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Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgsumwrev 14801 A sum in an opposite monoid is the regular sum of a reversed word. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Proof shortened by Mario Carneiro, 28-Feb-2016.)
 |-  B  =  ( Base `  M )   &    |-  O  =  (oppg `  M )   =>    |-  ( ( M  e.  Mnd  /\  W  e. Word  B )  ->  ( O  gsumg 
 W )  =  ( M  gsumg  (reverse `  W )
 ) )
 
10.2.9  p-Groups and Sylow groups; Sylow's theorems
 
Syntaxcod 14802 Extend class notation to include the order function on the elements of a group.
 class  od
 
Syntaxcgex 14803 Extend class notation to include the order function on the elements of a group.
 class gEx
 
Syntaxcpgp 14804 Extend class notation to include the class of all p-groups.
 class pGrp
 
Syntaxcslw 14805 Extend class notation to include the class of all Sylow p-subgroups of a group.
 class pSyl
 
Definitiondf-od 14806* Define the order of an element in a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.)
 |- 
 od  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g
 )  |->  [_ { n  e. 
 NN  |  ( n (.g `  g ) x )  =  ( 0g
 `  g ) }  /  i ]_ if (
 i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) ) )
 
Definitiondf-gex 14807* Define the exponent of a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.)
 |- gEx 
 =  ( g  e. 
 _V  |->  [_ { n  e. 
 NN  |  A. x  e.  ( Base `  g )
 ( n (.g `  g
 ) x )  =  ( 0g `  g
 ) }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
 
Definitiondf-pgp 14808* Define the set of p-groups, which are groups such that every element has a power of  p as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |- pGrp  =  { <. p ,  g >.  |  ( ( p  e.  Prime  /\  g  e. 
 Grp )  /\  A. x  e.  ( Base `  g ) E. n  e.  NN0  ( ( od
 `  g ) `  x )  =  ( p ^ n ) ) }
 
Definitiondf-slw 14809* Define the set of Sylow p-subgroups of a group  g. A Sylow p-subgroup is a p-group that is not a subgroup of any other p-groups in  g. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |- pSyl  =  ( p  e.  Prime ,  g  e.  Grp  |->  { h  e.  (SubGrp `  g )  |  A. k  e.  (SubGrp `  g ) ( ( h  C_  k  /\  p pGrp  ( gs  k ) )  <->  h  =  k
 ) } )
 
Theoremodfval 14810* Value of the order function. (Contributed by Mario Carneiro, 13-Jul-2014.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  O  =  ( od `  G )   =>    |-  O  =  ( x  e.  X  |->  [_ { y  e.  NN  |  ( y 
 .x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
 
Theoremodval 14811* Second substitution for the group order definition. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  O  =  ( od `  G )   &    |-  I  =  {
 y  e.  NN  |  ( y  .x.  A )  =  .0.  }   =>    |-  ( A  e.  X  ->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
 
Theoremodlem1 14812* The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  O  =  ( od `  G )   &    |-  I  =  {
 y  e.  NN  |  ( y  .x.  A )  =  .0.  }   =>    |-  ( A  e.  X  ->  ( ( ( O `  A )  =  0  /\  I  =  (/) )  \/  ( O `  A )  e.  I ) )
 
Theoremodcl 14813 The order of a group element is always a nonnegative integer. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
 
Theoremodf 14814 Functionality of the group element order. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  O : X --> NN0
 
Theoremodid 14815 Any element to the power of its order is the identity. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( A  e.  X  ->  ( ( O `  A )  .x.  A )  =  .0.  )
 
Theoremodlem2 14816 Any positive annihilator of a group element is an upper bound on the (positive) order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( A  e.  X  /\  N  e.  NN  /\  ( N  .x.  A )  =  .0.  )  ->  ( O `  A )  e.  ( 1 ... N ) )
 
Theoremodmodnn0 14817 Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  /\  ( O `  A )  e.  NN )  ->  ( ( N 
 mod  ( O `  A ) )  .x.  A )  =  ( N 
 .x.  A ) )
 
