:: Lattice of Subgroups of a Group. Frattini Subgroup
:: by Wojciech A. Trybulec
::
:: Received August 22, 1990
:: Copyright (c) 1990 Association of Mizar Users
:: deftheorem GROUP_4:def 1 :
canceled;
:: deftheorem defines @ GROUP_4:def 2 :
theorem :: GROUP_4:1
canceled;
theorem :: GROUP_4:2
canceled;
theorem Th3: :: GROUP_4:3
theorem :: GROUP_4:4
theorem Th5: :: GROUP_4:5
theorem Th6: :: GROUP_4:6
:: deftheorem defines Product GROUP_4:def 3 :
theorem :: GROUP_4:7
canceled;
theorem Th8: :: GROUP_4:8
theorem Th9: :: GROUP_4:9
theorem Th10: :: GROUP_4:10
theorem Th11: :: GROUP_4:11
theorem :: GROUP_4:12
theorem :: GROUP_4:13
theorem :: GROUP_4:14
theorem :: GROUP_4:15
theorem :: GROUP_4:16
Lm3:
for F1 being FinSequence
for y being Element of NAT st y in dom F1 holds
( ((len F1) - y) + 1 is Element of NAT & ((len F1) - y) + 1 >= 1 & ((len F1) - y) + 1 <= len F1 )
theorem Th17: :: GROUP_4:17
theorem :: GROUP_4:18
theorem :: GROUP_4:19
theorem :: GROUP_4:20
theorem Th21: :: GROUP_4:21
:: deftheorem Def4 defines |^ GROUP_4:def 4 :
theorem :: GROUP_4:22
canceled;
theorem :: GROUP_4:23
canceled;
theorem :: GROUP_4:24
canceled;
theorem Th25: :: GROUP_4:25
theorem Th26: :: GROUP_4:26
theorem Th27: :: GROUP_4:27
theorem Th28: :: GROUP_4:28
theorem Th29: :: GROUP_4:29
theorem :: GROUP_4:30
for
i1,
i2,
i3 being
Integer for
G being
Group for
a,
b,
c being
Element of
G holds
<*a,b,c*> |^ <*(@ i1),(@ i2),(@ i3)*> = <*(a |^ i1),(b |^ i2),(c |^ i3)*>
theorem :: GROUP_4:31
theorem :: GROUP_4:32
theorem :: GROUP_4:33
:: deftheorem Def5 defines gr GROUP_4:def 5 :
theorem :: GROUP_4:34
canceled;
theorem :: GROUP_4:35
canceled;
theorem :: GROUP_4:36
canceled;
theorem Th37: :: GROUP_4:37
theorem Th38: :: GROUP_4:38
theorem :: GROUP_4:39
theorem Th40: :: GROUP_4:40
theorem Th41: :: GROUP_4:41
theorem :: GROUP_4:42
theorem Th43: :: GROUP_4:43
theorem Th44: :: GROUP_4:44
:: deftheorem Def6 defines generating GROUP_4:def 6 :
theorem :: GROUP_4:45
canceled;
theorem :: GROUP_4:46
:: deftheorem Def7 defines maximal GROUP_4:def 7 :
theorem :: GROUP_4:47
canceled;
theorem Th48: :: GROUP_4:48
:: deftheorem Def8 defines Phi GROUP_4:def 8 :
theorem :: GROUP_4:49
canceled;
theorem :: GROUP_4:50
canceled;
theorem :: GROUP_4:51
canceled;
theorem Th52: :: GROUP_4:52
theorem :: GROUP_4:53
theorem Th54: :: GROUP_4:54
theorem Th55: :: GROUP_4:55
theorem :: GROUP_4:56
:: deftheorem defines * GROUP_4:def 9 :
theorem :: GROUP_4:57
theorem :: GROUP_4:58
canceled;
theorem :: GROUP_4:59
for
G being
Group for
H1,
H2,
H3 being
Subgroup of
G holds
(H1 * H2) * H3 = H1 * (H2 * H3)
theorem :: GROUP_4:60
theorem :: GROUP_4:61
theorem :: GROUP_4:62
theorem :: GROUP_4:63
:: deftheorem defines "\/" GROUP_4:def 10 :
theorem :: GROUP_4:64
canceled;
theorem :: GROUP_4:65
canceled;
theorem :: GROUP_4:66
canceled;
theorem :: GROUP_4:67
