GROUP_4: Lattice of Subgroups of a Group. Frattini Subgroup

:: Lattice of Subgroups of a Group. Frattini Subgroup
:: by Wojciech A. Trybulec
::
:: Received August 22, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let D be non empty set ;
let F be FinSequence of D;
let X be set ;
:: original: -
redefine func F - X -> FinSequence of D;
coherence by FINSEQ_3:86;

end;

scheme :: GROUP_4:sch 1
MeetSbgEx{ F₁() -> Group, P₁[ set ] } :

ex H being strict Subgroup of F₁() st the carrier of H = meet { A where A is Subset of F₁() : ex K being strict Subgroup of F₁() st
( A = the carrier of K & P₁[K] ) }

provided

A1: ex H being strict Subgroup of F₁() st P₁[H]

proof end;

scheme :: GROUP_4:sch 2
SubgrSep{ F₁() -> Group, P₁[ set ] } :

ex X being set st
( X c= Subgroups F₁() & ( for H being strict Subgroup of F₁() holds
( H in X iff P₁[H] ) ) )

proof end;

definition

let i be Integer;

func @ i -> Element of INT equals :: GROUP_4:def 1
i;

coherence by INT_1:def 2;

end;

:: deftheorem defines @ GROUP_4:def 1 :

theorem Th1: :: GROUP_4:1

for n being Nat
for G being Group
for a being Element of G
for H being Subgroup of G
for h being Element of H st a = h holds
a |^ n = h |^ n

proof end;

theorem :: GROUP_4:2

for i being Integer
for G being Group
for a being Element of G
for H being Subgroup of G
for h being Element of H st a = h holds
a |^ i = h |^ i

proof end;

theorem Th3: :: GROUP_4:3

for n being Nat
for G being Group
for a being Element of G
for H being Subgroup of G st a in H holds
a |^ n in H

proof end;

theorem Th4: :: GROUP_4:4

for i being Integer
for G being Group
for a being Element of G
for H being Subgroup of G st a in H holds
a |^ i in H

proof end;

definition

let G be non empty multMagma ;
let F be FinSequence of the carrier of G;

func Product F -> Element of G equals :: GROUP_4:def 2
the multF of G "**" F;

end;

:: deftheorem defines Product GROUP_4:def 2 :

theorem :: GROUP_4:5

for G being non empty unital associative multMagma
for F1, F2 being FinSequence of the carrier of G holds Product (F1 ^ F2) = (Product F1) * (Product F2) by FINSOP_1:5;

theorem :: GROUP_4:6

for G being non empty unital multMagma
for F being FinSequence of the carrier of G
for a being Element of G holds Product (F ^ <*a*>) = (Product F) * a by FINSOP_1:4;

theorem :: GROUP_4:7

for G being non empty unital associative multMagma
for F being FinSequence of the carrier of G
for a being Element of G holds Product (<*a*> ^ F) = a * (Product F) by FINSOP_1:6;

theorem Th8: :: GROUP_4:8

for G being non empty unital multMagma holds Product (<*> the carrier of G) = 1_ G

proof end;

theorem :: GROUP_4:9

for G being non empty multMagma
for a being Element of G holds Product <*a*> = a by FINSOP_1:11;

theorem :: GROUP_4:10

for G being non empty multMagma
for a, b being Element of G holds Product <*a,b*> = a * b by FINSOP_1:12;

theorem :: GROUP_4:11

for G being Group
for a, b, c being Element of G holds
( Product <*a,b,c*> = (a * b) * c & Product <*a,b,c*> = a * (b * c) )

proof end;

Lm1: now :: thesis:

let G be Group; :: thesis:
let a be Element of G; :: thesis:
thus Product (0 |-> a) = Product (<*> the carrier of G)
.= 1_ G by Th8
.= a |^ 0 by GROUP_1:25 ; :: thesis:

end;