Theoremmndodconglem 14818 Lemma for mndodcong 14819. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  ( O `  A )  e.  NN )   &    |-  ( ph  ->  M  e.  NN0 )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  M  <  ( O `  A ) )   &    |-  ( ph  ->  N  <  ( O `  A ) )   &    |-  ( ph  ->  ( M  .x.  A )  =  ( N  .x.  A )
 )   =>    |-  ( ( ph  /\  M  <_  N )  ->  M  =  N )
 
Theoremmndodcong 14819 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( ( G  e.  Mnd  /\  A  e.  X )  /\  ( M  e.  NN0  /\  N  e.  NN0 )  /\  ( O `
  A )  e. 
 NN )  ->  (
 ( O `  A )  ||  ( M  -  N )  <->  ( M  .x.  A )  =  ( N 
 .x.  A ) ) )
 
Theoremmndodcongi 14820 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. For monoids, the reverse implication is false for elements with infinite order. For example, the powers of  2 mod  10 are 1,2,4,8,6,2,4,8,6,... so that the identity 1 never repeats, which is infinite order by our definition, yet other numbers like 6 appear many times in the sequence. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  ( ( O `  A )  ||  ( M  -  N )  ->  ( M  .x.  A )  =  ( N  .x.  A ) ) )
 
Theoremoddvdsnn0 14821 The only multiples of  A that are equal to the identity are the multiples of the order of  A. (Contributed by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  N  e.  NN0 )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  )
 )
 
Theoremodnncl 14822 If a nonzero multiple of an element is zero, the element has positive order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  =/=  0  /\  ( N  .x.  A )  =  .0.  )
 )  ->  ( O `  A )  e.  NN )
 
Theoremodmod 14823 Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A )  e.  NN )  ->  ( ( N 
 mod  ( O `  A ) )  .x.  A )  =  ( N 
 .x.  A ) )
 
Theoremoddvds 14824 The only multiples of  A that are equal to the identity are the multiples of the order of  A. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( ( O `  A )  ||  N  <->  ( N  .x.  A )  =  .0.  )
 )
 
Theoremoddvdsi 14825 Any group element is annihilated by any multiple of its order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A ) 
 ||  N )  ->  ( N  .x.  A )  =  .0.  )
 
Theoremodcong 14826 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( M  e.  ZZ  /\  N  e.  ZZ )
 )  ->  ( ( O `  A )  ||  ( M  -  N ) 
 <->  ( M  .x.  A )  =  ( N  .x.  A ) ) )
 
Theoremodeq 14827* The oddvds 14824 property uniquely defines the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  NN0 )  ->  ( N  =  ( O `  A )  <->  A. y  e.  NN0  ( N  ||  y  <->  ( y  .x.  A )  =  .0.  )
 ) )
 
Theoremodval2 14828* A non-conditional definition of the group order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( O `  A )  =  ( iota_ x  e. 
 NN0 A. y  e.  NN0  ( x  ||  y  <->  ( y  .x.  A )  =  .0.  )
 ) )
 
Theoremodmulgid 14829 A relationship between the order of a multiple and the order of the basepoint. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  K  e.  ZZ )  ->  ( ( O `  ( N  .x.  A ) )  ||  K  <->  ( O `  A )  ||  ( K  x.  N ) ) )
 
Theoremodmulg2 14830 The order of a multiple divides the order of the base point. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  ( N  .x.  A ) )  ||  ( O `  A ) )
 
Theoremodmulg 14831 Relationship between the order of an element and that of a multiple. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( O `  A )  =  (
 ( N  gcd  ( O `  A ) )  x.  ( O `  ( N  .x.  A ) ) ) )
 
Theoremodmulgeq 14832 A multiple of a point of finite order only has the same order if the multiplier is relatively prime. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( O `  A )  e.  NN )  ->  ( ( O `  ( N  .x.  A ) )  =  ( O `
  A )  <->  ( N  gcd  ( O `  A ) )  =  1 ) )
 
Theoremodbezout 14833* If  N is coprime to the order of  A, there is a modular inverse  x to cancel multiplication by  N. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  /\  ( N  gcd  ( O `  A ) )  =  1 )  ->  E. x  e.  ZZ  ( x  .x.  ( N 
 .x.  A ) )  =  A )
 
Theoremod1 14834 The order of the group identity is one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
 |-  O  =  ( od
 `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  ( O `  .0.  )  =  1 )
 
