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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremisghmd 19101* Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
๐‘‹ = (Baseโ€˜๐‘†)    &   ๐‘Œ = (Baseโ€˜๐‘‡)    &    + = (+gโ€˜๐‘†)    &    โจฃ = (+gโ€˜๐‘‡)    &   (๐œ‘ โ†’ ๐‘† โˆˆ Grp)    &   (๐œ‘ โ†’ ๐‘‡ โˆˆ Grp)    &   (๐œ‘ โ†’ ๐น:๐‘‹โŸถ๐‘Œ)    &   ((๐œ‘ โˆง (๐‘ฅ โˆˆ ๐‘‹ โˆง ๐‘ฆ โˆˆ ๐‘‹)) โ†’ (๐นโ€˜(๐‘ฅ + ๐‘ฆ)) = ((๐นโ€˜๐‘ฅ) โจฃ (๐นโ€˜๐‘ฆ)))    โ‡’   (๐œ‘ โ†’ ๐น โˆˆ (๐‘† GrpHom ๐‘‡))
 
Theoremghmmhm 19102 A group homomorphism is a monoid homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(๐น โˆˆ (๐‘† GrpHom ๐‘‡) โ†’ ๐น โˆˆ (๐‘† MndHom ๐‘‡))
 
Theoremghmmhmb 19103 Group homomorphisms 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.)
((๐‘† โˆˆ Grp โˆง ๐‘‡ โˆˆ Grp) โ†’ (๐‘† GrpHom ๐‘‡) = (๐‘† MndHom ๐‘‡))
 
Theoremghmmulg 19104 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
๐ต = (Baseโ€˜๐บ)    &    ยท = (.gโ€˜๐บ)    &    ร— = (.gโ€˜๐ป)    โ‡’   ((๐น โˆˆ (๐บ GrpHom ๐ป) โˆง ๐‘ โˆˆ โ„ค โˆง ๐‘‹ โˆˆ ๐ต) โ†’ (๐นโ€˜(๐‘ ยท ๐‘‹)) = (๐‘ ร— (๐นโ€˜๐‘‹)))
 
Theoremghmrn 19105 The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
(๐น โˆˆ (๐‘† GrpHom ๐‘‡) โ†’ ran ๐น โˆˆ (SubGrpโ€˜๐‘‡))
 
Theorem0ghm 19106 The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
0 = (0gโ€˜๐‘)    &   ๐ต = (Baseโ€˜๐‘€)    โ‡’   ((๐‘€ โˆˆ Grp โˆง ๐‘ โˆˆ Grp) โ†’ (๐ต ร— { 0 }) โˆˆ (๐‘€ GrpHom ๐‘))
 
Theoremidghm 19107 The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
๐ต = (Baseโ€˜๐บ)    โ‡’   (๐บ โˆˆ Grp โ†’ ( I โ†พ ๐ต) โˆˆ (๐บ GrpHom ๐บ))
 
Theoremresghm 19108 Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
๐‘ˆ = (๐‘† โ†พs ๐‘‹)    โ‡’   ((๐น โˆˆ (๐‘† GrpHom ๐‘‡) โˆง ๐‘‹ โˆˆ (SubGrpโ€˜๐‘†)) โ†’ (๐น โ†พ ๐‘‹) โˆˆ (๐‘ˆ GrpHom ๐‘‡))
 
Theoremresghm2 19109 One direction of resghm2b 19110. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
๐‘ˆ = (๐‘‡ โ†พs ๐‘‹)    โ‡’   ((๐น โˆˆ (๐‘† GrpHom ๐‘ˆ) โˆง ๐‘‹ โˆˆ (SubGrpโ€˜๐‘‡)) โ†’ ๐น โˆˆ (๐‘† GrpHom ๐‘‡))
 
Theoremresghm2b 19110 Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
๐‘ˆ = (๐‘‡ โ†พs ๐‘‹)    โ‡’   ((๐‘‹ โˆˆ (SubGrpโ€˜๐‘‡) โˆง ran ๐น โŠ† ๐‘‹) โ†’ (๐น โˆˆ (๐‘† GrpHom ๐‘‡) โ†” ๐น โˆˆ (๐‘† GrpHom ๐‘ˆ)))
 
Theoremghmghmrn 19111 A group homomorphism from ๐บ to ๐ป is also a group homomorphism from ๐บ to its image in ๐ป. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by AV, 26-Aug-2021.)
๐‘ˆ = (๐‘‡ โ†พs ran ๐น)    โ‡’   (๐น โˆˆ (๐‘† GrpHom ๐‘‡) โ†’ ๐น โˆˆ (๐‘† GrpHom ๐‘ˆ))
 
Theoremghmco 19112 The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
((๐น โˆˆ (๐‘‡ GrpHom ๐‘ˆ) โˆง ๐บ โˆˆ (๐‘† GrpHom ๐‘‡)) โ†’ (๐น โˆ˜ ๐บ) โˆˆ (๐‘† GrpHom ๐‘ˆ))
 
Theoremghmima 19113 The image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
((๐น โˆˆ (๐‘† GrpHom ๐‘‡) โˆง ๐‘ˆ โˆˆ (SubGrpโ€˜๐‘†)) โ†’ (๐น โ€œ ๐‘ˆ) โˆˆ (SubGrpโ€˜๐‘‡))
 
