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Theorem List for Metamath Proof Explorer - 14701-14800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgictr 14701 Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( ( R  ~=ph𝑔  S  /\  S  ~=ph𝑔 
 T )  ->  R  ~=ph𝑔  T )
 
Theoremgicer 14702 Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- 
 ~=ph𝑔  Er  Grp
 
Theoremgicen 14703 Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  C  =  (
 Base `  S )   =>    |-  ( R  ~=ph𝑔  S  ->  B 
 ~~  C )
 
Theoremgicsubgen 14704 A less trivial example of a group invariant: cardinality of the subgroup lattice. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑔 
 S  ->  (SubGrp `  R )  ~~  (SubGrp `  S ) )
 
10.2.5  Group actions
 
Syntaxcga 14705 Extend class definition to include the class of group actions.
 class  GrpAct
 
Definitiondf-ga 14706* Define the class of all group actions. A group  G acts on a set  S if a permutation on  S is associated with every element of  G in such a way that the identity permutation on  S is associated with the neutral element of 
G, and the composition of the permutations associated with two elements of  G is identical with the permutation associated to the composition of these two elements (in the same order) in the group  G. (Contributed by Jeff Hankins, 10-Aug-2009.)
 |-  GrpAct  =  ( g  e. 
 Grp ,  s  e.  _V 
 |->  [_ ( Base `  g
 )  /  b ]_ { m  e.  (
 s  ^m  ( b  X.  s ) )  | 
 A. x  e.  s  ( ( ( 0g
 `  g ) m x )  =  x 
 /\  A. y  e.  b  A. z  e.  b  ( ( y (
 +g  `  g )
 z ) m x )  =  ( y m ( z m x ) ) ) } )
 
Theoremisga 14707* The predicate "is a (left) group action." The group  G is said to act on the base set  Y of the action, which is not assumed to have any special properties. There is a related notion of right group action, but as the Wikipedia article explains, it is not mathematically interesting. The way actions are usually thought of is that each element  g of  G is a permutation of the elements of  Y (see gapm 14722). Since group theory was classically about symmetry groups, it is therefore likely that the notion of group action was useful even in early group theory. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  (  .(+)  e.  ( G  GrpAct  Y )  <->  ( ( G  e.  Grp  /\  Y  e.  _V )  /\  (  .(+)  : ( X  X.  Y )
 --> Y  /\  A. x  e.  Y  ( (  .0.  .(+)  x )  =  x 
 /\  A. y  e.  X  A. z  e.  X  ( ( y  .+  z
 )  .(+)  x )  =  ( y  .(+)  ( z 
 .(+)  x ) ) ) ) ) )
 
Theoremgagrp 14708 The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
 
Theoremgaset 14709 The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)
 |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  Y  e.  _V )
 
Theoremgagrpid 14710 The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  (  .0.  .(+)  A )  =  A )
 
Theoremgaf 14711 The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
 
Theoremgafo 14712 A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) -onto-> Y )
 
Theoremgaass 14713 An "associative" property for group actions. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y ) 
 /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  Y ) )  ->  ( ( A  .+  B )  .(+)  C )  =  ( A 
 .(+)  ( B  .(+)  C ) ) )
 
Theoremga0 14714 The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  ( G  e.  Grp  ->  (/) 
 e.  ( G  GrpAct  (/) ) )
 
Theoremgaid 14715 The trivial action of a group on any set. Each group element corresponds to the identity permutation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( G  e.  Grp  /\  S  e.  V ) 
 ->  ( 2nd  |`  ( X  X.  S ) )  e.  ( G  GrpAct  S ) )
 
Theoremsubgga 14716* A subgroup acts on its parent group. (Contributed by Jeff Hankins, 13-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  H  =  ( Gs  Y )   &    |-  F  =  ( x  e.  Y ,  y  e.  X  |->  ( x 
 .+  y ) )   =>    |-  ( Y  e.  (SubGrp `  G )  ->  F  e.  ( H  GrpAct  X ) )
 