Theoremodeq1 14835 The group identity is the unique element of a group with order one. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
 |-  O  =  ( od
 `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( ( O `  A )  =  1  <->  A  =  .0.  ) )
 
Theoremodinv 14836 The order of the inverse of a group element. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   &    |-  I  =  ( inv g `  G )   &    |-  X  =  (
 Base `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( O `
  ( I `  A ) )  =  ( O `  A ) )
 
Theoremodf1 14837* The multiples of an element with infinite order form an infinite cyclic subgroup of  G. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( ( O `  A )  =  0  <->  F : ZZ -1-1-> X ) )
 
Theoremodinf 14838* The multiples of an element with infinite order form an infinite cyclic subgroup of  G. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0 )  ->  -.  ran  F  e.  Fin )
 
Theoremdfod2 14839* An alternative definition of the order of a group element is as the cardinality of the cyclic subgroup generated by the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( O `  A )  =  if ( ran  F  e.  Fin ,  ( # `  ran  F ) ,  0 )
 )
 
Theoremodcl2 14840 The order of an element of a finite group is finite. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  A  e.  X )  ->  ( O `  A )  e.  NN )
 
Theoremoddvds2 14841 The order of an element of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  A  e.  X )  ->  ( O `  A )  ||  ( # `  X ) )
 
Theoremsubmod 14842 The order of an element is the same in a subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  H  =  ( Gs  Y )   &    |-  O  =  ( od `  G )   &    |-  P  =  ( od `  H )   =>    |-  ( ( Y  e.  (SubMnd `  G )  /\  A  e.  Y )  ->  ( O `  A )  =  ( P `  A ) )
 
Theoremsubgod 14843 The order of an element is the same in a subgroup. (Contributed by Mario Carneiro, 14-Jan-2015.) (Proof shortened by Stefan O'Rear, 12-Sep-2015.)
 |-  H  =  ( Gs  Y )   &    |-  O  =  ( od `  G )   &    |-  P  =  ( od `  H )   =>    |-  ( ( Y  e.  (SubGrp `  G )  /\  A  e.  Y )  ->  ( O `  A )  =  ( P `  A ) )
 
Theoremodsubdvds 14844 The order of an element of a subgroup divides the order of the subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  O  =  ( od
 `  G )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  S  e.  Fin  /\  A  e.  S ) 
 ->  ( O `  A )  ||  ( # `  S ) )
 
Theoremodf1o1 14845* An element with zero order has infinitely many multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0
 )  ->  ( x  e.  ZZ  |->  ( x  .x.  A ) ) : ZZ -1-1-onto-> ( K `  { A }
 ) )
 
Theoremodf1o2 14846* An element with nonzero order has as many multiples as its order. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  ->  ( x  e.  ( 0..^ ( O `
  A ) ) 
 |->  ( x  .x.  A ) ) : ( 0..^ ( O `  A ) ) -1-1-onto-> ( K `
  { A }
 ) )
 
Theoremodhash 14847 An element of zero order generates an infinite subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  =  0
 )  ->  ( # `  ( K `  { A }
 ) )  =  +oo )
 
Theoremodhash2 14848 If an element has nonzero order, it generates a subgroup with size equal to the order. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  ->  ( # `  ( K `  { A }
 ) )  =  ( O `  A ) )
 
Theoremodhash3 14849 An element which generates a finite subgroup has order the size of that subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( K `  { A } )  e. 
 Fin )  ->  ( O `  A )  =  ( # `  ( K `  { A }
 ) ) )
 
Theoremodngen 14850* A cyclic subgroup of size  ( O `  A
) has  ( phi `  ( O `  A ) ) generators. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  ( O `  A )  e.  NN )  ->  ( # `  { x  e.  ( K `  { A } )  |  ( O `  x )  =  ( O `  A ) } )  =  ( phi `  ( O `  A ) ) )
 
Theoremgexval 14851* Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  E  =  (gEx `  G )   &    |-  I  =  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0.  }   =>    |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
 
Theoremgexlem1 14852* The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 23-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  E  =  (gEx `  G )   &    |-  I  =  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0.  }   =>    |-  ( G  e.  V  ->  ( ( E  =  0  /\  I  =  (/) )  \/  E  e.  I
 ) )
 
