HomeHome Metamath Proof Explorer
Theorem List (p. 149 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21492)
  Hilbert Space Explorer  Hilbert Space Explorer
(21493-23015)
  Users' Mathboxes  Users' Mathboxes
(23016-31423)
 

Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcntzi 14801 Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |- 
 .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( X  e.  ( Z `  S ) 
 /\  Y  e.  S )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremcntri 14802 Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntr `  M )   =>    |-  (
 ( X  e.  Z  /\  Y  e.  B ) 
 ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremresscntz 14803 Centralizer in a substructure. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  H  =  ( Gs  A )   &    |-  Z  =  (Cntz `  G )   &    |-  Y  =  (Cntz `  H )   =>    |-  ( ( A  e.  V  /\  S  C_  A )  ->  ( Y `  S )  =  (
 ( Z `  S )  i^i  A ) )
 
Theoremcntz2ss 14804 Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( S  C_  B  /\  T  C_  S )  ->  ( Z `  S )  C_  ( Z `
  T ) )
 
Theoremcntzrec 14805 Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( S  C_  B  /\  T  C_  B )  ->  ( S  C_  ( Z `  T )  <->  T  C_  ( Z `  S ) ) )
 
Theoremcntziinsn 14806* Express any centralizer as an intersection of singleton centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( S  C_  B  ->  ( Z `  S )  =  ( B  i^i  |^|_ x  e.  S  ( Z `  { x } ) ) )
 
Theoremcntzsubm 14807 Centralizers in a monoid are submonoids. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( M  e.  Mnd  /\  S  C_  B )  ->  ( Z `  S )  e.  (SubMnd `  M ) )
 
Theoremcntzsubg 14808 Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( ( M  e.  Grp  /\  S  C_  B )  ->  ( Z `  S )  e.  (SubGrp `  M ) )
 
Theoremcntzidss 14809 If the elements of  S commute, the elements of a subset 
T also commute. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  Z  =  (Cntz `  G )   =>    |-  ( ( S  C_  ( Z `  S ) 
 /\  T  C_  S )  ->  T  C_  ( Z `  T ) )
 
Theoremcntzmhm 14810 Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  Y  =  (Cntz `  H )   =>    |-  ( ( F  e.  ( G MndHom  H )  /\  A  e.  ( Z `  S ) )  ->  ( F `  A )  e.  ( Y `  ( F " S ) ) )
 
Theoremcntzmhm2 14811 Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  Y  =  (Cntz `  H )   =>    |-  ( ( F  e.  ( G MndHom  H )  /\  S  C_  ( Z `  T ) )  ->  ( F " S ) 
 C_  ( Y `  ( F " T ) ) )
 
Theoremcntrsubgnsg 14812 A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  Z  =  (Cntr `  M )   =>    |-  ( ( X  e.  (SubGrp `  M )  /\  X  C_  Z )  ->  X  e.  (NrmSGrp `  M ) )
 
Theoremcntrnsg 14813 The center of a group is a normal subgroup. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  Z  =  (Cntr `  M )   =>    |-  ( M  e.  Grp  ->  Z  e.  (NrmSGrp `  M ) )
 
10.2.8  The opposite group
 
Syntaxcoppg 14814 The opposite group operation.
 class oppg
 
Definitiondf-oppg 14815 Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 15401 does the same thing for multiplication. (Contributed by Stefan O'Rear, 25-Aug-2015.)
 |- oppg  =  ( w  e.  _V  |->  ( w sSet  <. ( +g  ` 
 ndx ) , tpos  ( +g  `  w ) >. ) )
 
Theoremoppgval 14816 Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
 |- 
 .+  =  ( +g  `  R )   &    |-  O  =  (oppg `  R )   =>    |-  O  =  ( R sSet  <. ( +g  `  ndx ) , tpos  .+  >. )
 
Theoremoppgplusfval 14817 Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
 |- 
 .+  =  ( +g  `  R )   &    |-  O  =  (oppg `  R )   &    |-  .+b  =  ( +g  `  O )   =>    |-  .+b  = tpos  .+
 