Theoremghmpreima 19114 The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
((๐น โˆˆ (๐‘† GrpHom ๐‘‡) โˆง ๐‘‰ โˆˆ (SubGrpโ€˜๐‘‡)) โ†’ (โ—ก๐น โ€œ ๐‘‰) โˆˆ (SubGrpโ€˜๐‘†))
 
Theoremghmeql 19115 The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
((๐น โˆˆ (๐‘† GrpHom ๐‘‡) โˆง ๐บ โˆˆ (๐‘† GrpHom ๐‘‡)) โ†’ dom (๐น โˆฉ ๐บ) โˆˆ (SubGrpโ€˜๐‘†))
 
Theoremghmnsgima 19116 The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
๐‘Œ = (Baseโ€˜๐‘‡)    โ‡’   ((๐น โˆˆ (๐‘† GrpHom ๐‘‡) โˆง ๐‘ˆ โˆˆ (NrmSGrpโ€˜๐‘†) โˆง ran ๐น = ๐‘Œ) โ†’ (๐น โ€œ ๐‘ˆ) โˆˆ (NrmSGrpโ€˜๐‘‡))
 
Theoremghmnsgpreima 19117 The inverse image of a normal subgroup under a homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
((๐น โˆˆ (๐‘† GrpHom ๐‘‡) โˆง ๐‘‰ โˆˆ (NrmSGrpโ€˜๐‘‡)) โ†’ (โ—ก๐น โ€œ ๐‘‰) โˆˆ (NrmSGrpโ€˜๐‘†))
 
Theoremghmker 19118 The kernel of a homomorphism is a normal subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
0 = (0gโ€˜๐‘‡)    โ‡’   (๐น โˆˆ (๐‘† GrpHom ๐‘‡) โ†’ (โ—ก๐น โ€œ { 0 }) โˆˆ (NrmSGrpโ€˜๐‘†))
 
Theoremghmeqker 19119 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.)
๐ต = (Baseโ€˜๐‘†)    &    0 = (0gโ€˜๐‘‡)    &   ๐พ = (โ—ก๐น โ€œ { 0 })    &    โˆ’ = (-gโ€˜๐‘†)    โ‡’   ((๐น โˆˆ (๐‘† GrpHom ๐‘‡) โˆง ๐‘ˆ โˆˆ ๐ต โˆง ๐‘‰ โˆˆ ๐ต) โ†’ ((๐นโ€˜๐‘ˆ) = (๐นโ€˜๐‘‰) โ†” (๐‘ˆ โˆ’ ๐‘‰) โˆˆ ๐พ))
 
Theorempwsdiagghm 19120* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
๐‘Œ = (๐‘… โ†‘s ๐ผ)    &   ๐ต = (Baseโ€˜๐‘…)    &   ๐น = (๐‘ฅ โˆˆ ๐ต โ†ฆ (๐ผ ร— {๐‘ฅ}))    โ‡’   ((๐‘… โˆˆ Grp โˆง ๐ผ โˆˆ ๐‘Š) โ†’ ๐น โˆˆ (๐‘… GrpHom ๐‘Œ))
 
Theoremghmf1 19121* 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.)
๐‘‹ = (Baseโ€˜๐‘†)    &   ๐‘Œ = (Baseโ€˜๐‘‡)    &    0 = (0gโ€˜๐‘†)    &   ๐‘ˆ = (0gโ€˜๐‘‡)    โ‡’   (๐น โˆˆ (๐‘† GrpHom ๐‘‡) โ†’ (๐น:๐‘‹โ€“1-1โ†’๐‘Œ โ†” โˆ€๐‘ฅ โˆˆ ๐‘‹ ((๐นโ€˜๐‘ฅ) = ๐‘ˆ โ†’ ๐‘ฅ = 0 )))
 
Theoremghmf1o 19122 A bijective group homomorphism is an isomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.)
๐‘‹ = (Baseโ€˜๐‘†)    &   ๐‘Œ = (Baseโ€˜๐‘‡)    โ‡’   (๐น โˆˆ (๐‘† GrpHom ๐‘‡) โ†’ (๐น:๐‘‹โ€“1-1-ontoโ†’๐‘Œ โ†” โ—ก๐น โˆˆ (๐‘‡ GrpHom ๐‘†)))
 
Theoremconjghm 19123* Conjugation is an automorphism of the group. (Contributed by Mario Carneiro, 13-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &    โˆ’ = (-gโ€˜๐บ)    &   ๐น = (๐‘ฅ โˆˆ ๐‘‹ โ†ฆ ((๐ด + ๐‘ฅ) โˆ’ ๐ด))    โ‡’   ((๐บ โˆˆ Grp โˆง ๐ด โˆˆ ๐‘‹) โ†’ (๐น โˆˆ (๐บ GrpHom ๐บ) โˆง ๐น:๐‘‹โ€“1-1-ontoโ†’๐‘‹))
 