Theoremgexcl 14853 The exponent of a group is a nonnegative integer. (Contributed by Mario Carneiro, 23-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( G  e.  V  ->  E  e.  NN0 )
 
Theoremgexid 14854 Any element to the power of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( A  e.  X  ->  ( E  .x.  A )  =  .0.  )
 
Theoremgexlem2 14855* Any positive annihilator of all the group elements is an upper bound on the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  V  /\  N  e.  NN  /\ 
 A. x  e.  X  ( N  .x.  x )  =  .0.  )  ->  E  e.  ( 1 ... N ) )
 
Theoremgexdvdsi 14856 Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  E  ||  N )  ->  ( N  .x.  A )  =  .0.  )
 
Theoremgexdvds 14857* The only  N that annihilate all the elements of the group are the multiples of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( E  ||  N  <->  A. x  e.  X  ( N  .x.  x )  =  .0.  ) )
 
Theoremgexdvds2 14858* An integer divides the group exponent iff it divides all the group orders. In other words, the group exponent is the LCM of the orders of all the elements. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( E  ||  N  <->  A. x  e.  X  ( O `  x ) 
 ||  N ) )
 
Theoremgexod 14859 Any group element is annihilated by any multiple of the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( O `  A )  ||  E )
 
Theoremgexcl3 14860* If the order of every group element is bounded by  N, the group has finite exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ...
 N ) )  ->  E  e.  NN )
 
Theoremgexnnod 14861 Every group element has finite order if the exponent is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Grp  /\  E  e.  NN  /\  A  e.  X )  ->  ( O `  A )  e.  NN )
 
Theoremgexcl2 14862 The exponent of a finite group is finite. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin )  ->  E  e.  NN )
 
Theoremgexdvds3 14863 The exponent of a finite group divides the order (cardinality) of the group. Corollary of Lagrange's theorem for the order of a subgroup. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin )  ->  E  ||  ( # `  X ) )
 
Theoremgex1 14864 A group or monoid has exponent 1 iff it is trivial. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( G  e.  Mnd  ->  ( E  =  1  <->  X 
 ~~  1o ) )
 
Theoremispgp 14865* A group is a  P-group if every element has some power of  P as its order. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( P pGrp  G  <->  ( P  e.  Prime  /\  G  e.  Grp  /\ 
 A. x  e.  X  E. n  e.  NN0  ( O `  x )  =  ( P ^ n ) ) )
 
Theorempgpprm 14866 Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( P pGrp  G  ->  P  e.  Prime )
 
Theorempgpgrp 14867 Reverse closure for the second argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( P pGrp  G  ->  G  e.  Grp )
 
Theorempgpfi1 14868 A finite group with order a power of a prime  P is a 
P-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  P  e.  Prime  /\  N  e.  NN0 )  ->  (
 ( # `  X )  =  ( P ^ N )  ->  P pGrp  G ) )
 
Theorempgp0 14869 The identity subgroup is a  P-group for every prime  P. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  P  e.  Prime )  ->  P pGrp  ( Gs  {  .0.  } ) )
 
Theoremsubgpgp 14870 A subgroup of a p-group is a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  ( ( P pGrp  G  /\  S  e.  (SubGrp `  G ) )  ->  P pGrp  ( Gs  S ) )
 
Theoremsylow1lem1 14871* Lemma for sylow1 14876. The p-adic valuation of the size of  S is equal to the number of excess powers of  P in  ( # `  X
)  /  ( P ^ N ). (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   &    |-  .+  =  ( +g  `  G )   &    |-  S  =  { s  e.  ~P X  |  ( # `  s
 )  =  ( P ^ N ) }   =>    |-  ( ph  ->  ( ( # `  S )  e.  NN  /\  ( P  pCnt  ( # `
  S ) )  =  ( ( P 
 pCnt  ( # `  X ) )  -  N ) ) )
 
Theoremsylow1lem2 14872* Lemma for sylow1 14876. The function  .(+) is a group action on  S. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   &    |-  .+  =  ( +g  `  G )   &    |-  S  =  { s  e.  ~P X  |  ( # `  s
 )  =  ( P ^ N ) }   &    |-  .(+)  =  ( x  e.  X ,  y  e.  S  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   =>    |-  ( ph  ->  .(+)  e.  ( G  GrpAct  S ) )
 
