P. 193 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19201-19300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	gafo 19201	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)–onto→𝑌)
Theorem	gaass 19202	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)    ⇒   ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 + 𝐵) ⊕ 𝐶) = (𝐴 ⊕ (𝐵 ⊕ 𝐶)))
Theorem	ga0 19203	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 ∅))
Theorem	gaid 19204	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 ∧ 𝑆 ∈ 𝑉) → (2nd ↾ (𝑋 × 𝑆)) ∈ (𝐺 GrpAct 𝑆))
Theorem	subgga 19205*	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 = (𝐺 ↾s 𝑌)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))    ⇒   ⊢ (𝑌 ∈ (SubGrp‘𝐺) → 𝐹 ∈ (𝐻 GrpAct 𝑋))
Theorem	gass 19206*	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) ∧ 𝑍 ⊆ 𝑌) → (( ⊕ ↾ (𝑋 × 𝑍)) ∈ (𝐺 GrpAct 𝑍) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑍 (𝑥 ⊕ 𝑦) ∈ 𝑍))
Theorem	gasubg 19207	The restriction of a group action to a subgroup is a group action. (Contributed by Mario Carneiro, 17-Jan-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ( ⊕ ↾ (𝑆 × 𝑌)) ∈ (𝐻 GrpAct 𝑌))
Theorem	gaid2 19208*	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 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (𝑥 + 𝑦))    ⇒   ⊢ (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpAct 𝑋))
Theorem	galcan 19209	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) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) ↔ 𝐵 = 𝐶))
Theorem	gacan 19210	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 = (invg‘𝐺)    ⇒   ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = 𝐶 ↔ ((𝑁‘𝐴) ⊕ 𝐶) = 𝐵))
Theorem	gapm 19211*	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 ∈ 𝑌 ↦ (𝐴 ⊕ 𝑥))    ⇒   ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑌–1-1-onto→𝑌)
Theorem	gaorb 19212*	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 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    ⇒   ⊢ (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵))
Theorem	gaorber 19213*	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 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    &   ⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∼ Er 𝑌)
Theorem	gastacl 19214*	The stabilizer subgroup in a group action. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴}    ⇒   ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐻 ∈ (SubGrp‘𝐺))
Theorem	gastacos 19215*	Write the coset relation for the stabilizer subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴}    &   ⊢ ∼ = (𝐺 ~QG 𝐻)    ⇒   ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵 ∼ 𝐶 ↔ (𝐵 ⊕ 𝐴) = (𝐶 ⊕ 𝐴)))
Theorem	orbstafun 19216*	Existence and uniqueness for the function of orbsta 19218. (Contributed by Mario Carneiro, 15-Jan-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑋 = (Base‘𝐺)    &   ⊢ 𝐻 = {𝑢 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴}    &   ⊢ ∼ = (𝐺 ~QG 𝐻)    &   ⊢ 𝐹 = ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩)    ⇒   ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → Fun 𝐹)
Theorem	orbstaval 19217*	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 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴}    &   ⊢ ∼ = (𝐺 ~QG 𝐻)    &   ⊢ 𝐹 = ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩)    ⇒   ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝐵 ∈ 𝑋) → (𝐹‘[𝐵] ∼ ) = (𝐵 ⊕ 𝐴))
Theorem	orbsta 19218*	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 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴}    &   ⊢ ∼ = (𝐺 ~QG 𝐻)    &   ⊢ 𝐹 = ran (𝑘 ∈ 𝑋 ↦ ⟨[𝑘] ∼ , (𝑘 ⊕ 𝐴)⟩)    &   ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    ⇒   ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → 𝐹:(𝑋 / ∼ )–1-1-onto→[𝐴]𝑂)
Theorem	orbsta2 19219*	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 ∈ 𝑋 ∣ (𝑢 ⊕ 𝐴) = 𝐴}    &   ⊢ ∼ = (𝐺 ~QG 𝐻)    &   ⊢ 𝑂 = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)}    ⇒   ⊢ ((( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) ∧ 𝑋 ∈ Fin) → (♯‘𝑋) = ((♯‘[𝐴]𝑂) · (♯‘𝐻)))
10.2.8  Centralizers and centers
Syntax	ccntz 19220	Syntax for the centralizer of a set in a monoid.
class Cntz
Syntax	ccntr 19221	Syntax for the centralizer of a monoid.