Lm2: now :: thesis:

let G be Group; :: thesis:
let a be Element of G; :: thesis:
let n be Nat; :: thesis:
assume A1: Product (n |-> a) = a |^ n ; :: thesis:
thus Product ((n + 1) |-> a) = Product ((n |-> a) ^ <*a*>) by FINSEQ_2:60
.= (Product (n |-> a)) * (Product <*a*>) by FINSOP_1:5
.= (a |^ n) * a by A1, FINSOP_1:11
.= a |^ (n + 1) by GROUP_1:34 ; :: thesis:

end;

theorem :: GROUP_4:12

for n being Nat
for G being Group
for a being Element of G holds Product (n |-> a) = a |^ n

proof end;

theorem :: GROUP_4:13

for G being Group
for F being FinSequence of the carrier of G holds Product (F - {(1_ G)}) = Product F

proof end;

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 )

proof end;

theorem Th14: :: GROUP_4:14

for G being Group
for F1, F2 being FinSequence of the carrier of G st len F1 = len F2 & ( for k being Nat st k in dom F1 holds
F2 . (((len F1) - k) + 1) = (F1 /. k) " ) holds
Product F1 = (Product F2) "

proof end;

theorem :: GROUP_4:15

for G being Group
for F1, F2 being FinSequence of the carrier of G st G is commutative Group holds
for P being Permutation of (dom F1) st F2 = F1 * P holds
Product F1 = Product F2

proof end;

theorem :: GROUP_4:16

for G being Group
for F1, F2 being FinSequence of the carrier of G st G is commutative Group & F1 is one-to-one & F2 is one-to-one & rng F1 = rng F2 holds
Product F1 = Product F2

proof end;

theorem :: GROUP_4:17

for G being Group
for F, F1, F2 being FinSequence of the carrier of G st G is commutative Group & len F = len F1 & len F = len F2 & ( for k being Nat st k in dom F holds
F . k = (F1 /. k) * (F2 /. k) ) holds
Product F = (Product F1) * (Product F2)

proof end;

theorem Th18: :: GROUP_4:18

for G being Group
for H being Subgroup of G
for F being FinSequence of the carrier of G st rng F c= carr H holds
Product F in H

proof end;

definition

let G be Group;
let I be FinSequence of INT ;
let F be FinSequence of the carrier of G;

func F |^ I -> FinSequence of the carrier of G means :Def3: :: GROUP_4:def 3
( len it = len F & ( for k being Nat st k in dom F holds
it . k = (F /. k) |^ (@ (I /. k)) ) );

proof end;

proof end;

end;

:: deftheorem Def3 defines |^ GROUP_4:def 3 :

theorem Th19: :: GROUP_4:19

for G being Group
for F1, F2 being FinSequence of the carrier of G
for I1, I2 being FinSequence of INT st len F1 = len I1 & len F2 = len I2 holds
(F1 ^ F2) |^ (I1 ^ I2) = (F1 |^ I1) ^ (F2 |^ I2)

proof end;

theorem Th20: :: GROUP_4:20

for G being Group
for H being Subgroup of G
for F being FinSequence of the carrier of G
for I being FinSequence of INT st rng F c= carr H holds
Product (F |^ I) in H

proof end;

theorem Th21: :: GROUP_4:21

for G being Group holds (<*> the carrier of G) |^ (<*> INT) = {}

proof end;

theorem Th22: :: GROUP_4:22

for i being Integer
for G being Group
for a being Element of G holds <*a*> |^ <*(@ i)*> = <*(a |^ i)*>

proof end;

theorem Th23: :: GROUP_4:23

for i, j being Integer
for G being Group
for a, b being Element of G holds <*a,b*> |^ <*(@ i),(@ j)*> = <*(a |^ i),(b |^ j)*>

proof end;

theorem :: GROUP_4:24

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)*>

proof end;

theorem :: GROUP_4:25

for G being Group
for F being FinSequence of the carrier of G holds F |^ ((len F) |-> (@ 1)) = F

proof end;

theorem :: GROUP_4:26

for G being Group
for F being FinSequence of the carrier of G holds F |^ ((len F) |-> (@ 0)) = (len F) |-> (1_ G)

proof end;

theorem :: GROUP_4:27

for n being Nat
for G being Group
for I being FinSequence of INT st len I = n holds
(n |-> (1_ G)) |^ I = n |-> (1_ G)

proof end;

definition

let G be Group;
let A be Subset of G;

func gr A -> strict Subgroup of G means :Def4: :: GROUP_4:def 4
( A c= the carrier of it & ( for H being strict Subgroup of G st A c= the carrier of H holds
it is Subgroup of H ) );

proof end;

proof end;

end;