Theoremsylow1lem3 14873* Lemma for sylow1 14876. One of the orbits of the group action has p-adic valuation less than the prime count of the set  S. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   &    |-  .+  =  ( +g  `  G )   &    |-  S  =  { s  e.  ~P X  |  ( # `  s
 )  =  ( P ^ N ) }   &    |-  .(+)  =  ( x  e.  X ,  y  e.  S  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  S  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  ( ph  ->  E. w  e.  S  ( P  pCnt  ( # `  [ w ]  .~  ) )  <_  ( ( P  pCnt  ( # `  X ) )  -  N ) )
 
Theoremsylow1lem4 14874* Lemma for sylow1 14876. The stabilizer subgroup of any element of  S is at most  P ^ N in size. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   &    |-  .+  =  ( +g  `  G )   &    |-  S  =  { s  e.  ~P X  |  ( # `  s
 )  =  ( P ^ N ) }   &    |-  .(+)  =  ( x  e.  X ,  y  e.  S  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  S  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   &    |-  ( ph  ->  B  e.  S )   &    |-  H  =  { u  e.  X  |  ( u 
 .(+)  B )  =  B }   =>    |-  ( ph  ->  ( # `
  H )  <_  ( P ^ N ) )
 
Theoremsylow1lem5 14875* Lemma for sylow1 14876. Using Lagrange's theorem and the orbit-stabilizer theorem, show that there is a subgroup with size exactly  P ^ N. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   &    |-  .+  =  ( +g  `  G )   &    |-  S  =  { s  e.  ~P X  |  ( # `  s
 )  =  ( P ^ N ) }   &    |-  .(+)  =  ( x  e.  X ,  y  e.  S  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  S  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   &    |-  ( ph  ->  B  e.  S )   &    |-  H  =  { u  e.  X  |  ( u 
 .(+)  B )  =  B }   &    |-  ( ph  ->  ( P  pCnt  ( # `  [ B ]  .~  ) )  <_  ( ( P  pCnt  ( # `  X ) )  -  N ) )   =>    |-  ( ph  ->  E. h  e.  (SubGrp `  G )
 ( # `  h )  =  ( P ^ N ) )
 
Theoremsylow1 14876* Sylow's first theorem. If  P ^ N is a prime power that divides the cardinality of  G, then  G has a supgroup with size  P ^ N. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( P ^ N ) 
 ||  ( # `  X ) )   =>    |-  ( ph  ->  E. g  e.  (SubGrp `  G )
 ( # `  g )  =  ( P ^ N ) )
 
Theoremodcau 14877* Cauchy's theorem for the order of an element in a group. A finite group whose order divides a prime 
P contains an element of order  P. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  P  ||  ( # `  X ) )  ->  E. g  e.  X  ( O `  g )  =  P )
 
Theorempgpfi 14878* The converse to pgpfi1 14868. A finite group is a  P-group iff it has size some power of  P. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin )  ->  ( P pGrp  G  <->  ( P  e.  Prime  /\  E. n  e. 
 NN0  ( # `  X )  =  ( P ^ n ) ) ) )
 
Theorempgpfi2 14879 Alternate version of pgpfi 14878. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin )  ->  ( P pGrp  G  <->  ( P  e.  Prime  /\  ( # `  X )  =  ( P ^ ( P  pCnt  ( # `  X ) ) ) ) ) )
 
Theorempgphash 14880 The order of a p-group. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( P pGrp  G  /\  X  e.  Fin )  ->  ( # `  X )  =  ( P ^ ( P  pCnt  ( # `  X ) ) ) )
 
Theoremisslw 14881* The property of being a Sylow subgroup. A Sylow  P-subgroup is a  P-group which has no proper supersets that are also  P-groups. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( H  e.  ( P pSyl  G )  <->  ( P  e.  Prime  /\  H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G ) ( ( H 
 C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k
 ) ) )
 
Theoremslwprm 14882 Reverse closure for the first argument of a Sylow  P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 2-May-2015.)
 |-  ( H  e.  ( P pSyl  G )  ->  P  e.  Prime )
 
