative multMagma holds
InnAutGroup b1,b1 ./. center b1 are_isomorphic;
:: AUTGROUP:th 32
theorem
for b1 being non empty strict Group-like associative multMagma
st b1 is non empty Group-like associative commutative multMagma
for b2 being Element of the carrier of InnAutGroup b1 holds
b2 = 1_ InnAutGroup b1;