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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembrgici 18401 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝑅𝑔 𝑆)

Theoremgicref 18402 Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝑅 ∈ Grp → 𝑅𝑔 𝑅)

Theoremgiclcl 18403 Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑔 𝑆𝑅 ∈ Grp)

Theoremgicrcl 18404 Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)
(𝑅𝑔 𝑆𝑆 ∈ Grp)

Theoremgicsym 18405 Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
(𝑅𝑔 𝑆𝑆𝑔 𝑅)

Theoremgictr 18406 Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
((𝑅𝑔 𝑆𝑆𝑔 𝑇) → 𝑅𝑔 𝑇)

Theoremgicer 18407 Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 1-May-2021.)
𝑔 Er Grp

Theoremgicen 18408 Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝑅𝑔 𝑆𝐵𝐶)

Theoremgicsubgen 18409 A less trivial example of a group invariant: cardinality of the subgroup lattice. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))

10.2.7  Group actions

Syntaxcga 18410 Extend class definition to include the class of group actions.
class GrpAct

Definitiondf-ga 18411* 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 with the composition of these two elements (in the same order) in the group 𝐺. (Contributed by Jeff Hankins, 10-Aug-2009.)
GrpAct = (𝑔 ∈ Grp, 𝑠 ∈ V ↦ (Base‘𝑔) / 𝑏{𝑚 ∈ (𝑠m (𝑏 × 𝑠)) ∣ ∀𝑥𝑠 (((0g𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦𝑏𝑧𝑏 ((𝑦(+g𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})

Theoremisga 18412* 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 18427). 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)    &    + = (+g𝐺)    &    0 = (0g𝐺)       ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦𝑋𝑧𝑋 ((𝑦 + 𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))

Theoremgagrp 18413 The left argument of a group action is a group. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 30-Apr-2015.)
( ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)

Theoremgaset 18414 The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)
( ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)

Theoremgagrpid 18415 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 = (0g𝐺)       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( 0 𝐴) = 𝐴)

Theoremgaf 18416 The mapping of the group action operation. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)       ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)⟶𝑌)

Theoremgafo 18417 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)       ( ∈ (𝐺 GrpAct 𝑌) → :(𝑋 × 𝑌)–onto𝑌)

Theoremgaass 18418 An "associative" property for group actions. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)       (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑋𝐶𝑌)) → ((𝐴 + 𝐵) 𝐶) = (𝐴 (𝐵 𝐶)))

Theoremga0 18419 The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
(𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))

Theoremgaid 18420 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 ∈ Grp ∧ 𝑆𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆))

Theoremsubgga 18421* 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)    &    + = (+g𝐺)    &   𝐻 = (𝐺s 𝑌)    &   𝐹 = (𝑥𝑌, 𝑦𝑋 ↦ (𝑥 + 𝑦))       (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))

Theoremgass 18422* 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)       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑍𝑌) → (( ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) ↔ ∀𝑥𝑋𝑦𝑍 (𝑥 𝑦) ∈ 𝑍))

Theoremgasubg 18423 The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.)
𝐻 = (𝐺s 𝑆)       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌))

Theoremgaid2 18424* 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)    &    + = (+g𝐺)    &   𝐹 = (𝑥𝑋, 𝑦𝑋 ↦ (𝑥 + 𝑦))       (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpAct 𝑋))

Theoremgalcan 18425 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)       (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = (𝐴 𝐶) ↔ 𝐵 = 𝐶))

Theoremgacan 18426 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 = (invg𝐺)       (( ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴𝑋𝐵𝑌𝐶𝑌)) → ((𝐴 𝐵) = 𝐶 ↔ ((𝑁𝐴) 𝐶) = 𝐵))

Theoremgapm 18427* 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 = (𝑥𝑌 ↦ (𝐴 𝑥))       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑋) → 𝐹:𝑌1-1-onto𝑌)

Theoremgaorb 18428* 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} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (𝐴 𝐵 ↔ (𝐴𝑌𝐵𝑌 ∧ ∃𝑋 ( 𝐴) = 𝐵))

Theoremgaorber 18429* 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} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}    &   𝑋 = (Base‘𝐺)       ( ∈ (𝐺 GrpAct 𝑌) → Er 𝑌)

Theoremgastacl 18430* The stabilizer subgroup in a group action. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐻 ∈ (SubGrp‘𝐺))

Theoremgastacos 18431* Write the coset relation for the stabilizer subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}    &    = (𝐺 ~QG 𝐻)       ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ (𝐵𝑋𝐶𝑋)) → (𝐵 𝐶 ↔ (𝐵 𝐴) = (𝐶 𝐴)))

