GROUP_10: The Sylow Theorems

proof end;

end;

:: ALG I.5.1 DEF 1

definition

let S be non empty unital multMagma ;
let E be set ;
mode LeftOperation of S,E is being_left_operation Action of the carrier of S,E;

end;

scheme :: GROUP_10:sch 1
ExLeftOperation{ F₁() -> set , F₂() -> non empty Group-like multMagma , F₃( Element of F₂()) -> Function of F₁(),F₁() } :

ex T being LeftOperation of F₂(),F₁() st
for s being Element of F₂() holds T . s = F₃(s)

provided

A1: F₃((1_ F₂())) = id F₁() and
A2: for s1, s2 being Element of F₂() holds F₃((s1 * s2)) = F₃(s1) * F₃(s2)

proof end;

registration

let E be non empty set ;
let S be non empty Group-like multMagma ;
let s be Element of S;
let LO be LeftOperation of S,E;

cluster ^ -> one-to-one for Function of E,E;

coherence

proof end;

end;

notation

let S be non empty multMagma ;
let s be Element of S;
synonym the_left_translation_of s for s * ;

end;

definition

let S be non empty Group-like associative multMagma ;

func the_left_operation_of S -> LeftOperation of S, the carrier of S means :Def2: :: GROUP_10:def 2
for s being Element of S holds it . s = the_left_translation_of s;

proof end;

proof end;

end;

:: deftheorem Def2 defines the_left_operation_of GROUP_10:def 2 :

definition

let E, n be set ;

func the_subsets_of_card (n,E) -> Subset-Family of E equals :: GROUP_10:def 3
{ X where X is Subset of E : card X = n } ;

proof end;

end;

:: deftheorem defines the_subsets_of_card GROUP_10:def 3 :

registration

let E be finite set ;
let n be set ;

cluster the_subsets_of_card (n,E) -> finite ;

correctness ;

end;

theorem Th1: :: GROUP_10:1

for n being Nat
for E being non empty set st card n c= card E holds
not the_subsets_of_card (n,E) is empty

proof end;

theorem Th2: :: GROUP_10:2

for E being non empty finite set
for k being Element of NAT
for x1, x2 being set st x1 <> x2 holds
card (Choose (E,k,x1,x2)) = card (the_subsets_of_card (k,E))

proof end;

definition

let E be non empty set ;
let n be Nat;
let S be non empty Group-like multMagma ;
let s be Element of S;
let LO be LeftOperation of S,E;
assume A1: card n c= card E ;

func the_extension_of_left_translation_of (n,s,LO) -> Function of (the_subsets_of_card (n,E)),(the_subsets_of_card (n,E)) means :Def4: :: GROUP_10:def 4
for X being Element of the_subsets_of_card (n,E) holds it . X = (LO ^ s) .: X;

proof end;

proof end;

end;

:: deftheorem Def4 defines the_extension_of_left_translation_of GROUP_10:def 4 :

definition

let E be non empty set ;
let n be Nat;
let S be non empty Group-like multMagma ;
let LO be LeftOperation of S,E;
assume A1: card n c= card E ;

func the_extension_of_left_operation_of (n,LO) -> LeftOperation of S,(the_subsets_of_card (n,E)) means :Def5: :: GROUP_10:def 5
for s being Element of S holds it . s = the_extension_of_left_translation_of (n,s,LO);

proof end;

proof end;

end;

:: deftheorem Def5 defines the_extension_of_left_operation_of GROUP_10:def 5 :

definition

let S be non empty multMagma ;
let s be Element of S;
let Z be non empty set ;

func the_left_translation_of (s,Z) -> Function of [: the carrier of S,Z:],[: the carrier of S,Z:] means :Def6: :: GROUP_10:def 6
for z1 being Element of [: the carrier of S,Z:] ex z2 being Element of [: the carrier of S,Z:] ex s1, s2 being Element of S ex z being Element of Z st
( z2 = it . z1 & s2 = s * s1 & z1 = [s1,z] & z2 = [s2,z] );

proof end;

proof end;

end;

:: deftheorem Def6 defines the_left_translation_of GROUP_10:def 6 :

definition

let S be non empty Group-like associative multMagma ;
let Z be non empty set ;

func the_left_operation_of (S,Z) -> LeftOperation of S,[: the carrier of S,Z:] means :Def7: :: GROUP_10:def 7
for s being Element of S holds it . s = the_left_translation_of (s,Z);

proof end;

proof end;

end;

:: deftheorem Def7 defines the_left_operation_of GROUP_10:def 7 :

definition

let G be Group;
let H, P be Subgroup of G;
let h be Element of H;

func the_left_translation_of (h,P) -> Function of (Left_Cosets P),(Left_Cosets P) means :Def8: :: GROUP_10:def 8
for P1 being Element of Left_Cosets P ex P2 being Element of Left_Cosets P ex A1, A2 being Subset of G ex g being Element of G st
( P2 = it . P1 & A2 = g * A1 & A1 = P1 & A2 = P2 & g = h );

proof end;

proof end;

end;