Theoremslwsubg 14883 A Sylow  P-subgroup is a subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  ( H  e.  ( P pSyl  G )  ->  H  e.  (SubGrp `  G )
 )
 
Theoremslwispgp 14884 Defining property of a Sylow  P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  S  =  ( Gs  K )   =>    |-  ( ( H  e.  ( P pSyl  G )  /\  K  e.  (SubGrp `  G ) )  ->  ( ( H  C_  K  /\  P pGrp  S )  <->  H  =  K ) )
 
Theoremslwpss 14885 A proper superset of a Sylow subgroup is not a  P-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  S  =  ( Gs  K )   =>    |-  ( ( H  e.  ( P pSyl  G )  /\  K  e.  (SubGrp `  G )  /\  H  C.  K )  ->  -.  P pGrp  S )
 
Theoremslwpgp 14886 A Sylow  P-subgroup is a  P-group. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  S  =  ( Gs  H )   =>    |-  ( H  e.  ( P pSyl  G )  ->  P pGrp  S )
 
Theorempgpssslw 14887* Every  P-subgroup is contained in a Sylow  P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  S  =  ( Gs  H )   &    |-  F  =  ( x  e.  { y  e.  (SubGrp `  G )  |  ( P pGrp  ( Gs  y )  /\  H  C_  y ) }  |->  ( # `  x ) )   =>    |-  ( ( H  e.  (SubGrp `  G )  /\  X  e.  Fin  /\  P pGrp  S )  ->  E. k  e.  ( P pSyl  G ) H  C_  k )
 
Theoremslwn0 14888 Every finite group contains a Sylow 
P-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  ->  ( P pSyl  G )  =/=  (/) )
 
Theoremsubgslw 14889 A Sylow subgroup that is contained in a larger subgroup is also Sylow with respect to the subgroup. (The converse may not be true, though.) (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  K  e.  ( P pSyl  G )  /\  K  C_  S )  ->  K  e.  ( P pSyl  H ) )
 
Theoremsylow2alem1 14890* Lemma for sylow2a 14892. An equivalence class of fixed points is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  .(+) 
 e.  ( G  GrpAct  Y ) )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  Y  e.  Fin )   &    |-  Z  =  { u  e.  Y  |  A. h  e.  X  ( h  .(+)  u )  =  u }   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  (
 ( ph  /\  A  e.  Z )  ->  [ A ]  .~  =  { A } )
 
Theoremsylow2alem2 14891* Lemma for sylow2a 14892. All the orbits which are not for fixed points have size  |  G  |  /  |  G x  | (where  G x is the stabilizer subgroup) and thus are powers of  P. And since they are all nontrivial (because any orbit which is a singleton is a fixed point), they all divide  P, and so does the sum of all of them. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  .(+) 
 e.  ( G  GrpAct  Y ) )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  Y  e.  Fin )   &    |-  Z  =  { u  e.  Y  |  A. h  e.  X  ( h  .(+)  u )  =  u }   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  ( ph  ->  P  ||  sum_ z  e.  ( ( Y /.  .~  )  \  ~P Z ) ( # `  z
 ) )
 
Theoremsylow2a 14892* A named lemma of Sylow's second and third theorems. If  G is a finite  P-group that acts on the finite set  Y, then the set  Z of all points of  Y fixed by every element of  G has cardinality equivalent to the cardinality of  Y, 
mod  P. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  .(+) 
 e.  ( G  GrpAct  Y ) )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  Y  e.  Fin )   &    |-  Z  =  { u  e.  Y  |  A. h  e.  X  ( h  .(+)  u )  =  u }   &    |-  .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  ( ph  ->  P  ||  (
 ( # `  Y )  -  ( # `  Z ) ) )
 
Theoremsylow2blem1 14893* Lemma for sylow2b 14896. Evaluate the group action on a left coset. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  K  e.  (SubGrp `  G ) )   &    |-  .+  =  ( +g  `  G )   &    |-  .~  =  ( G ~QG  K )   &    |-  .x.  =  ( x  e.  H ,  y  e.  ( X /.  .~  )  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   =>    |-  ( ( ph  /\  B  e.  H  /\  C  e.  X )  ->  ( B 
 .x.  [ C ]  .~  )  =  [ ( B  .+  C ) ]  .~  )
 