Theoremgass 14717* A subset of a group action is a group action iff it is closed under the group action operation. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y )  /\  Z  C_  Y )  ->  ( (  .(+)  |`  ( X  X.  Z ) )  e.  ( G  GrpAct  Z )  <->  A. x  e.  X  A. y  e.  Z  ( x  .(+)  y )  e.  Z ) )
 
Theoremgasubg 14718 The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y )  /\  S  e.  (SubGrp `  G ) )  ->  (  .(+)  |`  ( S  X.  Y ) )  e.  ( H 
 GrpAct  Y ) )
 
Theoremgaid2 14719* A group operation is a left group action of the group on itself. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  F  =  ( x  e.  X ,  y  e.  X  |->  ( x  .+  y ) )   =>    |-  ( G  e.  Grp  ->  F  e.  ( G  GrpAct  X ) )
 
Theoremgalcan 14720 The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
 )  ->  ( ( A  .(+)  B )  =  ( A  .(+)  C )  <->  B  =  C )
 )
 
Theoremgacan 14721 Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  B  e.  Y  /\  C  e.  Y )
 )  ->  ( ( A  .(+)  B )  =  C  <->  ( ( N `
  A )  .(+)  C )  =  B ) )
 
Theoremgapm 14722* The action of a particular group element is a permutation of the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  F  =  ( x  e.  Y  |->  ( A  .(+)  x )
 )   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  ->  F : Y -1-1-onto-> Y )
 
Theoremgaorb 14723* The orbit equivalence relation puts two points in the group action in the same equivalence class iff there is a group element that takes one element to the other. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |- 
 .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  (
 g  .(+)  x )  =  y ) }   =>    |-  ( A  .~  B 
 <->  ( A  e.  Y  /\  B  e.  Y  /\  E. h  e.  X  ( h  .(+)  A )  =  B ) )
 
Theoremgaorber 14724* The orbit equivalence relation is an equivalence relation on the target set of the group action. (Contributed by NM, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |- 
 .~  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  (
 g  .(+)  x )  =  y ) }   &    |-  X  =  ( Base `  G )   =>    |-  (  .(+) 
 e.  ( G  GrpAct  Y )  ->  .~  Er  Y )
 
Theoremgastacl 14725* The stabilizer subgroup in a group action. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y ) 
 /\  A  e.  Y )  ->  H  e.  (SubGrp `  G ) )
 
Theoremgastacos 14726* Write the coset relation for the stabilizer subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }   &    |-  .~  =  ( G ~QG  H )   =>    |-  ( ( (  .(+)  e.  ( G  GrpAct  Y ) 
 /\  A  e.  Y )  /\  ( B  e.  X  /\  C  e.  X ) )  ->  ( B 
 .~  C  <->  ( B  .(+)  A )  =  ( C 
 .(+)  A ) ) )
 
Theoremorbstafun 14727* Existence and uniqueness for the function of orbsta 14729. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  X  =  ( Base `  G )   &    |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }   &    |-  .~  =  ( G ~QG  H )   &    |-  F  =  ran  (  k  e.  X  |->  <. [ k ]  .~  ,  ( k  .(+)  A )
 >. )   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  ->  Fun  F )
 
Theoremorbstaval 14728* Value of the function at a given equivalence class element. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  X  =  ( Base `  G )   &    |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }   &    |-  .~  =  ( G ~QG  H )   &    |-  F  =  ran  (  k  e.  X  |->  <. [ k ]  .~  ,  ( k  .(+)  A )
 >. )   =>    |-  ( ( (  .(+)  e.  ( G  GrpAct  Y ) 
 /\  A  e.  Y )  /\  B  e.  X )  ->  ( F `  [ B ]  .~  )  =  ( B  .(+)  A ) )
 
Theoremorbsta 14729* The Orbit-Stabilizer theorem. The mapping  F is a bijection from the cosets of the stabilizer subgroup of  A to the orbit of  A. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }   &    |-  .~  =  ( G ~QG  H )   &    |-  F  =  ran  (  k  e.  X  |->  <. [ k ]  .~  ,  ( k  .(+)  A )
 >. )   &    |-  O  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  ( (  .(+)  e.  ( G  GrpAct  Y ) 
 /\  A  e.  Y )  ->  F : ( X /.  .~  ) -1-1-onto-> [ A ] O )
 