:: deftheorem Def8 defines the_left_translation_of GROUP_10:def 8 :

definition

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

func the_left_operation_of (H,P) -> LeftOperation of H,(Left_Cosets P) means :Def9: :: GROUP_10:def 9
for h being Element of H holds it . h = the_left_translation_of (h,P);

proof end;

proof end;

end;

:: deftheorem Def9 defines the_left_operation_of GROUP_10:def 9 :

:: stabilizer
:: ALG I.5.2 Definition 3

definition

let G be Group;
let E be non empty set ;
let T be LeftOperation of G,E;
let A be Subset of E;

func the_strict_stabilizer_of (A,T) -> strict Subgroup of G means :Def10: :: GROUP_10:def 10
the carrier of it = { g where g is Element of G : (T ^ g) .: A = A } ;

proof end;

uniqueness by GROUP_2:59;

end;

:: deftheorem Def10 defines the_strict_stabilizer_of GROUP_10:def 10 :

definition

let G be Group;
let E be non empty set ;
let T be LeftOperation of G,E;
let x be Element of E;

func the_strict_stabilizer_of (x,T) -> strict Subgroup of G equals :: GROUP_10:def 11
the_strict_stabilizer_of ({x},T);

correctness ;

end;

:: deftheorem defines the_strict_stabilizer_of GROUP_10:def 11 :

definition

let S be non empty unital multMagma ;
let E be set ;
let T be LeftOperation of S,E;
let x be Element of E;

pred x is_fixed_under T means :: GROUP_10:def 12
for s being Element of S holds x = (T ^ s) . x;

end;

:: deftheorem defines is_fixed_under GROUP_10:def 12 :

definition

let S be non empty unital multMagma ;
let E be set ;
let T be LeftOperation of S,E;

func the_fixed_points_of T -> Subset of E equals :Def13: :: GROUP_10:def 13
{ x where x is Element of E : x is_fixed_under T } if not E is empty
otherwise {} E;

proof end;

end;

:: deftheorem Def13 defines the_fixed_points_of GROUP_10:def 13 :

:: ALG I.5.4 Definition 5

definition

let S be non empty unital multMagma ;
let E be set ;
let T be LeftOperation of S,E;
let x, y be Element of E;

pred x,y are_conjugated_under T means :: GROUP_10:def 14
ex s being Element of S st y = (T ^ s) . x;

end;

:: deftheorem defines are_conjugated_under GROUP_10:def 14 :

theorem Th3: :: GROUP_10:3

for S being non empty unital multMagma
for E being non empty set
for x being Element of E
for T being LeftOperation of S,E holds x,x are_conjugated_under T

proof end;

theorem Th4: :: GROUP_10:4

for G being Group
for E being non empty set
for x, y being Element of E
for T being LeftOperation of G,E st x,y are_conjugated_under T holds
y,x are_conjugated_under T

proof end;

theorem Th5: :: GROUP_10:5

for S being non empty unital multMagma
for E being non empty set
for x, y, z being Element of E
for T being LeftOperation of S,E st x,y are_conjugated_under T & y,z are_conjugated_under T holds
x,z are_conjugated_under T

proof end;

definition

let S be non empty unital multMagma ;
let E be non empty set ;
let T be LeftOperation of S,E;
let x be Element of E;

func the_orbit_of (x,T) -> Subset of E equals :: GROUP_10:def 15
{ y where y is Element of E : x,y are_conjugated_under T } ;

proof end;

end;

:: deftheorem defines the_orbit_of GROUP_10:def 15 :

registration

let S be non empty unital multMagma ;
let E be non empty set ;
let x be Element of E;
let T be LeftOperation of S,E;

cluster the_orbit_of (x,T) -> non empty ;

coherence

proof end;

end;

theorem Th6: :: GROUP_10:6

for G being Group
for E being non empty set
for x, y being Element of E
for T being LeftOperation of G,E holds
( the_orbit_of (x,T) misses the_orbit_of (y,T) or the_orbit_of (x,T) = the_orbit_of (y,T) )

proof end;

theorem Th7: :: GROUP_10:7

for S being non empty unital multMagma
for E being non empty finite set
for x being Element of E
for T being LeftOperation of S,E st x is_fixed_under T holds
the_orbit_of (x,T) = {x}

proof end;

:: the orbit-stabilizer theorem

theorem Th8: :: GROUP_10:8

for G being Group
for E being non empty set
for a being Element of E
for T being LeftOperation of G,E holds card (the_orbit_of (a,T)) = Index (the_strict_stabilizer_of (a,T))

proof end;

definition

let G be Group;
let E be non empty set ;
let T be LeftOperation of G,E;

func the_orbits_of T -> a_partition of E equals :: GROUP_10:def 16
{ X where X is Subset of E : ex x being Element of E st X = the_orbit_of (x,T) } ;

proof end;

end;

:: deftheorem defines the_orbits_of GROUP_10:def 16 :

:: ALG I.6.5 Definition 9

definition

let p be Nat;
let G be Group;

attr G is p -group means :: GROUP_10:def 17
ex r being Nat st card G = p |^ r;

end;

:: deftheorem defines -group GROUP_10:def 17 :