class Cntr
Definition	df-cntz 19222*	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) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
Definition	df-cntr 19223	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)))
Theorem	cntrval 19224	Substitute definition of the center. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑍‘𝐵) = (Cntr‘𝑀)
Theorem	cntzfval 19225*	First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑀 ∈ 𝑉 → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
Theorem	cntzval 19226*	Definition substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
Theorem	elcntz 19227*	Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑆 ⊆ 𝐵 → (𝐴 ∈ (𝑍‘𝑆) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴))))
Theorem	cntzel 19228*	Membership in a centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ (𝑍‘𝑆) ↔ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋)))
Theorem	cntzsnval 19229*	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 ∈ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
Theorem	elcntzsn 19230	Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋))))
Theorem	sscntz 19231*	A centralizer expression for two sets elementwise commuting. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Theorem	cntzrcl 19232	Reverse closure for elements of the centralizer. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ 𝐵))
Theorem	cntzssv 19233	The centralizer is unconditionally a subset. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑍‘𝑆) ⊆ 𝐵
Theorem	cntzi 19234	Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	elcntr 19235*	Elementhood in the center of a magma. (Contributed by SN, 21-Mar-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ (𝐴 ∈ 𝑍 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
Theorem	cntrss 19236	The center is a subset of the base field. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (Cntr‘𝑀) ⊆ 𝐵
Theorem	cntri 19237	Defining property of the center of a group. (Contributed by Mario Carneiro, 22-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    &   ⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ ((𝑋 ∈ 𝑍 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Theorem	resscntz 19238	Centralizer in a substructure. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    &   ⊢ 𝑍 = (Cntz‘𝐺)    &   ⊢ 𝑌 = (Cntz‘𝐻)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑆 ⊆ 𝐴) → (𝑌‘𝑆) = ((𝑍‘𝑆) ∩ 𝐴))
Theorem	cntzsgrpcl 19239*	Centralizers are closed under the semigroup operation. (Contributed by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    &   ⊢ 𝐶 = (𝑍‘𝑆)    ⇒   ⊢ ((𝑀 ∈ Smgrp ∧ 𝑆 ⊆ 𝐵) → ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐶 (𝑦(+g‘𝑀)𝑧) ∈ 𝐶)
Theorem	cntz2ss 19240	Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝑆) → (𝑍‘𝑆) ⊆ (𝑍‘𝑇))
Theorem	cntzrec 19241	Reciprocity relationship for centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑇 ⊆ 𝐵) → (𝑆 ⊆ (𝑍‘𝑇) ↔ 𝑇 ⊆ (𝑍‘𝑆)))
Theorem	cntziinsn 19242*	Express any centralizer as an intersection of singleton centralizers. (Contributed by Stefan O'Rear, 5-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = (𝐵 ∩ ∩ 𝑥 ∈ 𝑆 (𝑍‘{𝑥})))
Theorem	cntzsubm 19243	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 ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubMnd‘𝑀))
Theorem	cntzsubg 19244	Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Grp ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubGrp‘𝑀))
Theorem	cntzidss 19245	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 ⊆ (𝑍‘𝑆) ∧ 𝑇 ⊆ 𝑆) → 𝑇 ⊆ (𝑍‘𝑇))
Theorem	cntzmhm 19246	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 ∈ (𝑍‘𝑆)) → (𝐹‘𝐴) ∈ (𝑌‘(𝐹 “ 𝑆)))
Theorem	cntzmhm2 19247	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 ⊆ (𝑍‘𝑇)) → (𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)))
Theorem	cntrsubgnsg 19248	A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.)
⊢ 𝑍 = (Cntr‘𝑀)    ⇒   ⊢ ((𝑋 ∈ (SubGrp‘𝑀) ∧ 𝑋 ⊆ 𝑍) → 𝑋 ∈ (NrmSGrp‘𝑀))
Theorem	cntrnsg 19249	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
Syntax	coppg 19250	The opposite group operation.
class oppg
Definition	df-oppg 19251	Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 20248 does the same thing for multiplication. (Contributed by Stefan O'Rear, 25-Aug-2015.)
