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Theorem List for Metamath Proof Explorer - 14701-14800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremresghm2b 14701 Restriction of a the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
 |-  U  =  ( Ts  X )   =>    |-  ( ( X  e.  (SubGrp `  T )  /\  ran 
 F  C_  X )  ->  ( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
 
Theoremghmco 14702 The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ( F  e.  ( T  GrpHom  U ) 
 /\  G  e.  ( S  GrpHom  T ) ) 
 ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )
 
Theoremghmima 14703 The image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  U  e.  (SubGrp `  S ) )  ->  ( F " U )  e.  (SubGrp `  T ) )
 
Theoremghmpreima 14704 The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  V  e.  (SubGrp `  T ) )  ->  ( `' F " V )  e.  (SubGrp `  S ) )
 
Theoremghmeql 14705 The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  G  e.  ( S  GrpHom  T ) ) 
 ->  dom  ( F  i^i  G )  e.  (SubGrp `  S ) )
 
Theoremghmnsgima 14706 The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  Y  =  ( Base `  T )   =>    |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ( F " U )  e.  (NrmSGrp `  T ) )
 
Theoremghmnsgpreima 14707 The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  V  e.  (NrmSGrp `  T ) )  ->  ( `' F " V )  e.  (NrmSGrp `  S ) )
 
Theoremghmker 14708 The kernel of a homomorphism is a normal subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |- 
 .0.  =  ( 0g `  T )   =>    |-  ( F  e.  ( S  GrpHom  T )  ->  ( `' F " {  .0.  } )  e.  (NrmSGrp `  S ) )
 
Theoremghmeqker 14709 Two source points map to the same destination point under a group homomorphism iff their difference belongs to the kernel. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  B  =  ( Base `  S )   &    |-  .0.  =  ( 0g `  T )   &    |-  K  =  ( `' F " {  .0.  }
 )   &    |-  .-  =  ( -g `  S )   =>    |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  U  e.  B  /\  V  e.  B ) 
 ->  ( ( F `  U )  =  ( F `  V )  <->  ( U  .-  V )  e.  K ) )
 
Theorempwsdiagghm 14710* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  R )   &    |-  F  =  ( x  e.  B  |->  ( I  X.  { x } ) )   =>    |-  ( ( R  e.  Grp  /\  I  e.  W )  ->  F  e.  ( R  GrpHom  Y ) )
 
Theoremghmf1 14711* Two ways of saying a group homomorphism is 1-1 into its codomain. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   &    |-  .0.  =  ( 0g `  S )   &    |-  U  =  ( 0g
 `  T )   =>    |-  ( F  e.  ( S  GrpHom  T ) 
 ->  ( F : X -1-1-> Y  <->  A. x  e.  X  ( ( F `  x )  =  U  ->  x  =  .0.  )
 ) )
 
Theoremghmf1o 14712 A bijective group homomorphism is an isomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   =>    |-  ( F  e.  ( S  GrpHom  T ) 
 ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( T 
 GrpHom  S ) ) )
 
Theoremconjghm 14713* Conjugation is an automorphism of the group. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  F  =  ( x  e.  X  |->  ( ( A 
 .+  x )  .-  A ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( F  e.  ( G  GrpHom  G )  /\  F : X
 -1-1-onto-> X ) )
 
Theoremconjsubg 14714* A conjugated subgroup is also a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  F  =  ( x  e.  S  |->  ( ( A 
 .+  x )  .-  A ) )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  ran  F  e.  (SubGrp `  G ) )
 
Theoremconjsubgen 14715* A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  F  =  ( x  e.  S  |->  ( ( A 
 .+  x )  .-  A ) )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  ~~  ran  F )
 
Theoremconjnmz 14716* A subgroup is unchanged under conjugation by an element of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  F  =  ( x  e.  S  |->  ( ( A 
 .+  x )  .-  A ) )   &    |-  N  =  { y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) }   =>    |-  (
 ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  S  =  ran  F )
 
Theoremconjnmzb 14717* Alternative condition for elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  F  =  ( x  e.  S  |->  ( ( A 
 .+  x )  .-  A ) )   &    |-  N  =  { y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) }   =>    |-  ( S  e.  (SubGrp `  G )  ->  ( A  e.  N 
 <->  ( A  e.  X  /\  S  =  ran  F ) ) )
 
Theoremconjnsg 14718* A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  F  =  ( x  e.  S  |->  ( ( A 
 .+  x )  .-  A ) )   =>    |-  ( ( S  e.  (NrmSGrp `  G )  /\  A  e.  X )  ->  S  =  ran  F )
 
Theoremdivsghm 14719* If  Y is a normal subgroup of  G, then the "natural map" from elements to their cosets is a group homomorphism from  G to  G  /  Y. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 18-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  H  =  ( G  /.s  ( G ~QG  Y ) )   &    |-  F  =  ( x  e.  X  |->  [ x ] ( G ~QG  Y ) )   =>    |-  ( Y  e.  (NrmSGrp `  G )  ->  F  e.  ( G  GrpHom  H ) )
 
Theoremghmpropd 14720* Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  J )
 )   &    |-  ( ph  ->  C  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ph  ->  C  =  ( Base `  M )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  J )
 y )  =  ( x ( +g  `  L ) y ) )   &    |-  ( ( ph  /\  ( x  e.  C  /\  y  e.  C )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  M ) y ) )   =>    |-  ( ph  ->  ( J  GrpHom  K )  =  ( L  GrpHom  M ) )
 