Theoremconjsubg 19124* A conjugated subgroup is also a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &    โˆ’ = (-gโ€˜๐บ)    &   ๐น = (๐‘ฅ โˆˆ ๐‘† โ†ฆ ((๐ด + ๐‘ฅ) โˆ’ ๐ด))    โ‡’   ((๐‘† โˆˆ (SubGrpโ€˜๐บ) โˆง ๐ด โˆˆ ๐‘‹) โ†’ ran ๐น โˆˆ (SubGrpโ€˜๐บ))
 
Theoremconjsubgen 19125* A conjugated subgroup is equinumerous to the original subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &    โˆ’ = (-gโ€˜๐บ)    &   ๐น = (๐‘ฅ โˆˆ ๐‘† โ†ฆ ((๐ด + ๐‘ฅ) โˆ’ ๐ด))    โ‡’   ((๐‘† โˆˆ (SubGrpโ€˜๐บ) โˆง ๐ด โˆˆ ๐‘‹) โ†’ ๐‘† โ‰ˆ ran ๐น)
 
Theoremconjnmz 19126* A subgroup is unchanged under conjugation by an element of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &    โˆ’ = (-gโ€˜๐บ)    &   ๐น = (๐‘ฅ โˆˆ ๐‘† โ†ฆ ((๐ด + ๐‘ฅ) โˆ’ ๐ด))    &   ๐‘ = {๐‘ฆ โˆˆ ๐‘‹ โˆฃ โˆ€๐‘ง โˆˆ ๐‘‹ ((๐‘ฆ + ๐‘ง) โˆˆ ๐‘† โ†” (๐‘ง + ๐‘ฆ) โˆˆ ๐‘†)}    โ‡’   ((๐‘† โˆˆ (SubGrpโ€˜๐บ) โˆง ๐ด โˆˆ ๐‘) โ†’ ๐‘† = ran ๐น)
 
Theoremconjnmzb 19127* Alternative condition for elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &    โˆ’ = (-gโ€˜๐บ)    &   ๐น = (๐‘ฅ โˆˆ ๐‘† โ†ฆ ((๐ด + ๐‘ฅ) โˆ’ ๐ด))    &   ๐‘ = {๐‘ฆ โˆˆ ๐‘‹ โˆฃ โˆ€๐‘ง โˆˆ ๐‘‹ ((๐‘ฆ + ๐‘ง) โˆˆ ๐‘† โ†” (๐‘ง + ๐‘ฆ) โˆˆ ๐‘†)}    โ‡’   (๐‘† โˆˆ (SubGrpโ€˜๐บ) โ†’ (๐ด โˆˆ ๐‘ โ†” (๐ด โˆˆ ๐‘‹ โˆง ๐‘† = ran ๐น)))
 
Theoremconjnsg 19128* A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &    โˆ’ = (-gโ€˜๐บ)    &   ๐น = (๐‘ฅ โˆˆ ๐‘† โ†ฆ ((๐ด + ๐‘ฅ) โˆ’ ๐ด))    โ‡’   ((๐‘† โˆˆ (NrmSGrpโ€˜๐บ) โˆง ๐ด โˆˆ ๐‘‹) โ†’ ๐‘† = ran ๐น)
 
Theoremqusghm 19129* If ๐‘Œ is a normal subgroup of ๐บ, then the "natural map" from elements to their cosets is a group homomorphism from ๐บ to ๐บ / ๐‘Œ. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 18-Sep-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &   ๐ป = (๐บ /s (๐บ ~QG ๐‘Œ))    &   ๐น = (๐‘ฅ โˆˆ ๐‘‹ โ†ฆ [๐‘ฅ](๐บ ~QG ๐‘Œ))    โ‡’   (๐‘Œ โˆˆ (NrmSGrpโ€˜๐บ) โ†’ ๐น โˆˆ (๐บ GrpHom ๐ป))
 
Theoremghmpropd 19130* Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
(๐œ‘ โ†’ ๐ต = (Baseโ€˜๐ฝ))    &   (๐œ‘ โ†’ ๐ถ = (Baseโ€˜๐พ))    &   (๐œ‘ โ†’ ๐ต = (Baseโ€˜๐ฟ))    &   (๐œ‘ โ†’ ๐ถ = (Baseโ€˜๐‘€))    &   ((๐œ‘ โˆง (๐‘ฅ โˆˆ ๐ต โˆง ๐‘ฆ โˆˆ ๐ต)) โ†’ (๐‘ฅ(+gโ€˜๐ฝ)๐‘ฆ) = (๐‘ฅ(+gโ€˜๐ฟ)๐‘ฆ))    &   ((๐œ‘ โˆง (๐‘ฅ โˆˆ ๐ถ โˆง ๐‘ฆ โˆˆ ๐ถ)) โ†’ (๐‘ฅ(+gโ€˜๐พ)๐‘ฆ) = (๐‘ฅ(+gโ€˜๐‘€)๐‘ฆ))    โ‡’   (๐œ‘ โ†’ (๐ฝ GrpHom ๐พ) = (๐ฟ GrpHom ๐‘€))
 
10.2.6  Isomorphisms of groups
 
Syntaxcgim 19131 The class of group isomorphism sets.
class GrpIso
 
Syntaxcgic 19132 The class of the group isomorphism relation.
class โ‰ƒ๐‘”
 