:: deftheorem Def4 defines gr GROUP_4:def 4 :

theorem Th28: :: GROUP_4:28

for G being Group
for a being Element of G
for A being Subset of G holds
( a in gr A iff ex F being FinSequence of the carrier of G ex I being FinSequence of INT st
( len F = len I & rng F c= A & Product (F |^ I) = a ) )

proof end;

theorem Th29: :: GROUP_4:29

for G being Group
for a being Element of G
for A being Subset of G st a in A holds
a in gr A

proof end;

theorem :: GROUP_4:30

for G being Group holds gr ({} the carrier of G) = (1). G

proof end;

theorem Th31: :: GROUP_4:31

for G being Group
for H being strict Subgroup of G holds gr (carr H) = H

proof end;

theorem Th32: :: GROUP_4:32

for G being Group
for A, B being Subset of G st A c= B holds
gr A is Subgroup of gr B

proof end;

theorem :: GROUP_4:33

for G being Group
for A, B being Subset of G holds gr (A /\ B) is Subgroup of (gr A) /\ (gr B)

proof end;

theorem Th34: :: GROUP_4:34

for G being Group
for A being Subset of G holds the carrier of (gr A) = meet { B where B is Subset of G : ex H being strict Subgroup of G st
( B = the carrier of H & A c= carr H ) }

proof end;

theorem Th35: :: GROUP_4:35

for G being Group
for A being Subset of G holds gr A = gr (A \ {(1_ G)})

proof end;

definition

let G be Group;
let a be Element of G;

attr a is generating means :: GROUP_4:def 5
ex A being Subset of G st
( gr A = multMagma(# the carrier of G, the multF of G #) & not gr (A \ {a}) = multMagma(# the carrier of G, the multF of G #) );

end;

:: deftheorem defines generating GROUP_4:def 5 :

theorem :: GROUP_4:36

for G being Group holds not 1_ G is generating by Th35;

definition

let G be Group;
let H be Subgroup of G;

attr H is maximal means :: GROUP_4:def 6
( multMagma(# the carrier of H, the multF of H #) <> multMagma(# the carrier of G, the multF of G #) & ( for K being strict Subgroup of G st multMagma(# the carrier of H, the multF of H #) <> K & H is Subgroup of K holds
K = multMagma(# the carrier of G, the multF of G #) ) );

end;

:: deftheorem defines maximal GROUP_4:def 6 :

theorem Th37: :: GROUP_4:37

for G being strict Group
for H being strict Subgroup of G
for a being Element of G st H is maximal & not a in H holds
gr ((carr H) \/ {a}) = G

proof end;

::
:: Frattini subgroup.
::

definition

let G be Group;

::

WP:

Frattini subgroup

func Phi G -> strict Subgroup of G means :Def7: :: GROUP_4:def 7
the carrier of it = meet { A where A is Subset of G : ex H being strict Subgroup of G st
( A = the carrier of H & H is maximal ) } if ex H being strict Subgroup of G st H is maximal
otherwise it = multMagma(# the carrier of G, the multF of G #);

proof end;

uniqueness by GROUP_2:59;
correctness ;

end;

:: deftheorem Def7 defines Phi GROUP_4:def 7 :

theorem Th38: :: GROUP_4:38

for G being Group
for a being Element of G
for G being Group st ex H being strict Subgroup of G st H is maximal holds
( a in Phi G iff for H being strict Subgroup of G st H is maximal holds
a in H )

proof end;

theorem :: GROUP_4:39

for G being Group
for a being Element of G st ( for H being strict Subgroup of G holds not H is maximal ) holds
a in Phi G

proof end;

theorem Th40: :: GROUP_4:40

for G being Group
for H being strict Subgroup of G st H is maximal holds
Phi G is Subgroup of H

proof end;

theorem Th41: :: GROUP_4:41

for G being strict Group holds the carrier of (Phi G) = { a where a is Element of G : not a is generating }

proof end;

theorem :: GROUP_4:42

for G being strict Group
for a being Element of G holds
( a in Phi G iff not a is generating )

proof end;

definition

let G be Group;
let H1, H2 be Subgroup of G;

func H1 * H2 -> Subset of G equals :: GROUP_4:def 8
(carr H1) * (carr H2);

end;