10.2.4  Isomorphisms of groups
 
Syntaxcgim 14721 The class of group isomorphism sets.
 class GrpIso
 
Syntaxcgic 14722 The class of the group isomorphism relation.
 class  ~=ph𝑔
 
Definitiondf-gim 14723* An isomorphism of groups is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group operation. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |- GrpIso  =  ( s  e.  Grp ,  t  e.  Grp  |->  { g  e.  ( s  GrpHom  t )  |  g : (
 Base `  s ) -1-1-onto-> ( Base `  t ) } )
 
Definitiondf-gic 14724 Two groups are said to be isomorphic iff they are connected by at least one isomorphism. Isomophic groups share all global group properties, but to relate local properties requires knowledge of a specific isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |- 
 ~=ph𝑔  =  ( `' GrpIso  " ( _V  \  1o ) )
 
Theoremgimfn 14725 The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |- GrpIso  Fn  ( Grp  X.  Grp )
 
Theoremisgim 14726 An isomorphism of groups is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  C  =  (
 Base `  S )   =>    |-  ( F  e.  ( R GrpIso  S )  <->  ( F  e.  ( R  GrpHom  S ) 
 /\  F : B -1-1-onto-> C ) )
 
Theoremgimf1o 14727 An isomorphism of groups is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  B  =  ( Base `  R )   &    |-  C  =  (
 Base `  S )   =>    |-  ( F  e.  ( R GrpIso  S )  ->  F : B -1-1-onto-> C )
 
Theoremgimghm 14728 An isomorphism of groups is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( F  e.  ( R GrpIso  S )  ->  F  e.  ( R  GrpHom  S ) )
 
Theoremisgim2 14729 A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 17450. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( F  e.  ( R GrpIso  S )  <->  ( F  e.  ( R  GrpHom  S ) 
 /\  `' F  e.  ( S  GrpHom  R ) ) )
 
Theoremsubggim 14730 Behavior of subgroups under isomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
 |-  B  =  ( Base `  R )   =>    |-  ( ( F  e.  ( R GrpIso  S )  /\  A  C_  B )  ->  ( A  e.  (SubGrp `  R )  <->  ( F " A )  e.  (SubGrp `  S ) ) )
 
Theoremgimcnv 14731 The converse of a bijective group homomorphism is a bijective group homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( F  e.  ( S GrpIso  T )  ->  `' F  e.  ( T GrpIso  S )
 )
 
Theoremgimco 14732 The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( ( F  e.  ( T GrpIso  U )  /\  G  e.  ( S GrpIso  T ) )  ->  ( F  o.  G )  e.  ( S GrpIso  U )
 )
 
Theorembrgic 14733 The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑔 
 S 
 <->  ( R GrpIso  S )  =/= 
 (/) )
 
Theorembrgici 14734 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( F  e.  ( R GrpIso  S )  ->  R  ~=ph𝑔  S )
 
Theoremgicref 14735 Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( R  e.  Grp  ->  R  ~=ph𝑔 
 R )
 
Theoremgiclcl 14736 Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑔 
 S  ->  R  e.  Grp )
 
Theoremgicrcl 14737 Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( R  ~=ph𝑔 
 S  ->  S  e.  Grp )
 
Theoremgicsym 14738 Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( R  ~=ph𝑔 
 S  ->  S  ~=ph𝑔  R )
 
Theoremgictr 14739 Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( ( R  ~=ph𝑔  S  /\  S  ~=ph𝑔 
 T )  ->  R  ~=ph𝑔  T )
 
Theoremgicer 14740 Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- 
 ~=ph𝑔  Er  Grp
 
Theoremgicen 14741 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 14742 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 14743 Extend class definition to include the class of group actions.
 class  GrpAct
 
Definitiondf-ga 14744* 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 14745* 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 14760). 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 14746 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 14747 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 14748 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 14749 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 14750 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 14751 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 14752 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 14753 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 14754* 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 14755* 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 14756 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 14757* 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 14758 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 14759 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 14760* 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 14761* 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 14762* 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 14763* 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 14764* 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 14765* Existence and uniqueness for the function of orbsta 14767. (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 14766* 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 14767* 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 14768* 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 14769 Extend class notation to include the class of symmetry groups.
 class  SymGrp
 
Definitiondf-symg 14770* 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 14771* 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 14772* 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 14773 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 14774 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 14775 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 14776* 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 14777 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 14778 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 14779 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 14780 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 14781 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 14782 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 14783* 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 14784* The converse of galactghm 14783. 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 14785 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 14786* 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 14787* Lemma for cayley 14789. (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 14788* Lemma for cayley 14789. (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 14789* 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 14790* 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 14791 Syntax for the centralizer of a set in a monoid.
 class Cntz
 
Syntaxccntr 14792 Syntax for the centralizer of a monoid.
 class Cntr
 
Definitiondf-cntz 14793* 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 14794 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 14795 Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  M )   &    |-  Z  =  (Cntz `  M )   =>    |-  ( Z `  B )  =  (Cntr `  M )
 
Theoremcntzfval 14796* 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 14797* 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 14798* 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 14799* 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 14800* 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 ) } )
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