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Theorem List for Metamath Proof Explorer - 14501-14600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeqgval 14501 Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   &    |- 
 .+  =  ( +g  `  G )   &    |-  R  =  ( G ~QG 
 S )   =>    |-  ( ( G  e.  V  /\  S  C_  X )  ->  ( A R B 
 <->  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A )  .+  B )  e.  S ) ) )
 
Theoremeqger 14502 The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   =>    |-  ( Y  e.  (SubGrp `  G )  ->  .~  Er  X )
 
Theoremeqglact 14503* A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A 
 .+  x ) )
 " Y ) )
 
Theoremeqgid 14504 The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( Y  e.  (SubGrp `  G )  ->  [  .0.  ] 
 .~  =  Y )
 
Theoremeqgen 14505 Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   =>    |-  ( ( Y  e.  (SubGrp `  G )  /\  A  e.  ( X /.  .~  ) )  ->  Y  ~~  A )
 
Theoremeqgcpbl 14506 The subgroup coset equivalence relation is compatible with addition when the subgroup is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( Y  e.  (NrmSGrp `  G )  ->  ( ( A  .~  C  /\  B  .~  D )  ->  ( A  .+  B )  .~  ( C 
 .+  D ) ) )
 
Theoremdivsgrp 14507 If  Y is a normal subgroup of  G, then  H  =  G  /  Y is a group, called the quotient of  G by  Y. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   =>    |-  ( S  e.  (NrmSGrp `  G )  ->  H  e.  Grp )
 
Theoremdivseccl 14508 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   &    |-  V  =  ( Base `  G )   &    |-  B  =  ( Base `  H )   =>    |-  (
 ( S  e.  (NrmSGrp `  G )  /\  X  e.  V )  ->  [ X ] ( G ~QG  S )  e.  B )
 
Theoremdivsadd 14509 Value of the group operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   &    |-  V  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .+b  =  ( +g  `  H )   =>    |-  (
 ( S  e.  (NrmSGrp `  G )  /\  X  e.  V  /\  Y  e.  V )  ->  ( [ X ] ( G ~QG  S )  .+b  [ Y ] ( G ~QG  S ) )  =  [
 ( X  .+  Y ) ] ( G ~QG  S )
 )
 
Theoremdivs0 14510 Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( S  e.  (NrmSGrp `  G )  ->  [  .0.  ] ( G ~QG  S )  =  ( 0g `  H ) )
 
Theoremdivsinv 14511 Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   &    |-  V  =  ( Base `  G )   &    |-  I  =  ( inv g `  G )   &    |-  N  =  ( inv g `  H )   =>    |-  ( ( S  e.  (NrmSGrp `  G )  /\  X  e.  V )  ->  ( N `  [ X ] ( G ~QG  S )
 )  =  [ ( I `  X ) ]
 ( G ~QG  S ) )
 
Theoremdivssub 14512 Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   &    |-  V  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( -g `  H )   =>    |-  ( ( S  e.  (NrmSGrp `  G )  /\  X  e.  V  /\  Y  e.  V )  ->  ( [ X ]
 ( G ~QG  S ) N [ Y ] ( G ~QG  S )
 )  =  [ ( X  .-  Y ) ]
 ( G ~QG  S ) )
 
Theoremlagsubg2 14513 Lagrange's theorem for finite groups. Call the "order" of a group the cardinal number of the basic set of the group, and "index of a subgroup" the cardinal number of the set of left (or right, this is the same) cosets of this subgroup. Then the order of the group is the (cardinal) product of the order of any of its subgroups by the index of this subgroup. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   &    |-  ( ph  ->  Y  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  X  e.  Fin )   =>    |-  ( ph  ->  ( # `  X )  =  ( ( # `  ( X /.  .~  ) )  x.  ( # `  Y ) ) )
 
Theoremlagsubg 14514 Lagrange theorem for Groups: the order of any subgroup of a finite group is a divisor of the order of the group. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( Y  e.  (SubGrp `  G )  /\  X  e.  Fin )  ->  ( # `  Y ) 
 ||  ( # `  X ) )
 
10.2.3  Elementary theory of group homomorphisms
 
Syntaxcghm 14515 Extend class notation with the generator of group hom-sets.
 class  GrpHom
 
Definitiondf-ghm 14516* A homomorphism of groups is a map between two structures which preserves the group operation. Requiring both sides to be groups simplifies most theorems at the cost of complicating the theorem which pushes forward a group structure. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  GrpHom  =  ( s  e. 
 Grp ,  t  e.  Grp  |->  { g  |  [. ( Base `  s )  /  w ]. ( g : w --> ( Base `  t )  /\  A. x  e.  w  A. y  e.  w  (
 g `  ( x ( +g  `  s )
 y ) )  =  ( ( g `  x ) ( +g  `  t ) ( g `
  y ) ) ) } )
 
Theoremreldmghm 14517 Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |- 
 Rel  dom  GrpHom
 
Theoremisghm 14518* Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  ( F  e.  ( S  GrpHom  T )  <->  ( ( S  e.  Grp  /\  T  e.  Grp )  /\  ( F : X --> Y  /\  A. u  e.  X  A. v  e.  X  ( F `  ( u  .+  v ) )  =  ( ( F `  u )  .+^  ( F `
  v ) ) ) ) )
 
Theoremisghm3 14519* Property of a group homomorphism, similar to ismhm 14252. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  (
 ( S  e.  Grp  /\  T  e.  Grp )  ->  ( F  e.  ( S  GrpHom  T )  <->  ( F : X
 --> Y  /\  A. u  e.  X  A. v  e.  X  ( F `  ( u  .+  v ) )  =  ( ( F `  u )  .+^  ( F `  v
 ) ) ) ) )
 
