P. 195 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	brgic 19401	The relation "is isomorphic to" for groups. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑔 𝑆 ↔ (𝑅 GrpIso 𝑆) ≠ ∅)
Theorem	brgici 19402	Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝐹 ∈ (𝑅 GrpIso 𝑆) → 𝑅 ≃𝑔 𝑆)
Theorem	gicref 19403	Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ (𝑅 ∈ Grp → 𝑅 ≃𝑔 𝑅)
Theorem	giclcl 19404	Isomorphism implies the left side is a group. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑔 𝑆 → 𝑅 ∈ Grp)
Theorem	gicrcl 19405	Isomorphism implies the right side is a group. (Contributed by Mario Carneiro, 6-May-2015.)
⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ∈ Grp)
Theorem	gicsym 19406	Isomorphism is symmetric. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ (𝑅 ≃𝑔 𝑆 → 𝑆 ≃𝑔 𝑅)
Theorem	gictr 19407	Isomorphism is transitive. (Contributed by Mario Carneiro, 21-Apr-2016.)
⊢ ((𝑅 ≃𝑔 𝑆 ∧ 𝑆 ≃𝑔 𝑇) → 𝑅 ≃𝑔 𝑇)
Theorem	gicer 19408	Isomorphism is an equivalence relation on groups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 1-May-2021.)
⊢ ≃𝑔 Er Grp
Theorem	gicen 19409	Isomorphic groups have equinumerous base sets. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝑅 ≃𝑔 𝑆 → 𝐵 ≈ 𝐶)
Theorem	gicsubgen 19410	A less trivial example of a group invariant: cardinality of the subgroup lattice. (Contributed by Stefan O'Rear, 25-Jan-2015.)
⊢ (𝑅 ≃𝑔 𝑆 → (SubGrp‘𝑅) ≈ (SubGrp‘𝑆))
10.2.6.1  The first isomorphism theorem of groups
Theorem	ghmqusnsglem1 19411*	Lemma for ghmqusnsg 19413. (Contributed by Thierry Arnoux, 13-May-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝑁 ⊆ 𝐾)    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))    ⇒   ⊢ (𝜑 → (𝐽‘[𝑋](𝐺 ~QG 𝑁)) = (𝐹‘𝑋))
Theorem	ghmqusnsglem2 19412*	Lemma for ghmqusnsg 19413. (Contributed by Thierry Arnoux, 13-May-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝑁 ⊆ 𝐾)    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑄))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥))
Theorem	ghmqusnsg 19413*	The mapping 𝐻 induced by a surjective group homomorphism 𝐹 from the quotient group 𝑄 over a normal subgroup 𝑁 of 𝐹's kernel 𝐾 is a group isomorphism. In this case, one says that 𝐹 factors through 𝑄, which is also called the factor group. (Contributed by Thierry Arnoux, 13-May-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝑁 ⊆ 𝐾)    &   ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺))    ⇒   ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
Theorem	ghmquskerlem1 19414*	Lemma for ghmqusker 19418. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺))    ⇒   ⊢ (𝜑 → (𝐽‘[𝑋](𝐺 ~QG 𝐾)) = (𝐹‘𝑋))
Theorem	ghmquskerco 19415*	In the case of theorem ghmqusker 19418, the composition of the natural homomorphism 𝐿 with the constructed homomorphism 𝐽 equals the original homomorphism 𝐹. One says that 𝐹 factors through 𝑄. (Proposed by Saveliy Skresanov, 15-Feb-2025.) (Contributed by Thierry Arnoux, 15-Feb-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝐿 = (𝑥 ∈ 𝐵 ↦ [𝑥](𝐺 ~QG 𝐾))    ⇒   ⊢ (𝜑 → 𝐹 = (𝐽 ∘ 𝐿))
Theorem	ghmquskerlem2 19416*	Lemma for ghmqusker 19418. (Contributed by Thierry Arnoux, 14-Feb-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → 𝑌 ∈ (Base‘𝑄))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 (𝐽‘𝑌) = (𝐹‘𝑥))
Theorem	ghmquskerlem3 19417*	The mapping 𝐻 induced by a surjective group homomorphism 𝐹 from the quotient group 𝑄 over 𝐹's kernel 𝐾 is a group isomorphism. In this case, one says that 𝐹 factors through 𝑄, which is also called the factor group. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    ⇒   ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpHom 𝐻))
Theorem	ghmqusker 19418*	A surjective group homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 15-Feb-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ 𝐽 = (𝑞 ∈ (Base‘𝑄) ↦ ∪ (𝐹 “ 𝑞))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))    ⇒   ⊢ (𝜑 → 𝐽 ∈ (𝑄 GrpIso 𝐻))
Theorem	gicqusker 19419	The image 𝐻 of a group homomorphism 𝐹 is isomorphic with the quotient group 𝑄 over 𝐹's kernel 𝐾. Together with ghmker 19373 and ghmima 19368, this is sometimes called the first isomorphism theorem for groups. (Contributed by Thierry Arnoux, 10-Mar-2025.)