Theoremorbstafun 18432* Existence and uniqueness for the function of orbsta 18434. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
𝑋 = (Base‘𝐺)    &   𝐻 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}    &    = (𝐺 ~QG 𝐻)    &   𝐹 = ran (𝑘𝑋 ↦ ⟨[𝑘] , (𝑘 𝐴)⟩)       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → Fun 𝐹)

Theoremorbstaval 18433* 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 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}    &    = (𝐺 ~QG 𝐻)    &   𝐹 = ran (𝑘𝑋 ↦ ⟨[𝑘] , (𝑘 𝐴)⟩)       ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝐵𝑋) → (𝐹‘[𝐵] ) = (𝐵 𝐴))

Theoremorbsta 18434* 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 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}    &    = (𝐺 ~QG 𝐻)    &   𝐹 = ran (𝑘𝑋 ↦ ⟨[𝑘] , (𝑘 𝐴)⟩)    &   𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → 𝐹:(𝑋 / )–1-1-onto→[𝐴]𝑂)

Theoremorbsta2 18435* 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 = {𝑢𝑋 ∣ (𝑢 𝐴) = 𝐴}    &    = (𝐺 ~QG 𝐻)    &   𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔𝑋 (𝑔 𝑥) = 𝑦)}       ((( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝐴]𝑂) · (♯‘𝐻)))

10.2.8  Centralizers and centers

Syntaxccntz 18436 Syntax for the centralizer of a set in a monoid.
class Cntz

Syntaxccntr 18437 Syntax for the centralizer of a monoid.
class Cntr

Definitiondf-cntz 18438* 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 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦𝑠 (𝑥(+g𝑚)𝑦) = (𝑦(+g𝑚)𝑥)}))

Definitiondf-cntr 18439 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 ∈ V ↦ ((Cntz‘𝑚)‘(Base‘𝑚)))

Theoremcntrval 18440 Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑍𝐵) = (Cntr‘𝑀)

Theoremcntzfval 18441* First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑀𝑉𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥𝐵 ∣ ∀𝑦𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))

Theoremcntzval 18442* Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑆𝐵 → (𝑍𝑆) = {𝑥𝐵 ∣ ∀𝑦𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})

Theoremelcntz 18443* Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑆𝐵 → (𝐴 ∈ (𝑍𝑆) ↔ (𝐴𝐵 ∧ ∀𝑦𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴))))

Theoremcntzel 18444* Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑍 = (Cntz‘𝑀)       ((𝑆𝐵𝑋𝐵) → (𝑋 ∈ (𝑍𝑆) ↔ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))

Theoremcntzsnval 18445* Special substitution for the centralizer of a singleton. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑌𝐵 → (𝑍‘{𝑌}) = {𝑥𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})

Theoremelcntzsn 18446 Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑌𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋))))

Theoremsscntz 18447* A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑍 = (Cntz‘𝑀)       ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ ∀𝑥𝑆𝑦𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))

Theoremcntzrcl 18448 Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑋 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆𝐵))

Theoremcntzssv 18449 The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑍𝑆) ⊆ 𝐵

Theoremcntzi 18450 Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
+ = (+g𝑀)    &   𝑍 = (Cntz‘𝑀)       ((𝑋 ∈ (𝑍𝑆) ∧ 𝑌𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremcntrss 18451 The center is a subset of the base field. (Contributed by Thierry Arnoux, 21-Aug-2023.)
𝐵 = (Base‘𝑀)       (Cntr‘𝑀) ⊆ 𝐵

Theoremcntri 18452 Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   𝑍 = (Cntr‘𝑀)       ((𝑋𝑍𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theoremresscntz 18453 Centralizer in a substructure. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐻 = (𝐺s 𝐴)    &   𝑍 = (Cntz‘𝐺)    &   𝑌 = (Cntz‘𝐻)       ((𝐴𝑉𝑆𝐴) → (𝑌𝑆) = ((𝑍𝑆) ∩ 𝐴))

Theoremcntz2ss 18454 Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       ((𝑆𝐵𝑇𝑆) → (𝑍𝑆) ⊆ (𝑍𝑇))

Theoremcntzrec 18455 Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       ((𝑆𝐵𝑇𝐵) → (𝑆 ⊆ (𝑍𝑇) ↔ 𝑇 ⊆ (𝑍𝑆)))

Theoremcntziinsn 18456* Express any centralizer as an intersection of singleton centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       (𝑆𝐵 → (𝑍𝑆) = (𝐵 𝑥𝑆 (𝑍‘{𝑥})))

Theoremcntzsubm 18457 Centralizers in a monoid are submonoids. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       ((𝑀 ∈ Mnd ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubMnd‘𝑀))

Theoremcntzsubg 18458 Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝐵 = (Base‘𝑀)    &   𝑍 = (Cntz‘𝑀)       ((𝑀 ∈ Grp ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubGrp‘𝑀))