theorem :: GROUP_4:68
theorem Th69: :: GROUP_4:69
theorem :: GROUP_4:70
theorem Th71: :: GROUP_4:71
theorem :: GROUP_4:72
theorem :: GROUP_4:73
theorem :: GROUP_4:74
Lm4:
for G being Group
for H1, H2 being Subgroup of G holds H1 is Subgroup of H1 "\/" H2
Lm5:
for G being Group
for H1, H2, H3 being Subgroup of G holds (H1 "\/" H2) "\/" H3 is Subgroup of H1 "\/" (H2 "\/" H3)
theorem Th75: :: GROUP_4:75
theorem :: GROUP_4:76
theorem Th77: :: GROUP_4:77
theorem Th78: :: GROUP_4:78
theorem Th79: :: GROUP_4:79
theorem :: GROUP_4:80
theorem :: GROUP_4:81
theorem :: GROUP_4:82
theorem :: GROUP_4:83
theorem Th84: :: GROUP_4:84
theorem Th85: :: GROUP_4:85
theorem :: GROUP_4:86
definition
let G be
Group;
func SubJoin G -> BinOp of
Subgroups G means :
Def11:
:: GROUP_4:def 11
for
S1,
S2 being
Element of
Subgroups G for
H1,
H2 being
Subgroup of
G st
S1 = H1 &
S2 = H2 holds
it . S1,
S2 = H1 "\/" H2;
existence
ex b1 being BinOp of Subgroups G st
for S1, S2 being Element of Subgroups G
for H1, H2 being Subgroup of G st S1 = H1 & S2 = H2 holds
b1 . S1,S2 = H1 "\/" H2
uniqueness
for b1, b2 being BinOp of Subgroups G st ( for S1, S2 being Element of Subgroups G
for H1, H2 being Subgroup of G st S1 = H1 & S2 = H2 holds
b1 . S1,S2 = H1 "\/" H2 ) & ( for S1, S2 being Element of Subgroups G
for H1, H2 being Subgroup of G st S1 = H1 & S2 = H2 holds
b2 . S1,S2 = H1 "\/" H2 ) holds
b1 = b2
end;
:: deftheorem Def11 defines SubJoin GROUP_4:def 11 :
definition
let G be
Group;
func SubMeet G -> BinOp of
Subgroups G means :
Def12:
:: GROUP_4:def 12
for
S1,
S2 being
Element of
Subgroups G for
H1,
H2 being
Subgroup of
G st
S1 = H1 &
S2 = H2 holds
it . S1,
S2 = H1 /\ H2;
existence
ex b1 being BinOp of Subgroups G st
for S1, S2 being Element of Subgroups G
for H1, H2 being Subgroup of G st S1 = H1 & S2 = H2 holds
b1 . S1,S2 = H1 /\ H2
uniqueness
for b1, b2 being BinOp of Subgroups G st ( for S1, S2 being Element of Subgroups G
for H1, H2 being Subgroup of G st S1 = H1 & S2 = H2 holds
b1 . S1,S2 = H1 /\ H2 ) & ( for S1, S2 being Element of Subgroups G
for H1, H2 being Subgroup of G st S1 = H1 & S2 = H2 holds
b2 . S1,S2 = H1 /\ H2 ) holds
b1 = b2
end;
:: deftheorem Def12 defines SubMeet GROUP_4:def 12 :
Lm6:
for G being Group holds
( LattStr(# (Subgroups G),(SubJoin G),(SubMeet G) #) is Lattice & LattStr(# (Subgroups G),(SubJoin G),(SubMeet G) #) is 0_Lattice & LattStr(# (Subgroups G),(SubJoin G),(SubMeet G) #) is 1_Lattice )
:: deftheorem defines lattice GROUP_4:def 13 :
theorem :: GROUP_4:87
canceled;
theorem :: GROUP_4:88
canceled;
theorem :: GROUP_4:89
canceled;
theorem :: GROUP_4:90
canceled;
theorem :: GROUP_4:91
canceled;
theorem :: GROUP_4:92
theorem :: GROUP_4:93
theorem :: GROUP_4:94
theorem :: GROUP_4:95
canceled;
theorem :: GROUP_4:96
canceled;
theorem :: GROUP_4:97
canceled;
theorem :: GROUP_4:98
theorem :: GROUP_4:99