Definitiondf-gim 19133* 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 = (๐‘  โˆˆ Grp, ๐‘ก โˆˆ Grp โ†ฆ {๐‘” โˆˆ (๐‘  GrpHom ๐‘ก) โˆฃ ๐‘”:(Baseโ€˜๐‘ )โ€“1-1-ontoโ†’(Baseโ€˜๐‘ก)})
 
Definitiondf-gic 19134 Two groups are said to be isomorphic iff they are connected by at least one isomorphism. Isomorphic 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.)
โ‰ƒ๐‘” = (โ—ก GrpIso โ€œ (V โˆ– 1o))
 
Theoremgimfn 19135 The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
GrpIso Fn (Grp ร— Grp)
 
Theoremisgim 19136 An isomorphism of groups is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
๐ต = (Baseโ€˜๐‘…)    &   ๐ถ = (Baseโ€˜๐‘†)    โ‡’   (๐น โˆˆ (๐‘… GrpIso ๐‘†) โ†” (๐น โˆˆ (๐‘… GrpHom ๐‘†) โˆง ๐น:๐ตโ€“1-1-ontoโ†’๐ถ))
 
Theoremgimf1o 19137 An isomorphism of groups is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
๐ต = (Baseโ€˜๐‘…)    &   ๐ถ = (Baseโ€˜๐‘†)    โ‡’   (๐น โˆˆ (๐‘… GrpIso ๐‘†) โ†’ ๐น:๐ตโ€“1-1-ontoโ†’๐ถ)
 
Theoremgimghm 19138 An isomorphism of groups is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
(๐น โˆˆ (๐‘… GrpIso ๐‘†) โ†’ ๐น โˆˆ (๐‘… GrpHom ๐‘†))
 
Theoremisgim2 19139 A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo 23263. (Contributed by Mario Carneiro, 6-May-2015.)
(๐น โˆˆ (๐‘… GrpIso ๐‘†) โ†” (๐น โˆˆ (๐‘… GrpHom ๐‘†) โˆง โ—ก๐น โˆˆ (๐‘† GrpHom ๐‘…)))
 
Theoremsubggim 19140 Behavior of subgroups under isomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
๐ต = (Baseโ€˜๐‘…)    โ‡’   ((๐น โˆˆ (๐‘… GrpIso ๐‘†) โˆง ๐ด โŠ† ๐ต) โ†’ (๐ด โˆˆ (SubGrpโ€˜๐‘…) โ†” (๐น โ€œ ๐ด) โˆˆ (SubGrpโ€˜๐‘†)))
 
Theoremgimcnv 19141 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.)
(๐น โˆˆ (๐‘† GrpIso ๐‘‡) โ†’ โ—ก๐น โˆˆ (๐‘‡ GrpIso ๐‘†))
 
Theoremgimco 19142 The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
((๐น โˆˆ (๐‘‡ GrpIso ๐‘ˆ) โˆง ๐บ โˆˆ (๐‘† GrpIso ๐‘‡)) โ†’ (๐น โˆ˜ ๐บ) โˆˆ (๐‘† GrpIso ๐‘ˆ))
 
Theorembrgic 19143 The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(๐‘… โ‰ƒ๐‘” ๐‘† โ†” (๐‘… GrpIso ๐‘†) โ‰  โˆ…)
 
Theorembrgici 19144 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(๐น โˆˆ (๐‘… GrpIso ๐‘†) โ†’ ๐‘… โ‰ƒ๐‘” ๐‘†)
 
Theoremgicref 19145 Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
(๐‘… โˆˆ Grp โ†’ ๐‘… โ‰ƒ๐‘” ๐‘…)
 
Theoremgiclcl 19146 Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(๐‘… โ‰ƒ๐‘” ๐‘† โ†’ ๐‘… โˆˆ Grp)
 
Theoremgicrcl 19147 Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)
(๐‘… โ‰ƒ๐‘” ๐‘† โ†’ ๐‘† โˆˆ Grp)
 
Theoremgicsym 19148 Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
(๐‘… โ‰ƒ๐‘” ๐‘† โ†’ ๐‘† โ‰ƒ๐‘” ๐‘…)
 
Theoremgictr 19149 Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
((๐‘… โ‰ƒ๐‘” ๐‘† โˆง ๐‘† โ‰ƒ๐‘” ๐‘‡) โ†’ ๐‘… โ‰ƒ๐‘” ๐‘‡)
 
Theoremgicer 19150 Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 1-May-2021.)
โ‰ƒ๐‘” Er Grp
 
Theoremgicen 19151 Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.)
๐ต = (Baseโ€˜๐‘…)    &   ๐ถ = (Baseโ€˜๐‘†)    โ‡’   (๐‘… โ‰ƒ๐‘” ๐‘† โ†’ ๐ต โ‰ˆ ๐ถ)
 
Theoremgicsubgen 19152 A less trivial example of a group invariant: cardinality of the subgroup lattice. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(๐‘… โ‰ƒ๐‘” ๐‘† โ†’ (SubGrpโ€˜๐‘…) โ‰ˆ (SubGrpโ€˜๐‘†))
 