:: deftheorem defines * GROUP_4:def 8 :

theorem :: GROUP_4:43

for G being Group
for H1, H2 being Subgroup of G holds
( H1 * H2 = (carr H1) * (carr H2) & H1 * H2 = H1 * (carr H2) & H1 * H2 = (carr H1) * H2 ) ;

theorem :: GROUP_4:44

for G being Group
for H1, H2, H3 being Subgroup of G holds (H1 * H2) * H3 = H1 * (H2 * H3)

proof end;

theorem :: GROUP_4:45

for G being Group
for a being Element of G
for H1, H2 being Subgroup of G holds (a * H1) * H2 = a * (H1 * H2)

proof end;

theorem :: GROUP_4:46

for G being Group
for a being Element of G
for H1, H2 being Subgroup of G holds (H1 * H2) * a = H1 * (H2 * a)

proof end;

theorem :: GROUP_4:47

for G being Group
for A being Subset of G
for H1, H2 being Subgroup of G holds (A * H1) * H2 = A * (H1 * H2)

proof end;

theorem :: GROUP_4:48

for G being Group
for A being Subset of G
for H1, H2 being Subgroup of G holds (H1 * H2) * A = H1 * (H2 * A)

proof end;

definition

let G be Group;
let H1, H2 be Subgroup of G;

func H1 "\/" H2 -> strict Subgroup of G equals :: GROUP_4:def 9
gr ((carr H1) \/ (carr H2));

end;

:: deftheorem defines "\/" GROUP_4:def 9 :

theorem :: GROUP_4:49

for G being Group
for a being Element of G
for H1, H2 being Subgroup of G holds
( a in H1 "\/" H2 iff ex F being FinSequence of the carrier of G ex I being FinSequence of INT st
( len F = len I & rng F c= (carr H1) \/ (carr H2) & a = Product (F |^ I) ) ) by Th28;

theorem :: GROUP_4:50

for G being Group
for H1, H2 being Subgroup of G holds H1 "\/" H2 = gr (H1 * H2)

proof end;

theorem Th51: :: GROUP_4:51

for G being Group
for H1, H2 being Subgroup of G st H1 * H2 = H2 * H1 holds
the carrier of (H1 "\/" H2) = H1 * H2

proof end;

theorem :: GROUP_4:52

for G being Group
for H1, H2 being Subgroup of G st G is commutative Group holds
the carrier of (H1 "\/" H2) = H1 * H2

proof end;

theorem Th53: :: GROUP_4:53

for G being Group
for N1, N2 being strict normal Subgroup of G holds the carrier of (N1 "\/" N2) = N1 * N2

proof end;

theorem :: GROUP_4:54

for G being Group
for N1, N2 being strict normal Subgroup of G holds N1 "\/" N2 is normal Subgroup of G

proof end;

theorem :: GROUP_4:55

for G being Group
for H being strict Subgroup of G holds H "\/" H = H by Th31;

theorem :: GROUP_4:56

for G being Group
for H1, H2 being Subgroup of G holds H1 "\/" H2 = H2 "\/" H1 ;

Lm4: for G being Group
for H1, H2 being Subgroup of G holds H1 is Subgroup of H1 "\/" H2

proof end;