Theoremorbsta2 14730* Relation between the size of the orbit and the size of the stabilizer of a point in a finite group action. (Contributed by Mario Carneiro, 16-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  { u  e.  X  |  ( u  .(+)  A )  =  A }   &    |-  .~  =  ( G ~QG  H )   &    |-  O  =  { <. x ,  y >.  |  ( { x ,  y }  C_  Y  /\  E. g  e.  X  ( g  .(+)  x )  =  y ) }   =>    |-  ( ( ( 
 .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  Y )  /\  X  e.  Fin )  ->  ( # `  X )  =  ( ( # `
  [ A ] O )  x.  ( # `
  H ) ) )
 
10.2.6  Symmetry groups and Cayley's Theorem
 
Syntaxcsymg 14731 Extend class notation to include the class of symmetry groups.
 class  SymGrp
 
Definitiondf-symg 14732* Define the symmetry group on set  x. We represent the group as the set of 1-1-onto functions from  x to itself under function composition, and topologize it as a function space assuming the set is discrete. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  SymGrp  =  ( x  e. 
 _V  |->  [_ { h  |  h : x -1-1-onto-> x }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
 <. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( f  o.  g ) ) >. , 
 <. (TopSet `  ndx ) ,  ( Xt_ `  ( x  X.  { ~P x } ) ) >. } )
 
Theoremsymgval 14733* The value of the symmetry group function at  A. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  { x  |  x : A
 -1-1-onto-> A }   &    |-  .+  =  (
 f  e.  B ,  g  e.  B  |->  ( f  o.  g ) )   &    |-  J  =  ( Xt_ `  ( A  X.  { ~P A } ) )   =>    |-  ( A  e.  V  ->  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  J >. } )
 
Theoremsymgbas 14734* The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   =>    |-  B  =  { x  |  x : A
 -1-1-onto-> A }
 
Theoremelsymgbas2 14735 Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Mario Carneiro, 28-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   =>    |-  ( F  e.  V  ->  ( F  e.  B 
 <->  F : A -1-1-onto-> A ) )
 
Theoremelsymgbas 14736 Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   =>    |-  ( A  e.  V  ->  ( F  e.  B 
 <->  F : A -1-1-onto-> A ) )
 
Theoremsymghash 14737 The symmetric group on  n objects has cardinality  n !. (Contributed by Mario Carneiro, 22-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   =>    |-  ( A  e.  Fin 
 ->  ( # `  B )  =  ( ! `  ( # `  A ) ) )
 
Theoremsymgplusg 14738* The value of the symmetry group function at  A. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  .+  =  ( f  e.  B ,  g  e.  B  |->  ( f  o.  g
 ) )
 
Theoremsymgov 14739 The value of the group operation of the symmetry group on  A. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+  Y )  =  ( X  o.  Y ) )
 
Theoremsymgcl 14740 The group operation of the symmetry group on  A is closed, i.e. a magma. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by Mario Carneiro, 28-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+  Y )  e.  B )
 
Theoremsymgtset 14741 The topology of the symmetry group on  A. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just bijections - but the definition of  TopOpen ensures that it is trimmed down before it gets use. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  ( Xt_ `  ( A  X.  { ~P A } ) )  =  (TopSet `  G )
 )
 
Theoremsymggrp 14742 The symmetry group on  A is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  G  e.  Grp )
 
Theoremsymgid 14743 The value of the identity element of the symmetry group on  A (Contributed by Paul Chapman, 25-Jul-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  G  =  ( SymGrp `  A )   =>    |-  ( A  e.  V  ->  (  _I  |`  A )  =  ( 0g `  G ) )
 
Theoremsymginv 14744 The group inverse in the symmetric group corresponds to the functional inverse. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
 |-  G  =  ( SymGrp `  A )   &    |-  B  =  (
 Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( F  e.  B  ->  ( N `  F )  =  `' F )
 