10.2.7  Group actions
 
Syntaxcga 19153 Extend class definition to include the class of group actions.
class GrpAct
 
Definitiondf-ga 19154* Define the class of all group actions. A group ๐บ acts on a set ๐‘† if a permutation on ๐‘† is associated with every element of ๐บ in such a way that the identity permutation on ๐‘† is associated with the neutral element of ๐บ, and the composition of the permutations associated with two elements of ๐บ is identical with the permutation associated with the composition of these two elements (in the same order) in the group ๐บ. (Contributed by Jeff Hankins, 10-Aug-2009.)
GrpAct = (๐‘” โˆˆ Grp, ๐‘  โˆˆ V โ†ฆ โฆ‹(Baseโ€˜๐‘”) / ๐‘โฆŒ{๐‘š โˆˆ (๐‘  โ†‘m (๐‘ ร— ๐‘ )) โˆฃ โˆ€๐‘ฅ โˆˆ ๐‘  (((0gโ€˜๐‘”)๐‘š๐‘ฅ) = ๐‘ฅ โˆง โˆ€๐‘ฆ โˆˆ ๐‘ โˆ€๐‘ง โˆˆ ๐‘ ((๐‘ฆ(+gโ€˜๐‘”)๐‘ง)๐‘š๐‘ฅ) = (๐‘ฆ๐‘š(๐‘ง๐‘š๐‘ฅ)))})
 
Theoremisga 19155* The predicate "is a (left) group action". The group ๐บ is said to act on the base set ๐‘Œ 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 ๐‘” of ๐บ is a permutation of the elements of ๐‘Œ (see gapm 19170). 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.)
๐‘‹ = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &    0 = (0gโ€˜๐บ)    โ‡’   ( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โ†” ((๐บ โˆˆ Grp โˆง ๐‘Œ โˆˆ V) โˆง ( โŠ• :(๐‘‹ ร— ๐‘Œ)โŸถ๐‘Œ โˆง โˆ€๐‘ฅ โˆˆ ๐‘Œ (( 0 โŠ• ๐‘ฅ) = ๐‘ฅ โˆง โˆ€๐‘ฆ โˆˆ ๐‘‹ โˆ€๐‘ง โˆˆ ๐‘‹ ((๐‘ฆ + ๐‘ง) โŠ• ๐‘ฅ) = (๐‘ฆ โŠ• (๐‘ง โŠ• ๐‘ฅ))))))
 
Theoremgagrp 19156 The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.)
( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โ†’ ๐บ โˆˆ Grp)
 
Theoremgaset 19157 The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)
( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โ†’ ๐‘Œ โˆˆ V)
 
Theoremgagrpid 19158 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 = (0gโ€˜๐บ)    โ‡’   (( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง ๐ด โˆˆ ๐‘Œ) โ†’ ( 0 โŠ• ๐ด) = ๐ด)
 
Theoremgaf 19159 The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    โ‡’   ( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โ†’ โŠ• :(๐‘‹ ร— ๐‘Œ)โŸถ๐‘Œ)
 
Theoremgafo 19160 A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    โ‡’   ( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โ†’ โŠ• :(๐‘‹ ร— ๐‘Œ)โ€“ontoโ†’๐‘Œ)
 
Theoremgaass 19161 An "associative" property for group actions. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    โ‡’   (( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง (๐ด โˆˆ ๐‘‹ โˆง ๐ต โˆˆ ๐‘‹ โˆง ๐ถ โˆˆ ๐‘Œ)) โ†’ ((๐ด + ๐ต) โŠ• ๐ถ) = (๐ด โŠ• (๐ต โŠ• ๐ถ)))
 
Theoremga0 19162 The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
(๐บ โˆˆ Grp โ†’ โˆ… โˆˆ (๐บ GrpAct โˆ…))
 
Theoremgaid 19163 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.)
๐‘‹ = (Baseโ€˜๐บ)    โ‡’   ((๐บ โˆˆ Grp โˆง ๐‘† โˆˆ ๐‘‰) โ†’ (2nd โ†พ (๐‘‹ ร— ๐‘†)) โˆˆ (๐บ GrpAct ๐‘†))
 
Theoremsubgga 19164* A subgroup acts on its parent group. (Contributed by Jeff Hankins, 13-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &   ๐ป = (๐บ โ†พs ๐‘Œ)    &   ๐น = (๐‘ฅ โˆˆ ๐‘Œ, ๐‘ฆ โˆˆ ๐‘‹ โ†ฆ (๐‘ฅ + ๐‘ฆ))    โ‡’   (๐‘Œ โˆˆ (SubGrpโ€˜๐บ) โ†’ ๐น โˆˆ (๐ป GrpAct ๐‘‹))
 
Theoremgass 19165* 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.)
๐‘‹ = (Baseโ€˜๐บ)    โ‡’   (( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง ๐‘ โŠ† ๐‘Œ) โ†’ (( โŠ• โ†พ (๐‘‹ ร— ๐‘)) โˆˆ (๐บ GrpAct ๐‘) โ†” โˆ€๐‘ฅ โˆˆ ๐‘‹ โˆ€๐‘ฆ โˆˆ ๐‘ (๐‘ฅ โŠ• ๐‘ฆ) โˆˆ ๐‘))
 