Lm5: for G being Group
for H1, H2, H3 being Subgroup of G holds (H1 "\/" H2) "\/" H3 is Subgroup of H1 "\/" (H2 "\/" H3)

proof end;

theorem Th57: :: GROUP_4:57

for G being Group
for H1, H2, H3 being Subgroup of G holds (H1 "\/" H2) "\/" H3 = H1 "\/" (H2 "\/" H3)

proof end;

theorem :: GROUP_4:58

for G being Group
for H being strict Subgroup of G holds
( ((1). G) "\/" H = H & H "\/" ((1). G) = H )

proof end;

theorem Th59: :: GROUP_4:59

for G being Group
for H being Subgroup of G holds
( ((Omega). G) "\/" H = (Omega). G & H "\/" ((Omega). G) = (Omega). G )

proof end;

theorem Th60: :: GROUP_4:60

for G being Group
for H1, H2 being Subgroup of G holds
( H1 is Subgroup of H1 "\/" H2 & H2 is Subgroup of H1 "\/" H2 )

proof end;

theorem Th61: :: GROUP_4:61

for G being Group
for H1 being Subgroup of G
for H2 being strict Subgroup of G holds
( H1 is Subgroup of H2 iff H1 "\/" H2 = H2 )

proof end;

theorem :: GROUP_4:62

for G being Group
for H1, H2, H3 being Subgroup of G st H1 is Subgroup of H2 holds
H1 is Subgroup of H2 "\/" H3

proof end;

theorem :: GROUP_4:63

for G being Group
for H1, H2 being Subgroup of G
for H3 being strict Subgroup of G st H1 is Subgroup of H3 & H2 is Subgroup of H3 holds
H1 "\/" H2 is Subgroup of H3

proof end;

theorem :: GROUP_4:64

for G being Group
for H1 being Subgroup of G
for H3, H2 being strict Subgroup of G st H1 is Subgroup of H2 holds
H1 "\/" H3 is Subgroup of H2 "\/" H3

proof end;

theorem :: GROUP_4:65

for G being Group
for H1, H2 being Subgroup of G holds H1 /\ H2 is Subgroup of H1 "\/" H2

proof end;

theorem Th66: :: GROUP_4:66

for G being Group
for H1 being Subgroup of G
for H2 being strict Subgroup of G holds (H1 /\ H2) "\/" H2 = H2

proof end;

theorem Th67: :: GROUP_4:67

for G being Group
for H2 being Subgroup of G
for H1 being strict Subgroup of G holds H1 /\ (H1 "\/" H2) = H1

proof end;

theorem :: GROUP_4:68

for G being Group
for H1, H2 being strict Subgroup of G holds
( H1 "\/" H2 = H2 iff H1 /\ H2 = H1 )

proof end;

definition

let G be Group;

func SubJoin G -> BinOp of (Subgroups G) means :Def10: :: GROUP_4:def 10
for H1, H2 being strict Subgroup of G holds it . (H1,H2) = H1 "\/" H2;

proof end;

proof end;

end;

:: deftheorem Def10 defines SubJoin GROUP_4:def 10 :

definition

let G be Group;

func SubMeet G -> BinOp of (Subgroups G) means :Def11: :: GROUP_4:def 11
for H1, H2 being strict Subgroup of G holds it . (H1,H2) = H1 /\ H2;

proof end;

proof end;

end;

:: deftheorem Def11 defines SubMeet GROUP_4:def 11 :

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 )

proof end;

definition

let G be Group;

func lattice G -> strict Lattice equals :: GROUP_4:def 12
LattStr(# (Subgroups G),(SubJoin G),(SubMeet G) #);

coherence by Lm6;

end;

:: deftheorem defines lattice GROUP_4:def 12 :

theorem :: GROUP_4:69

for G being Group holds the carrier of (lattice G) = Subgroups G ;

theorem :: GROUP_4:70

for G being Group holds the L_join of (lattice G) = SubJoin G ;

theorem :: GROUP_4:71

for G being Group holds the L_meet of (lattice G) = SubMeet G ;

registration

let G be Group;

cluster lattice G -> strict lower-bounded upper-bounded ;

coherence by Lm6;

end;

theorem :: GROUP_4:72

for G being Group holds Bottom (lattice G) = (1). G

proof end;

theorem :: GROUP_4:73

for G being Group holds Top (lattice G) = (Omega). G

proof end;