Theoremgalactghm 14745* The currying of a group action is a group homomorphism between the group  G and the symetry group  ( SymGrp `  Y
). (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  (
 SymGrp `  Y )   &    |-  F  =  ( x  e.  X  |->  ( y  e.  Y  |->  ( x  .(+)  y ) ) )   =>    |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  F  e.  ( G  GrpHom  H ) )
 
Theoremlactghmga 14746* The converse of galactghm 14745. The uncurrying of a homomorphism into  ( SymGrp `  Y
) is a group action. Thus group actions and group homomorphisms into a symmetric group are essentially equivalent notions. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  (
 SymGrp `  Y )   &    |-  .(+)  =  ( x  e.  X ,  y  e.  Y  |->  ( ( F `  x ) `
  y ) )   =>    |-  ( F  e.  ( G  GrpHom  H )  ->  .(+) 
 e.  ( G  GrpAct  Y ) )
 
Theoremsymgtopn 14747 The topology of the symmetry group on  A. (Contributed by Mario Carneiro, 29-Aug-2015.)
 |-  G  =  ( SymGrp `  X )   &    |-  B  =  (
 Base `  G )   =>    |-  ( X  e.  V  ->  ( ( Xt_ `  ( X  X.  { ~P X } ) )t  B )  =  ( TopOpen `  G ) )
 
Theoremsymgga 14748* The symmetric group induces a group action on its base set. (Contributed by Mario Carneiro, 24-Jan-2015.)
 |-  G  =  ( SymGrp `  X )   &    |-  B  =  (
 Base `  G )   &    |-  F  =  ( f  e.  B ,  x  e.  X  |->  ( f `  x ) )   =>    |-  ( X  e.  V  ->  F  e.  ( G 
 GrpAct  X ) )
 
Theoremcayleylem1 14749* Lemma for cayley 14751. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  H  =  ( SymGrp `  X )   &    |-  S  =  (
 Base `  H )   &    |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a ) ) )   =>    |-  ( G  e.  Grp  ->  F  e.  ( G  GrpHom  H ) )
 
Theoremcayleylem2 14750* Lemma for cayley 14751. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  H  =  ( SymGrp `  X )   &    |-  S  =  (
 Base `  H )   &    |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a ) ) )   =>    |-  ( G  e.  Grp  ->  F : X -1-1-> S )
 
Theoremcayley 14751* Cayley's Theorem (constructive version): given group  G,  F is an isomorphism between  G and the subgroup  S of the symmetry group  H on the underlying set  X of  G. (Contributed by Paul Chapman, 3-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  (
 SymGrp `  X )   &    |-  .+  =  ( +g  `  G )   &    |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a ) ) )   &    |-  S  =  ran  F   =>    |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  H )  /\  F  e.  ( G  GrpHom  ( Hs  S ) )  /\  F : X -1-1-onto-> S ) )
 
Theoremcayleyth 14752* Cayley's Theorem (existence version): every group  G is isomorphic to a subgroup of the symmetry group on the underlying set of  G. (For any group  G there exists an isomorphism  f between  G and a subgroup  h of the symmetry group on the underlying set of  G.) (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  (
 SymGrp `  X )   =>    |-  ( G  e.  Grp 
 ->  E. s  e.  (SubGrp `  H ) E. f  e.  ( G  GrpHom  ( Hs  s ) ) f : X -1-1-onto-> s )
 
10.2.7  Centralizers and centers
 
Syntaxccntz 14753 Syntax for the centralizer of a set in a monoid.
 class Cntz
 
Syntaxccntr 14754 Syntax for the centralizer of a monoid.
 class Cntr
 
Definitiondf-cntz 14755* Define the centralizer of a subset of a magma, which is the set of elements each of which commutes with each element of the given subset. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- Cntz  =  ( m  e.  _V  |->  ( s  e.  ~P ( Base `  m )  |->  { x  e.  ( Base `  m )  | 
 A. y  e.  s  ( x ( +g  `  m ) y )  =  ( y ( +g  `  m ) x ) } ) )
 