Theoremgasubg 19166 The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.)
๐ป = (๐บ โ†พs ๐‘†)    โ‡’   (( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง ๐‘† โˆˆ (SubGrpโ€˜๐บ)) โ†’ ( โŠ• โ†พ (๐‘† ร— ๐‘Œ)) โˆˆ (๐ป GrpAct ๐‘Œ))
 
Theoremgaid2 19167* 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.)
๐‘‹ = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &   ๐น = (๐‘ฅ โˆˆ ๐‘‹, ๐‘ฆ โˆˆ ๐‘‹ โ†ฆ (๐‘ฅ + ๐‘ฆ))    โ‡’   (๐บ โˆˆ Grp โ†’ ๐น โˆˆ (๐บ GrpAct ๐‘‹))
 
Theoremgalcan 19168 The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    โ‡’   (( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง (๐ด โˆˆ ๐‘‹ โˆง ๐ต โˆˆ ๐‘Œ โˆง ๐ถ โˆˆ ๐‘Œ)) โ†’ ((๐ด โŠ• ๐ต) = (๐ด โŠ• ๐ถ) โ†” ๐ต = ๐ถ))
 
Theoremgacan 19169 Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &   ๐‘ = (invgโ€˜๐บ)    โ‡’   (( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง (๐ด โˆˆ ๐‘‹ โˆง ๐ต โˆˆ ๐‘Œ โˆง ๐ถ โˆˆ ๐‘Œ)) โ†’ ((๐ด โŠ• ๐ต) = ๐ถ โ†” ((๐‘โ€˜๐ด) โŠ• ๐ถ) = ๐ต))
 
Theoremgapm 19170* 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.)
๐‘‹ = (Baseโ€˜๐บ)    &   ๐น = (๐‘ฅ โˆˆ ๐‘Œ โ†ฆ (๐ด โŠ• ๐‘ฅ))    โ‡’   (( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง ๐ด โˆˆ ๐‘‹) โ†’ ๐น:๐‘Œโ€“1-1-ontoโ†’๐‘Œ)
 
Theoremgaorb 19171* 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.)
โˆผ = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ({๐‘ฅ, ๐‘ฆ} โŠ† ๐‘Œ โˆง โˆƒ๐‘” โˆˆ ๐‘‹ (๐‘” โŠ• ๐‘ฅ) = ๐‘ฆ)}    โ‡’   (๐ด โˆผ ๐ต โ†” (๐ด โˆˆ ๐‘Œ โˆง ๐ต โˆˆ ๐‘Œ โˆง โˆƒโ„Ž โˆˆ ๐‘‹ (โ„Ž โŠ• ๐ด) = ๐ต))
 
Theoremgaorber 19172* 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.)
โˆผ = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ({๐‘ฅ, ๐‘ฆ} โŠ† ๐‘Œ โˆง โˆƒ๐‘” โˆˆ ๐‘‹ (๐‘” โŠ• ๐‘ฅ) = ๐‘ฆ)}    &   ๐‘‹ = (Baseโ€˜๐บ)    โ‡’   ( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โ†’ โˆผ Er ๐‘Œ)
 
Theoremgastacl 19173* The stabilizer subgroup in a group action. (Contributed by Mario Carneiro, 15-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &   ๐ป = {๐‘ข โˆˆ ๐‘‹ โˆฃ (๐‘ข โŠ• ๐ด) = ๐ด}    โ‡’   (( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง ๐ด โˆˆ ๐‘Œ) โ†’ ๐ป โˆˆ (SubGrpโ€˜๐บ))
 
Theoremgastacos 19174* Write the coset relation for the stabilizer subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &   ๐ป = {๐‘ข โˆˆ ๐‘‹ โˆฃ (๐‘ข โŠ• ๐ด) = ๐ด}    &    โˆผ = (๐บ ~QG ๐ป)    โ‡’   ((( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง ๐ด โˆˆ ๐‘Œ) โˆง (๐ต โˆˆ ๐‘‹ โˆง ๐ถ โˆˆ ๐‘‹)) โ†’ (๐ต โˆผ ๐ถ โ†” (๐ต โŠ• ๐ด) = (๐ถ โŠ• ๐ด)))
 
Theoremorbstafun 19175* Existence and uniqueness for the function of orbsta 19177. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
๐‘‹ = (Baseโ€˜๐บ)    &   ๐ป = {๐‘ข โˆˆ ๐‘‹ โˆฃ (๐‘ข โŠ• ๐ด) = ๐ด}    &    โˆผ = (๐บ ~QG ๐ป)    &   ๐น = ran (๐‘˜ โˆˆ ๐‘‹ โ†ฆ โŸจ[๐‘˜] โˆผ , (๐‘˜ โŠ• ๐ด)โŸฉ)    โ‡’   (( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง ๐ด โˆˆ ๐‘Œ) โ†’ Fun ๐น)
 