Theoremsylow2blem2 14894* Lemma for sylow2b 14896. Left multiplication in a subgroup  H is a group action on the set of all left cosets of  K. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  K  e.  (SubGrp `  G ) )   &    |-  .+  =  ( +g  `  G )   &    |-  .~  =  ( G ~QG  K )   &    |-  .x.  =  ( x  e.  H ,  y  e.  ( X /.  .~  )  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   =>    |-  ( ph  ->  .x.  e.  ( ( Gs  H ) 
 GrpAct  ( X /.  .~  ) ) )
 
Theoremsylow2blem3 14895* Sylow's second theorem. Putting together the results of sylow2a 14892 and the orbit-stabilizer theorem to show that  P does not divide the set of all fixed points under the group action, we get that there is a fixed point of the group action, so that there is some  g  e.  X with  h g K  =  g K for all  h  e.  H. This implies that  inv g ( g ) h g  e.  K, so  h is in the conjugated subgroup  g K inv g ( g ). (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  K  e.  (SubGrp `  G ) )   &    |-  .+  =  ( +g  `  G )   &    |-  .~  =  ( G ~QG  K )   &    |-  .x.  =  ( x  e.  H ,  y  e.  ( X /.  .~  )  |->  ran  (  z  e.  y  |->  ( x  .+  z ) ) )   &    |-  ( ph  ->  P pGrp 
 ( Gs  H ) )   &    |-  ( ph  ->  ( # `  K )  =  ( P ^ ( P  pCnt  ( # `  X ) ) ) )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ph  ->  E. g  e.  X  H  C_ 
 ran  (  x  e.  K  |->  ( ( g 
 .+  x )  .-  g ) ) )
 
Theoremsylow2b 14896* Sylow's second theorem. Any  P-group  H is a subgroup of a conjugated  P-group  K of order  P ^ n  ||  ( # `  X
) with  n maximal. This is usually stated under the assumption that  K is a Sylow subgroup, but we use a slightly different definition, whose equivalence to this one requires this theorem. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  K  e.  (SubGrp `  G ) )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  P pGrp  ( Gs  H ) )   &    |-  ( ph  ->  ( # `  K )  =  ( P ^ ( P  pCnt  ( # `  X ) ) ) )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ph  ->  E. g  e.  X  H  C_  ran  (  x  e.  K  |->  ( ( g  .+  x ) 
 .-  g ) ) )
 
Theoremslwhash 14897 A sylow subgroup has cardinality equal to the maximum power of  P dividing the group. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  ( P pSyl  G ) )   =>    |-  ( ph  ->  ( # `  H )  =  ( P ^ ( P  pCnt  ( # `  X ) ) ) )
 
Theoremfislw 14898 The sylow subgroups of a finite group are exactly the groups which have cardinality equal to the maximum power of  P dividing the group. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  ->  ( H  e.  ( P pSyl  G )  <->  ( H  e.  (SubGrp `  G )  /\  ( # `  H )  =  ( P ^
 ( P  pCnt  ( # `
  X ) ) ) ) ) )
 
Theoremsylow2 14899* Sylow's second theorem. See also sylow2b 14896 for the "hard" part of the proof. Any two Sylow  P-subgroups are conjugate to one another, and hence the same size, namely 
P ^ ( P 
pCnt  |  X  | 
) (see fislw 14898). (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  H  e.  ( P pSyl  G ) )   &    |-  ( ph  ->  K  e.  ( P pSyl  G ) )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ph  ->  E. g  e.  X  H  =  ran  (  x  e.  K  |->  ( ( g  .+  x )  .-  g ) ) )
 
Theoremsylow3lem1 14900* Lemma for sylow3 14906, first part. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  Fin )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .(+) 
 =  ( x  e.  X ,  y  e.  ( P pSyl  G ) 
 |->  ran  (  z  e.  y  |->  ( ( x 
 .+  z )  .-  x ) ) )   =>    |-  ( ph  ->  .(+)  e.  ( G  GrpAct  ( P pSyl  G ) ) )
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