Theoremcntzidss 18459 If the elements of 𝑆 commute, the elements of a subset 𝑇 also commute. (Contributed by Mario Carneiro, 25-Apr-2016.)
𝑍 = (Cntz‘𝐺)       ((𝑆 ⊆ (𝑍𝑆) ∧ 𝑇𝑆) → 𝑇 ⊆ (𝑍𝑇))

Theoremcntzmhm 18460 Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &   𝑌 = (Cntz‘𝐻)       ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝐴 ∈ (𝑍𝑆)) → (𝐹𝐴) ∈ (𝑌‘(𝐹𝑆)))

Theoremcntzmhm2 18461 Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)
𝑍 = (Cntz‘𝐺)    &   𝑌 = (Cntz‘𝐻)       ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → (𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)))

Theoremcntrsubgnsg 18462 A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑍 = (Cntr‘𝑀)       ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))

Theoremcntrnsg 18463 The center of a group is a normal subgroup. (Contributed by Stefan O'Rear, 6-Sep-2015.)
𝑍 = (Cntr‘𝑀)       (𝑀 ∈ Grp → 𝑍 ∈ (NrmSGrp‘𝑀))

10.2.9  The opposite group

Syntaxcoppg 18464 The opposite group operation.
class oppg

Definitiondf-oppg 18465 Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 19367 does the same thing for multiplication. (Contributed by Stefan O'Rear, 25-Aug-2015.)
oppg = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), tpos (+g𝑤)⟩))

Theoremoppgval 18466 Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
+ = (+g𝑅)    &   𝑂 = (oppg𝑅)       𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)

Theoremoppgplusfval 18467 Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
+ = (+g𝑅)    &   𝑂 = (oppg𝑅)    &    = (+g𝑂)        = tpos +

Theoremoppgplus 18468 Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
+ = (+g𝑅)    &   𝑂 = (oppg𝑅)    &    = (+g𝑂)       (𝑋 𝑌) = (𝑌 + 𝑋)

Theoremoppglem 18469 Lemma for oppgbas 18470. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 2       (𝐸𝑅) = (𝐸𝑂)

Theoremoppgbas 18470 Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)    &   𝐵 = (Base‘𝑅)       𝐵 = (Base‘𝑂)

Theoremoppgtset 18471 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝑅)    &   𝐽 = (TopSet‘𝑅)       𝐽 = (TopSet‘𝑂)

Theoremoppgtopn 18472 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝑅)    &   𝐽 = (TopOpen‘𝑅)       𝐽 = (TopOpen‘𝑂)

Theoremoppgmnd 18473 The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Mnd → 𝑂 ∈ Mnd)

Theoremoppgmndb 18474 Bidirectional form of oppgmnd 18473. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)

Theoremoppgid 18475 Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
𝑂 = (oppg𝑅)    &    0 = (0g𝑅)        0 = (0g𝑂)

Theoremoppggrp 18476 The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Grp → 𝑂 ∈ Grp)

Theoremoppggrpb 18477 Bidirectional form of oppggrp 18476. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp)

Theoremoppginv 18478 Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)    &   𝐼 = (invg𝑅)       (𝑅 ∈ Grp → 𝐼 = (invg𝑂))

Theoreminvoppggim 18479 The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso 𝑂))

Theoremoppggic 18480 Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝐺)       (𝐺 ∈ Grp → 𝐺𝑔 𝑂)

Theoremoppgsubm 18481 Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (SubMnd‘𝐺) = (SubMnd‘𝑂)

Theoremoppgsubg 18482 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (SubGrp‘𝐺) = (SubGrp‘𝑂)

Theoremoppgcntz 18483 A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑂 = (oppg𝐺)    &   𝑍 = (Cntz‘𝐺)       (𝑍𝐴) = ((Cntz‘𝑂)‘𝐴)

Theoremoppgcntr 18484 The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑂 = (oppg𝐺)    &   𝑍 = (Cntr‘𝐺)       𝑍 = (Cntr‘𝑂)

Theoremgsumwrev 18485 A sum in an opposite monoid is the regular sum of a reversed word. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Proof shortened by Mario Carneiro, 28-Feb-2016.)
𝐵 = (Base‘𝑀)    &   𝑂 = (oppg𝑀)       ((𝑀 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵) → (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊)))

10.2.10  Symmetric groups

10.2.10.1  Definition and basic properties

According to Wikipedia ("Symmetric group", 09-Mar-2019, https://en.wikipedia.org/wiki/symmetric_group) "In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions." and according to Encyclopedia of Mathematics ("Symmetric group", 09-Mar-2019, https://www.encyclopediaofmath.org/index.php/Symmetric_group) "The group of all permutations (self-bijections) of a set with the operation of composition (see Permutation group).". In [Rotman] p. 27 "If X is a nonempty set, a permutation of X is a function a : X -> X that is a one-to-one correspondence." and "If X is a nonempty set, the symmetric group on X, denoted SX, is the group whose elements are the permutations of X and whose binary operation is composition of functions.". Therefore, we define the symmetric group on a set 𝐴 as the set of one-to-one onto functions from 𝐴 to itself under function composition, see df-symg 18487. However, the set is allowed to be empty, see symgbas0 18508. Hint: The symmetric groups should not be confused with "symmetry groups" which is a different topic in group theory.