⊢ oppg = (𝑤 ∈ V ↦ (𝑤 sSet ⟨(+g‘ndx), tpos (+g‘𝑤)⟩))
Theorem	oppgval 19252	Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
⊢ + = (+g‘𝑅)    &   ⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), tpos + ⟩)
Theorem	oppgplusfval 19253	Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
⊢ + = (+g‘𝑅)    &   ⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ✚ = (+g‘𝑂)    ⇒   ⊢ ✚ = tpos +
Theorem	oppgplus 19254	Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
⊢ + = (+g‘𝑅)    &   ⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ✚ = (+g‘𝑂)    ⇒   ⊢ (𝑋 ✚ 𝑌) = (𝑌 + 𝑋)
Theorem	setsplusg 19255	The other components of an extensible structure remain unchanged if the +g component is set/substituted. (Contributed by Stefan O'Rear, 26-Aug-2015.) Generalisation of the former oppglem and mgplem. (Revised by AV, 18-Oct-2024.)
⊢ 𝑂 = (𝑅 sSet ⟨(+g‘ndx), 𝑆⟩)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (+g‘ndx)    ⇒   ⊢ (𝐸‘𝑅) = (𝐸‘𝑂)
Theorem	oppgbas 19256	Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ 𝐵 = (Base‘𝑂)
Theorem	oppgtset 19257	Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐽 = (TopSet‘𝑅)    ⇒   ⊢ 𝐽 = (TopSet‘𝑂)
Theorem	oppgtopn 19258	Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    ⇒   ⊢ 𝐽 = (TopOpen‘𝑂)
Theorem	oppgmnd 19259	The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Mnd → 𝑂 ∈ Mnd)
Theorem	oppgmndb 19260	Bidirectional form of oppgmnd 19259. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)
Theorem	oppgid 19261	Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ 0 = (0g‘𝑂)
Theorem	oppggrp 19262	The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Grp → 𝑂 ∈ Grp)
Theorem	oppggrpb 19263	Bidirectional form of oppggrp 19262. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp)
Theorem	oppginv 19264	Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐼 = (invg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Grp → 𝐼 = (invg‘𝑂))
Theorem	invoppggim 19265	The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝐺)    &   ⊢ 𝐼 = (invg‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso 𝑂))
Theorem	oppggic 19266	Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (𝐺 ∈ Grp → 𝐺 ≃𝑔 𝑂)
Theorem	oppgsubm 19267	Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂)
Theorem	oppgsubg 19268	Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (SubGrp‘𝐺) = (SubGrp‘𝑂)
Theorem	oppgcntz 19269	A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝑂 = (oppg‘𝐺)    &   ⊢ 𝑍 = (Cntz‘𝐺)    ⇒   ⊢ (𝑍‘𝐴) = ((Cntz‘𝑂)‘𝐴)
Theorem	oppgcntr 19270	The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ 𝑂 = (oppg‘𝐺)    &   ⊢ 𝑍 = (Cntr‘𝐺)    ⇒   ⊢ 𝑍 = (Cntr‘𝑂)
Theorem	gsumwrev 19271	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 = (oppg‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵) → (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊)))
Theorem	oppgle 19272	less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ≤ = (le‘𝑅)    ⇒   ⊢ ≤ = (le‘𝑂)
Theorem	oppglt 19273	less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ < = (lt‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → < = (lt‘𝑂))
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 19275. However, the set is allowed to be empty, see symgbas0 19294. 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 19277), 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 19277). This means that a symmetric group is a permutation group, and each permutation group is a subgroup of a symmetric group (see pgrpsubgsymgbi 19313 and pgrpsubgsymg 19314). 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 19316, which is a (proper) subgroup of a symmetric group, see idressubgsymg 19315. As in [Rotman] p. 28 "Let 𝑥 ∈ 𝑋 and 𝑝 ∈ SymGrp(𝑋); we say 𝑝 fixes 𝑥 if (𝑝‘𝑥) = 𝑥; otherwise 𝑝 moves 𝑥.". The theorems starting with symgfix2 19321 are about fixed/moved elements.
Syntax	csymg 19274	Extend class notation to include the class of symmetric groups.
