:: AUTGROUP semantic presentation
Lemma1:
for b1 being strict Group
for b2 being Subgroup of b1 holds
( ( for b3, b4 being Element of b1 holds
( b4 is Element of b2 implies b4 |^ b3 in b2 ) ) implies b2 is normal )
Lemma2:
for b1 being strict Group
for b2 being Subgroup of b1 holds
( b2 is normal implies for b3, b4 being Element of b1 holds
( b4 is Element of b2 implies b4 |^ b3 in b2 ) )
theorem Th1: :: AUTGROUP:1
definition
let c
1 be
strict Group;
func Aut c
1 -> FUNCTION_DOMAIN of the
carrier of a
1,the
carrier of a
1 means :
Def1:
:: AUTGROUP:def 1
( ( for b
1 being
Element of a
2 holds
b
1 is
Homomorphism of a
1,a
1 ) & ( for b
1 being
Homomorphism of a
1,a
1 holds
( b
1 in a
2 iff ( b
1 is
one-to-one & b
1 is_epimorphism ) ) ) );
existence
ex b1 being FUNCTION_DOMAIN of the carrier of c1,the carrier of c1 st
( ( for b2 being Element of b1 holds
b2 is Homomorphism of c1,c1 ) & ( for b2 being Homomorphism of c1,c1 holds
( b2 in b1 iff ( b2 is one-to-one & b2 is_epimorphism ) ) ) )
uniqueness
for b1, b2 being FUNCTION_DOMAIN of the carrier of c1,the carrier of c1 holds
( ( for b3 being Element of b1 holds
b3 is Homomorphism of c1,c1 ) & ( for b3 being Homomorphism of c1,c1 holds
( b3 in b1 iff ( b3 is one-to-one & b3 is_epimorphism ) ) ) & ( for b3 being Element of b2 holds
b3 is Homomorphism of c1,c1 ) & ( for b3 being Homomorphism of c1,c1 holds
( b3 in b2 iff ( b3 is one-to-one & b3 is_epimorphism ) ) ) implies b1 = b2 )
end;
:: deftheorem Def1 defines Aut AUTGROUP:def 1 :
theorem Th2: :: AUTGROUP:2
canceled;
theorem Th3: :: AUTGROUP:3
theorem Th4: :: AUTGROUP:4
theorem Th5: :: AUTGROUP:5
Lemma6:
for b1 being strict Group
for b2 being Element of Aut b1 holds
( dom b2 = rng b2 & dom b2 = the carrier of b1 )
theorem Th6: :: AUTGROUP:6
theorem Th7: :: AUTGROUP:7
theorem Th8: :: AUTGROUP:8
:: deftheorem Def2 defines AutComp AUTGROUP:def 2 :
:: deftheorem Def3 defines AutGroup AUTGROUP:def 3 :
theorem Th9: :: AUTGROUP:9
theorem Th10: :: AUTGROUP:10
theorem Th11: :: AUTGROUP:11
:: deftheorem Def4 defines InnAut AUTGROUP:def 4 :
theorem Th12: :: AUTGROUP:12
theorem Th13: :: AUTGROUP:13
theorem Th14: :: AUTGROUP:14
theorem Th15: :: AUTGROUP:15
theorem Th16: :: AUTGROUP:16
theorem Th17: :: AUTGROUP:17
theorem Th18: :: AUTGROUP:18
:: deftheorem Def5 defines InnAutGroup AUTGROUP:def 5 :
theorem Th19: :: AUTGROUP:19
canceled;
theorem Th20: :: AUTGROUP:20
theorem Th21: :: AUTGROUP:21
theorem Th22: :: AUTGROUP:22
:: deftheorem Def6 defines Conjugate AUTGROUP:def 6 :
theorem Th23: :: AUTGROUP:23
theorem Th24: :: AUTGROUP:24
theorem Th25: :: AUTGROUP:25
theorem Th26: :: AUTGROUP:26
theorem Th27: :: AUTGROUP:27
theorem Th28: :: AUTGROUP:28
theorem Th29: :: AUTGROUP:29
theorem Th30: :: AUTGROUP:30
theorem Th31: :: AUTGROUP:31
theorem Th32: :: AUTGROUP:32