Theoremorbstaval 19176* 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.)
๐‘‹ = (Baseโ€˜๐บ)    &   ๐ป = {๐‘ข โˆˆ ๐‘‹ โˆฃ (๐‘ข โŠ• ๐ด) = ๐ด}    &    โˆผ = (๐บ ~QG ๐ป)    &   ๐น = ran (๐‘˜ โˆˆ ๐‘‹ โ†ฆ โŸจ[๐‘˜] โˆผ , (๐‘˜ โŠ• ๐ด)โŸฉ)    โ‡’   ((( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง ๐ด โˆˆ ๐‘Œ) โˆง ๐ต โˆˆ ๐‘‹) โ†’ (๐นโ€˜[๐ต] โˆผ ) = (๐ต โŠ• ๐ด))
 
Theoremorbsta 19177* The Orbit-Stabilizer theorem. The mapping ๐น is a bijection from the cosets of the stabilizer subgroup of ๐ด to the orbit of ๐ด. (Contributed by Mario Carneiro, 15-Jan-2015.)
๐‘‹ = (Baseโ€˜๐บ)    &   ๐ป = {๐‘ข โˆˆ ๐‘‹ โˆฃ (๐‘ข โŠ• ๐ด) = ๐ด}    &    โˆผ = (๐บ ~QG ๐ป)    &   ๐น = ran (๐‘˜ โˆˆ ๐‘‹ โ†ฆ โŸจ[๐‘˜] โˆผ , (๐‘˜ โŠ• ๐ด)โŸฉ)    &   ๐‘‚ = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ({๐‘ฅ, ๐‘ฆ} โŠ† ๐‘Œ โˆง โˆƒ๐‘” โˆˆ ๐‘‹ (๐‘” โŠ• ๐‘ฅ) = ๐‘ฆ)}    โ‡’   (( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง ๐ด โˆˆ ๐‘Œ) โ†’ ๐น:(๐‘‹ / โˆผ )โ€“1-1-ontoโ†’[๐ด]๐‘‚)
 
Theoremorbsta2 19178* 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.)
๐‘‹ = (Baseโ€˜๐บ)    &   ๐ป = {๐‘ข โˆˆ ๐‘‹ โˆฃ (๐‘ข โŠ• ๐ด) = ๐ด}    &    โˆผ = (๐บ ~QG ๐ป)    &   ๐‘‚ = {โŸจ๐‘ฅ, ๐‘ฆโŸฉ โˆฃ ({๐‘ฅ, ๐‘ฆ} โŠ† ๐‘Œ โˆง โˆƒ๐‘” โˆˆ ๐‘‹ (๐‘” โŠ• ๐‘ฅ) = ๐‘ฆ)}    โ‡’   ((( โŠ• โˆˆ (๐บ GrpAct ๐‘Œ) โˆง ๐ด โˆˆ ๐‘Œ) โˆง ๐‘‹ โˆˆ Fin) โ†’ (โ™ฏโ€˜๐‘‹) = ((โ™ฏโ€˜[๐ด]๐‘‚) ยท (โ™ฏโ€˜๐ป)))
 
10.2.8  Centralizers and centers
 
Syntaxccntz 19179 Syntax for the centralizer of a set in a monoid.
class Cntz
 
Syntaxccntr 19180 Syntax for the centralizer of a monoid.
class Cntr
 
Definitiondf-cntz 19181* 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 = (๐‘š โˆˆ V โ†ฆ (๐‘  โˆˆ ๐’ซ (Baseโ€˜๐‘š) โ†ฆ {๐‘ฅ โˆˆ (Baseโ€˜๐‘š) โˆฃ โˆ€๐‘ฆ โˆˆ ๐‘  (๐‘ฅ(+gโ€˜๐‘š)๐‘ฆ) = (๐‘ฆ(+gโ€˜๐‘š)๐‘ฅ)}))
 
Definitiondf-cntr 19182 Define the center of a magma, which is the elements that commute with all others. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Cntr = (๐‘š โˆˆ V โ†ฆ ((Cntzโ€˜๐‘š)โ€˜(Baseโ€˜๐‘š)))
 
Theoremcntrval 19183 Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   (๐‘โ€˜๐ต) = (Cntrโ€˜๐‘€)
 
Theoremcntzfval 19184* First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   (๐‘€ โˆˆ ๐‘‰ โ†’ ๐‘ = (๐‘  โˆˆ ๐’ซ ๐ต โ†ฆ {๐‘ฅ โˆˆ ๐ต โˆฃ โˆ€๐‘ฆ โˆˆ ๐‘  (๐‘ฅ + ๐‘ฆ) = (๐‘ฆ + ๐‘ฅ)}))
 
Theoremcntzval 19185* Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   (๐‘† โŠ† ๐ต โ†’ (๐‘โ€˜๐‘†) = {๐‘ฅ โˆˆ ๐ต โˆฃ โˆ€๐‘ฆ โˆˆ ๐‘† (๐‘ฅ + ๐‘ฆ) = (๐‘ฆ + ๐‘ฅ)})
 
Theoremelcntz 19186* Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   (๐‘† โŠ† ๐ต โ†’ (๐ด โˆˆ (๐‘โ€˜๐‘†) โ†” (๐ด โˆˆ ๐ต โˆง โˆ€๐‘ฆ โˆˆ ๐‘† (๐ด + ๐‘ฆ) = (๐‘ฆ + ๐ด))))
 