class SymGrp
Definition	df-symg 19275*	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 {ℎ ∣ ℎ:𝑥–1-1-onto→𝑥}))
Theorem	symgval 19276*	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 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}    ⇒   ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
Theorem	symgbas 19277*	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 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
Theorem	elsymgbas2 19278	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 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴))
Theorem	elsymgbas 19279	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 ∈ 𝑉 → (𝐹 ∈ 𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐴))
Theorem	symgbasf1o 19280	Elements in the symmetric group are 1-1 onto functions. (Contributed by SO, 9-Jul-2018.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴–1-1-onto→𝐴)
Theorem	symgbasf 19281	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)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐹:𝐴⟶𝐴)
Theorem	symgbasmap 19282	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)    ⇒   ⊢ (𝐹 ∈ 𝐵 → 𝐹 ∈ (𝐴 ↑m 𝐴))
Theorem	symghash 19283	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)))
Theorem	symgbasfi 19284	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)
Theorem	symgfv 19285	The function value of a permutation. (Contributed by AV, 1-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ 𝐴)
Theorem	symgfvne 19286	The function values of a permutation for different arguments are different. (Contributed by AV, 8-Jan-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹‘𝑋) = 𝑍 → (𝑌 ≠ 𝑋 → (𝐹‘𝑌) ≠ 𝑍)))
Theorem	symgressbas 19287	The symmetric group on 𝐴 characterized as structure restriction of the monoid of endofunctions on 𝐴 to its base set. (Contributed by AV, 30-Mar-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑀 = (EndoFMnd‘𝐴)    ⇒   ⊢ 𝐺 = (𝑀 ↾s 𝐵)
Theorem	symgplusg 19288*	The group operation of a symmetric group is the function composition. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) (Proof shortened by AV, 19-Feb-2024.) (Revised by AV, 29-Mar-2024.) (Proof shortened by AV, 14-Aug-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (𝐴 ↑m 𝐴)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ + = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘ 𝑔))
Theorem	symgov 19289	The value of the group operation of the symmetric group on 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) (Revised by AV, 30-Mar-2024.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌))
Theorem	symgcl 19290	The group operation of the symmetric 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 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Theorem	idresperm 19291	The identity function restricted to a set is a permutation of this set. (Contributed by AV, 17-Mar-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺))
Theorem	symgmov1 19292*	For a permutation of a set, each element of the set replaces an(other) element of the set. (Contributed by AV, 2-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    ⇒   ⊢ (𝑄 ∈ 𝑃 → ∀𝑛 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑛) = 𝑘)
Theorem	symgmov2 19293*	For a permutation of a set, each element of the set is replaced by an(other) element of the set. (Contributed by AV, 2-Jan-2019.)
⊢ 𝑃 = (Base‘(SymGrp‘𝑁))    ⇒   ⊢ (𝑄 ∈ 𝑃 → ∀𝑛 ∈ 𝑁 ∃𝑘 ∈ 𝑁 (𝑄‘𝑘) = 𝑛)
Theorem	symgbas0 19294	The base set of the symmetric group on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Feb-2019.)
⊢ (Base‘(SymGrp‘∅)) = {∅}
Theorem	symg1hash 19295	The symmetric group on a singleton has cardinality 1. (Contributed by AV, 9-Dec-2018.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼}    ⇒   ⊢ (𝐼 ∈ 𝑉 → (♯‘𝐵) = 1)
Theorem	symg1bas 19296	The symmetric group on a singleton is the symmetric group S1 consisting of the identity only. (Contributed by AV, 9-Dec-2018.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝐵 = {{⟨𝐼, 𝐼⟩}})
Theorem	symg2hash 19297	The symmetric group on a (proper) pair has cardinality 2. (Contributed by AV, 9-Dec-2018.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼, 𝐽}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊 ∧ 𝐼 ≠ 𝐽) → (♯‘𝐵) = 2)
Theorem	symg2bas 19298	The symmetric group on a pair is the symmetric group S2 consisting of the identity and the transposition. Notice that this statement is valid for proper pairs only. In the case that both elements are identical, i.e., the pairs are actually singletons, this theorem would be about S1, see Theorem symg1bas 19296. (Contributed by AV, 9-Dec-2018.) (Proof shortened by AV, 16-Jun-2022.)
⊢ 𝐺 = (SymGrp‘𝐴)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐴 = {𝐼, 𝐽}    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐽 ∈ 𝑊) → 𝐵 = {{⟨𝐼, 𝐼⟩, ⟨𝐽, 𝐽⟩}, {⟨𝐼, 𝐽⟩, ⟨𝐽, 𝐼⟩}})
Theorem	0symgefmndeq 19299	The symmetric group on the empty set is identical with the monoid of endofunctions on the empty set. (Contributed by AV, 30-Mar-2024.)
⊢ (EndoFMnd‘∅) = (SymGrp‘∅)
Theorem	snsymgefmndeq 19300	The symmetric group on a singleton 𝐴 is identical with the monoid of endofunctions on 𝐴. (Contributed by AV, 31-Mar-2024.)
⊢ (𝐴 = {𝑋} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))