Lm1: for n being non zero Nat holds card (INT.Group n) = n

proof end;

registration

let p be non zero Nat;

cluster INT.Group p -> p -group ;

coherence

proof end;

end;

registration

let p be non zero Nat;

cluster non empty finite strict unital Group-like associative commutative cyclic p -group for multMagma ;

proof end;

end;

:: like WEDDWITT:39
:: ALG I.6.5 PRO 11

theorem Th9: :: GROUP_10:9

for E being non empty finite set
for G being finite Group
for p being prime Nat
for T being LeftOperation of G,E st G is p -group holds
(card (the_fixed_points_of T)) mod p = (card E) mod p

proof end;

:: ALG I.6.6 Definition 10

definition

let p be Nat;
let G be Group;
let P be Subgroup of G;

pred P is_Sylow_p-subgroup_of_prime p means :: GROUP_10:def 18
( P is p -group & not p divides index P );

end;

:: deftheorem defines is_Sylow_p-subgroup_of_prime GROUP_10:def 18 :

Lm2: for n being Nat
for p being prime Nat st n <> 0 holds
ex m, r being Nat st
( n = (p |^ r) * m & not p divides m )

proof end;

Lm3: for X, Y being non empty set holds card { [:X,{y}:] where y is Element of Y : verum } = card Y

proof end;

Lm4: for G being finite Group
for E, T being non empty set
for LO being LeftOperation of G,E
for n being Nat st LO = the_left_operation_of (G,T) & n = card G & E = [: the carrier of G,T:] & card n c= card E holds
the_fixed_points_of (the_extension_of_left_operation_of (n,LO)) = { [: the carrier of G,{t}:] where t is Element of T : verum }

proof end;

Lm5: for n, m, r being Nat
for p being prime Nat st n = (p |^ r) * m & not p divides m holds
(n choose (p |^ r)) mod p <> 0

proof end;

Lm6: for r, n, m1, m2 being Nat
for p being prime Nat st (p |^ r) * n = m1 * m2 & not p divides n & not p divides m2 holds
p |^ r divides m1

proof end;

Lm7: for G being Group
for A being non empty Subset of G
for x being Element of G holds card A = card (A * x)

proof end;

:: ALG I.6.6 Theorem 2
:: The first Sylow theorem
:: WP:

First Sylow Theorem

theorem Th10: :: GROUP_10:10

for G being finite Group
for p being prime Nat ex P being strict Subgroup of G st P is_Sylow_p-subgroup_of_prime p

proof end;

Lm8: for n, r being Nat
for p being prime Nat st n divides p |^ r & n > 1 holds
p divides n

proof end;

:: ALG I.6.6 Corollary
:: The Cauchy theorem

theorem :: GROUP_10:11

for G being finite Group
for p being prime Nat st p divides card G holds
ex g being Element of G st ord g = p

proof end;

:: ALG I.6.6 Theorem 3
:: The second Sylow theorem
:: WP:

Second Sylow Theorem

theorem Th12: :: GROUP_10:12

for G being finite Group
for p being prime Nat holds
( ( for H being Subgroup of G st H is p -group holds
ex P being Subgroup of G st
( P is_Sylow_p-subgroup_of_prime p & H is Subgroup of P ) ) & ( for P1, P2 being Subgroup of G st P1 is_Sylow_p-subgroup_of_prime p & P2 is_Sylow_p-subgroup_of_prime p holds
P1,P2 are_conjugated ) )

proof end;

definition

let G be Group;
let p be Nat;

func the_sylow_p-subgroups_of_prime (p,G) -> Subset of (Subgroups G) equals :: GROUP_10:def 19
{ H where H is Element of Subgroups G : ex P being strict Subgroup of G st
( P = H & P is_Sylow_p-subgroup_of_prime p ) } ;

proof end;

end;

:: deftheorem defines the_sylow_p-subgroups_of_prime GROUP_10:def 19 :

registration

let G be finite Group;
let p be prime Nat;

cluster the_sylow_p-subgroups_of_prime (p,G) -> non empty finite ;

proof end;

end;

definition

let G be finite Group;
let p be prime Nat;
let H be Subgroup of G;
let h be Element of H;

func the_left_translation_of (h,p) -> Function of (the_sylow_p-subgroups_of_prime (p,G)),(the_sylow_p-subgroups_of_prime (p,G)) means :Def20: :: GROUP_10:def 20
for P1 being Element of the_sylow_p-subgroups_of_prime (p,G) ex P2 being Element of the_sylow_p-subgroups_of_prime (p,G) ex H1, H2 being strict Subgroup of G ex g being Element of G st
( P2 = it . P1 & P1 = H1 & P2 = H2 & h " = g & H2 = H1 |^ g );

proof end;

proof end;

end;

:: deftheorem Def20 defines the_left_translation_of GROUP_10:def 20 :

definition

let G be finite Group;
let p be prime Nat;
let H be Subgroup of G;

func the_left_operation_of (H,p) -> LeftOperation of H,(the_sylow_p-subgroups_of_prime (p,G)) means :Def21: :: GROUP_10:def 21
for h being Element of H holds it . h = the_left_translation_of (h,p);

proof end;