Theoremghmgrp1 14520 A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
 
Theoremghmgrp2 14521 A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
 
Theoremghmf 14522 A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   =>    |-  ( F  e.  ( S  GrpHom  T ) 
 ->  F : X --> Y )
 
Theoremghmlin 14523 A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  X  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  (
 ( F  e.  ( S  GrpHom  T )  /\  U  e.  X  /\  V  e.  X )  ->  ( F `  ( U  .+  V ) )  =  ( ( F `
  U )  .+^  ( F `  V ) ) )
 
Theoremghmid 14524 A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  Y  =  ( 0g
 `  S )   &    |-  .0.  =  ( 0g `  T )   =>    |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  =  .0.  )
 
Theoremghminv 14525 A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  B  =  ( Base `  S )   &    |-  M  =  ( inv g `  S )   &    |-  N  =  ( inv
 g `  T )   =>    |-  (
 ( F  e.  ( S  GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( M `  X ) )  =  ( N `  ( F `  X ) ) )
 
Theoremghmsub 14526 Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  B  =  ( Base `  S )   &    |-  .-  =  ( -g `  S )   &    |-  N  =  ( -g `  T )   =>    |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  U  e.  B  /\  V  e.  B ) 
 ->  ( F `  ( U  .-  V ) )  =  ( ( F `
  U ) N ( F `  V ) ) )
 
Theoremisghmd 14527* Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   &    |-  ( ph  ->  S  e.  Grp )   &    |-  ( ph  ->  T  e.  Grp )   &    |-  ( ph  ->  F : X --> Y )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( F `  ( x  .+  y
 ) )  =  ( ( F `  x )  .+^  ( F `  y ) ) )   =>    |-  ( ph  ->  F  e.  ( S  GrpHom  T ) )
 
Theoremghmmhm 14528 A group homorphism is a monoid homorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )
 
Theoremghmmhmb 14529 Group homorphisms and monoid homomorphisms coincide. (Thus,  GrpHom is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( S  GrpHom  T )  =  ( S MndHom  T ) )
 
Theoremghmmulg 14530 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .X.  =  (.g `  H )   =>    |-  ( ( F  e.  ( G  GrpHom  H ) 
 /\  N  e.  ZZ  /\  X  e.  B ) 
 ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X ) ) )
 
Theoremghmrn 14531 The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  e.  (SubGrp `  T ) )
 
Theorem0ghm 14532 The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- 
 .0.  =  ( 0g `  N )   &    |-  B  =  (
 Base `  M )   =>    |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M  GrpHom  N ) )
 
Theoremidghm 14533 The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  (  _I  |`  B )  e.  ( G  GrpHom  G ) )
 
Theoremresghm 14534 Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  U  =  ( Ss  X )   =>    |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  X  e.  (SubGrp `  S ) )  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
 
Theoremresghm2 14535 One direction of resghm2b 14536. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
 |-  U  =  ( Ts  X )   =>    |-  ( ( F  e.  ( S  GrpHom  U ) 
 /\  X  e.  (SubGrp `  T ) )  ->  F  e.  ( S  GrpHom  T ) )
 
Theoremresghm2b 14536 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 14537 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 14538 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 14539 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 14540 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 14541 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 14542 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 14543 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 14544 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 14545* 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 14546* 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 14547 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 14548* 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 14549* 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 14550* 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 14551* 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 14552* 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 14553* 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 14554* 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 14555* 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 14556 The class of group isomorphism sets.
 class GrpIso
 
Syntaxcgic 14557 The class of the group isomorphism relation.
 class  ~=ph𝑔
 
Definitiondf-gim 14558* 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 14559 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 14560 The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
 |- GrpIso  Fn  ( Grp  X.  Grp )
 
Theoremisgim 14561 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 14562 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 14563 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 14564 A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 17282. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( F  e.  ( R GrpIso  S )  <->  ( F  e.  ( R  GrpHom  S ) 
 /\  `' F  e.  ( S  GrpHom  R ) ) )
 
Theoremsubggim 14565 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 14566 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 14567 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 14568 The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑔 
 S 
 <->  ( R GrpIso  S )  =/= 
 (/) )
 
Theorembrgici 14569 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( F  e.  ( R GrpIso  S )  ->  R  ~=ph𝑔  S )
 
Theoremgicref 14570 Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( R  e.  Grp  ->  R  ~=ph𝑔 
 R )
 
Theoremgiclcl 14571 Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)
 |-  ( R  ~=ph𝑔 
 S  ->  R  e.  Grp )
 
Theoremgicrcl 14572 Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( R  ~=ph𝑔 
 S  ->  S  e.  Grp )
 
Theoremgicsym 14573 Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( R  ~=ph𝑔 
 S  ->  S  ~=ph𝑔  R )
 
Theoremgictr 14574 Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  ( ( R  ~=ph𝑔  S  /\  S  ~=ph𝑔 
 T )  ->  R  ~=ph𝑔  T )
 
Theoremgicer 14575 Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- 
 ~=ph𝑔  Er  Grp
 
Theoremgicen 14576 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 14577 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 14578 Extend class definition to include the class of group actions.
 class  GrpAct
 
Definitiondf-ga 14579* 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 14580* 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 14595). 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 14581 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 14582 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 14583 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 14584 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 14585 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 14586 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 14587 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 14588 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 14589* 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 14590* 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 14591 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 14592* 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 14593 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 14594 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 14595* 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 14596* 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 14597* 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 14598* 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 14599* 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 14600* Existence and uniqueness for the function of orbsta 14602. (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 )
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