Theoremcntzel 19187* Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   ((๐‘† โŠ† ๐ต โˆง ๐‘‹ โˆˆ ๐ต) โ†’ (๐‘‹ โˆˆ (๐‘โ€˜๐‘†) โ†” โˆ€๐‘ฆ โˆˆ ๐‘† (๐‘‹ + ๐‘ฆ) = (๐‘ฆ + ๐‘‹)))
 
Theoremcntzsnval 19188* Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   (๐‘Œ โˆˆ ๐ต โ†’ (๐‘โ€˜{๐‘Œ}) = {๐‘ฅ โˆˆ ๐ต โˆฃ (๐‘ฅ + ๐‘Œ) = (๐‘Œ + ๐‘ฅ)})
 
Theoremelcntzsn 19189 Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   (๐‘Œ โˆˆ ๐ต โ†’ (๐‘‹ โˆˆ (๐‘โ€˜{๐‘Œ}) โ†” (๐‘‹ โˆˆ ๐ต โˆง (๐‘‹ + ๐‘Œ) = (๐‘Œ + ๐‘‹))))
 
Theoremsscntz 19190* A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   ((๐‘† โŠ† ๐ต โˆง ๐‘‡ โŠ† ๐ต) โ†’ (๐‘† โŠ† (๐‘โ€˜๐‘‡) โ†” โˆ€๐‘ฅ โˆˆ ๐‘† โˆ€๐‘ฆ โˆˆ ๐‘‡ (๐‘ฅ + ๐‘ฆ) = (๐‘ฆ + ๐‘ฅ)))
 
Theoremcntzrcl 19191 Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   (๐‘‹ โˆˆ (๐‘โ€˜๐‘†) โ†’ (๐‘€ โˆˆ V โˆง ๐‘† โŠ† ๐ต))
 
Theoremcntzssv 19192 The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   (๐‘โ€˜๐‘†) โŠ† ๐ต
 
Theoremcntzi 19193 Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
+ = (+gโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   ((๐‘‹ โˆˆ (๐‘โ€˜๐‘†) โˆง ๐‘Œ โˆˆ ๐‘†) โ†’ (๐‘‹ + ๐‘Œ) = (๐‘Œ + ๐‘‹))
 
Theoremelcntr 19194* Elementhood in the center of a magma. (Contributed by SN, 21-Mar-2025.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &   ๐‘ = (Cntrโ€˜๐‘€)    โ‡’   (๐ด โˆˆ ๐‘ โ†” (๐ด โˆˆ ๐ต โˆง โˆ€๐‘ฆ โˆˆ ๐ต (๐ด + ๐‘ฆ) = (๐‘ฆ + ๐ด)))
 
Theoremcntrss 19195 The center is a subset of the base field. (Contributed by Thierry Arnoux, 21-Aug-2023.)
๐ต = (Baseโ€˜๐‘€)    โ‡’   (Cntrโ€˜๐‘€) โŠ† ๐ต
 
Theoremcntri 19196 Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &   ๐‘ = (Cntrโ€˜๐‘€)    โ‡’   ((๐‘‹ โˆˆ ๐‘ โˆง ๐‘Œ โˆˆ ๐ต) โ†’ (๐‘‹ + ๐‘Œ) = (๐‘Œ + ๐‘‹))
 
Theoremresscntz 19197 Centralizer in a substructure. (Contributed by Mario Carneiro, 3-Oct-2015.)
๐ป = (๐บ โ†พs ๐ด)    &   ๐‘ = (Cntzโ€˜๐บ)    &   ๐‘Œ = (Cntzโ€˜๐ป)    โ‡’   ((๐ด โˆˆ ๐‘‰ โˆง ๐‘† โŠ† ๐ด) โ†’ (๐‘Œโ€˜๐‘†) = ((๐‘โ€˜๐‘†) โˆฉ ๐ด))
 
Theoremcntzsgrpcl 19198* Centralizers are closed under the semigroup operation. (Contributed by AV, 17-Feb-2025.)
๐ต = (Baseโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    &   ๐ถ = (๐‘โ€˜๐‘†)    โ‡’   ((๐‘€ โˆˆ Smgrp โˆง ๐‘† โŠ† ๐ต) โ†’ โˆ€๐‘ฆ โˆˆ ๐ถ โˆ€๐‘ง โˆˆ ๐ถ (๐‘ฆ(+gโ€˜๐‘€)๐‘ง) โˆˆ ๐ถ)
 
Theoremcntz2ss 19199 Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
๐ต = (Baseโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   ((๐‘† โŠ† ๐ต โˆง ๐‘‡ โŠ† ๐‘†) โ†’ (๐‘โ€˜๐‘†) โŠ† (๐‘โ€˜๐‘‡))
 
Theoremcntzrec 19200 Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &   ๐‘ = (Cntzโ€˜๐‘€)    โ‡’   ((๐‘† โŠ† ๐ต โˆง ๐‘‡ โŠ† ๐ต) โ†’ (๐‘† โŠ† (๐‘โ€˜๐‘‡) โ†” ๐‘‡ โŠ† (๐‘โ€˜๐‘†)))
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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47852
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