In this context, the one-to-one onto functions are called permutations for short. Since the base set of symmetric groups on a set 𝐴 is the set of all permutations of 𝐴 (see symgbas 18490), we can formally say 𝑃 ∈ (SymGrp‘𝐴) expressing "𝑃 is a permutation of 𝐴" if we are not interested in the group (or topology) structure.

In general, a permutation group "... is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself)." (see Wikipedia "Permutation group", 17-Mar-2019, https://en.wikipedia.org/wiki/Permutation_group 18490). This means that a symmetric group is a permutation group, and each permutation group is a subgroup of a symmetric group (see pgrpsubgsymgbi 18527 and pgrpsubgsymg 18528). For example, the structure with the singleton containing only the identity function restricted to a set as base set and the function composition as group operation is a permutation group (group consisting of permutations), see idrespermg 18530, which is a (proper) subgroup of a symmetric group, see idressubgsymg 18529.

As in [Rotman] p. 28 "Let 𝑥𝑋 and 𝑝 ∈ SymGrp(𝑋); we say 𝑝 fixes 𝑥 if (𝑝𝑥) = 𝑥; otherwise 𝑝 moves 𝑥.". The theorems starting with symgfix2 18535 are about fixed/moved elements.

Syntaxcsymg 18486 Extend class notation to include the class of symmetric groups.
class SymGrp

Definitiondf-symg 18487* Define the symmetric group on set 𝑥. We represent the group as the set of one-to-one onto functions from 𝑥 to itself under function composition, and topologize it as a function space assuming the set is discrete. This definition is based on the fact that a symmetric group is a restriction of the monoid of endofunctions. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 28-Mar-2024.)
SymGrp = (𝑥 ∈ V ↦ ((EndoFMnd‘𝑥) ↾s {:𝑥1-1-onto𝑥}))

Theoremsymgval 18488* The value of the symmetric group function at 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 28-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}       𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)

Theorempermsetex 18489* The set of permutations of a set 𝐴 exists. (Contributed by AV, 30-Mar-2024.)
(𝐴𝑉 → {𝑓𝑓:𝐴1-1-onto𝐴} ∈ V)

Theoremsymgbas 18490* The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.) (Proof shortened by AV, 29-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}

Theoremsymgbasex 18491 The base set of the symmetric group over a set 𝐴 exists. (Contributed by AV, 30-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉𝐵 ∈ V)

Theoremelsymgbas2 18492 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)       (𝐹𝑉 → (𝐹𝐵𝐹:𝐴1-1-onto𝐴))

Theoremelsymgbas 18493 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)       (𝐴𝑉 → (𝐹𝐵𝐹:𝐴1-1-onto𝐴))

Theoremsymgbasf1o 18494 Elements in the symmetric group are 1-1 onto functions. (Contributed by SO, 9-Jul-2018.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐹𝐵𝐹:𝐴1-1-onto𝐴)

Theoremsymgbasf 18495 A permutation (element of the symmetric group) is a function from a set into itself. (Contributed by AV, 1-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐹𝐵𝐹:𝐴𝐴)

Theoremsymgbasmap 18496 A permutation (element of the symmetric group) is a mapping (or set exponentiation) from a set into itself. (Contributed by AV, 30-Mar-2024.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐹𝐵𝐹 ∈ (𝐴m 𝐴))

Theoremsymghash 18497 The symmetric group on 𝑛 objects has cardinality 𝑛!. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴 ∈ Fin → (♯‘𝐵) = (!‘(♯‘𝐴)))

Theoremsymgbasfi 18498 The symmetric group on a finite index set is finite. (Contributed by SO, 9-Jul-2018.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴 ∈ Fin → 𝐵 ∈ Fin)

Theoremsymgfv 18499 The function value of a permutation. (Contributed by AV, 1-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐹𝐵𝑋𝐴) → (𝐹𝑋) ∈ 𝐴)

Theoremsymgfvne 18500 The function values of a permutation for different arguments are different. (Contributed by AV, 8-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐹𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = 𝑍 → (𝑌𝑋 → (𝐹𝑌) ≠ 𝑍)))

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