Theoremoppgplus 14818 Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
 |- 
 .+  =  ( +g  `  R )   &    |-  O  =  (oppg `  R )   &    |-  .+b  =  ( +g  `  O )   =>    |-  ( X  .+b  Y )  =  ( Y 
 .+  X )
 
Theoremoppglem 14819 Lemma for oppgbas 14820. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  =/=  2   =>    |-  ( E `  R )  =  ( E `  O )
 
Theoremoppgbas 14820 Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   &    |-  B  =  ( Base `  R )   =>    |-  B  =  ( Base `  O )
 
Theoremoppgtset 14821 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  R )   &    |-  J  =  (TopSet `  R )   =>    |-  J  =  (TopSet `  O )
 
Theoremoppgtopn 14822 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  J  =  ( TopOpen `  O )
 
Theoremoppgmnd 14823 The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  O  =  (oppg `  R )   =>    |-  ( R  e.  Mnd  ->  O  e.  Mnd )
 
Theoremoppgmndb 14824 Bidirectional form of oppgmnd 14823. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   =>    |-  ( R  e.  Mnd  <->  O  e.  Mnd )
 
Theoremoppgid 14825 Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  O  =  (oppg `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |- 
 .0.  =  ( 0g `  O )
 
Theoremoppggrp 14826 The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   =>    |-  ( R  e.  Grp  ->  O  e.  Grp )
 
Theoremoppggrpb 14827 Bidirectional form of oppggrp 14826. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   =>    |-  ( R  e.  Grp  <->  O  e.  Grp )
 
Theoremoppginv 14828 Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   &    |-  I  =  ( inv
 g `  R )   =>    |-  ( R  e.  Grp  ->  I  =  ( inv g `  O ) )
 
Theoreminvoppggim 14829 The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  G )   &    |-  I  =  ( inv
 g `  G )   =>    |-  ( G  e.  Grp  ->  I  e.  ( G GrpIso  O )
 )
 
Theoremoppggic 14830 Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  G )   =>    |-  ( G  e.  Grp  ->  G  ~=ph𝑔 
 O )
 
Theoremoppgsubm 14831 Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  G )   =>    |-  (SubMnd `  G )  =  (SubMnd `  O )
 
Theoremoppgsubg 14832 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  G )   =>    |-  (SubGrp `  G )  =  (SubGrp `  O )
 
Theoremoppgcntz 14833 A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  O  =  (oppg `  G )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( Z `  A )  =  ( (Cntz `  O ) `  A )
 
Theoremoppgcntr 14834 The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  O  =  (oppg `  G )   &    |-  Z  =  (Cntr `  G )   =>    |-  Z  =  (Cntr `  O )
 
Theoremgsumwrev 14835 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 14836 Extend class notation to include the order function on the elements of a group.
 class  od
 
Syntaxcgex 14837 Extend class notation to include the order function on the elements of a group.
 class gEx
 
Syntaxcpgp 14838 Extend class notation to include the class of all p-groups.
 class pGrp
 
Syntaxcslw 14839 Extend class notation to include the class of all Sylow p-subgroups of a group.
 class pSyl
 
Definitiondf-od 14840* 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 14841* 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 14842* 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 14843* 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 14844* 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 14845* 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 14846* 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 14847 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 14848 Functionality of the group element order. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  O  =  ( od `  G )   =>    |-  O : X --> NN0
 
Theoremodid 14849 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 14850 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 14851 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 14852 Lemma for mndodcong 14853. (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 14853 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 14854 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 14855 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 14856 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 14857 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 14858 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 14859 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 14860 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 14861* The oddvds 14858 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 14862* 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 14863 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 14864 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 14865 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 14866 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 14867* 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 14868 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 14869 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 14870 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 14871* 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 14872* 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 14873* 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 14874 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 14875 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 14876 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 14877 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 14878 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 14879* 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 14880* 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 14881 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 14882 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 14883 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 14884* 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 14885* 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 14886* 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 14887 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 14888 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 14889* 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 14890 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 14891* 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 14892* 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 14893 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 14894* 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 14895 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 14896 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 14897 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 14898 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 14899* 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 14900 Reverse closure for the first argument of pGrp. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  ( P pGrp  G  ->  P  e.  Prime )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31423
  Copyright terms: Public domain < Previous  Next >