Definitiondf-cntr 14756 Define the center of a magma, which is the elements that commute with all others. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- Cntr  =  ( m  e.  _V  |->  ( (Cntz `  m ) `  ( Base `  m )
 ) )
 
Theoremcntrval 14757 Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( Z `  B )  =  (Cntr `  M )
 
Theoremcntzfval 14758* First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( M  e.  V  ->  Z  =  ( s  e. 
 ~P B  |->  { x  e.  B  |  A. y  e.  s  ( x  .+  y )  =  ( y  .+  x ) } ) )
 
Theoremcntzval 14759* Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( S  C_  B  ->  ( Z `  S )  =  { x  e.  B  |  A. y  e.  S  ( x  .+  y )  =  ( y  .+  x ) } )
 
Theoremelcntz 14760* Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( S  C_  B  ->  ( A  e.  ( Z `  S )  <->  ( A  e.  B  /\  A. y  e.  S  ( A  .+  y )  =  (
 y  .+  A )
 ) ) )
 
Theoremcntzel 14761* Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  (
 ( S  C_  B  /\  X  e.  B ) 
 ->  ( X  e.  ( Z `  S )  <->  A. y  e.  S  ( X  .+  y )  =  ( y  .+  X ) ) )
 
Theoremcntzsnval 14762* Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( Y  e.  B  ->  ( Z `  { Y } )  =  { x  e.  B  |  ( x  .+  Y )  =  ( Y  .+  x ) } )
 
Theoremelcntzsn 14763 Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( Y  e.  B  ->  ( X  e.  ( Z `
  { Y }
 ) 
 <->  ( X  e.  B  /\  ( X  .+  Y )  =  ( Y  .+  X ) ) ) )
 
Theoremsscntz 14764* A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  (
 ( S  C_  B  /\  T  C_  B )  ->  ( S  C_  ( Z `  T )  <->  A. x  e.  S  A. y  e.  T  ( x  .+  y )  =  ( y  .+  x ) ) )
 
Theoremcntzrcl 14765 Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( X  e.  ( Z `  S )  ->  ( M  e.  _V  /\  S  C_  B )
 )
 
Theoremcntzssv 14766 The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( Z `  S )  C_  B
 
Theoremcntzi 14767 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 14768 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 14769 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 14770 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 14771 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 14772* 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 14773 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 14774 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 14775 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 14776 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 14777 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 14778 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 14779 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 14780 The opposite group operation.
 class oppg
 
Definitiondf-oppg 14781 Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 15367 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 14782 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 14783 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 14784 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 14785 Lemma for oppgbas 14786. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  =/=  2   =>    |-  ( E `  R )  =  ( E `  O )
 
Theoremoppgbas 14786 Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   &    |-  B  =  ( Base `  R )   =>    |-  B  =  ( Base `  O )
 
Theoremoppgtset 14787 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  R )   &    |-  J  =  (TopSet `  R )   =>    |-  J  =  (TopSet `  O )
 
Theoremoppgtopn 14788 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  J  =  ( TopOpen `  O )
 
Theoremoppgmnd 14789 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 14790 Bidirectional form of oppgmnd 14789. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   =>    |-  ( R  e.  Mnd  <->  O  e.  Mnd )
 
Theoremoppgid 14791 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 14792 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 14793 Bidirectional form of oppggrp 14792. (Contributed by Stefan O'Rear, 26-Aug-2015.)
 |-  O  =  (oppg `  R )   =>    |-  ( R  e.  Grp  <->  O  e.  Grp )
 
Theoremoppginv 14794 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 14795 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 14796 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 14797 Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  G )   =>    |-  (SubMnd `  G )  =  (SubMnd `  O )
 
Theoremoppgsubg 14798 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  O  =  (oppg `  G )   =>    |-  (SubGrp `  G )  =  (SubGrp `  O )
 
Theoremoppgcntz 14799 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 14800 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 )
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