⊢ 0 = (0g‘𝐻)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐺 GrpHom 𝐻))    &   ⊢ 𝐾 = (◡𝐹 “ { 0 })    &   ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))    &   ⊢ (𝜑 → ran 𝐹 = (Base‘𝐻))    ⇒   ⊢ (𝜑 → 𝑄 ≃𝑔 𝐻)
10.2.7  Group actions
Syntax	cga 19420	Extend class definition to include the class of group actions.
class GrpAct
Definition	df-ga 19421*	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 ∈ (𝑠 ↑m (𝑏 × 𝑠)) ∣ ∀𝑥 ∈ 𝑠 (((0g‘𝑔)𝑚𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 ((𝑦(+g‘𝑔)𝑧)𝑚𝑥) = (𝑦𝑚(𝑧𝑚𝑥)))})
Theorem	isga 19422*	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 19437). 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)    ⇒   ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :(𝑋 × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
Theorem	gagrp 19423	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)
Theorem	gaset 19424	The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V)
Theorem	gagrpid 19425	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)    ⇒   ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴)
Theorem	gaf 19426	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)
Theorem	gafo 19427	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 19428	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 19429	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 19430	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 19431*	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 19432*	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 19433	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 19434*	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 19435	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 19436	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 19437*	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 19438*	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 19439*	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 19440*	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 19441*	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 19442*	Existence and uniqueness for the function of orbsta 19444. (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 19443*	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 19444*	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 19445*	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 19446	Syntax for the centralizer of a set in a monoid.
class Cntz
Syntax	ccntr 19447	Syntax for the centralizer of a monoid.
class Cntr
Definition	df-cntz 19448*	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 19449	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 19450	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 19451*	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 19452*	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 19453*	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 19454*	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 19455*	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 19456	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 19457*	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 19458	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 19459	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 19460	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 19461*	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 19462	The center is a subset of the base field. (Contributed by Thierry Arnoux, 21-Aug-2023.)
⊢ 𝐵 = (Base‘𝑀)    ⇒   ⊢ (Cntr‘𝑀) ⊆ 𝐵
Theorem	cntri 19463	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 19464	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 19465*	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 19466	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 19467	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 19468*	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 19469	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 19470	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 19471	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 19472	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 19473	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 19474	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 19475	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 19476	The opposite group operation.
class oppg
Definition	df-oppg 19477	Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 20482 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 19478	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 19479	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 19480	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 19481	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 19482	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 19483	Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐽 = (TopSet‘𝑅)    ⇒   ⊢ 𝐽 = (TopSet‘𝑂)
Theorem	oppgtopn 19484	Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    ⇒   ⊢ 𝐽 = (TopOpen‘𝑂)
Theorem	oppgmnd 19485	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 19486	Bidirectional form of oppgmnd 19485. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)
Theorem	oppgid 19487	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 19488	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 19489	Bidirectional form of oppggrp 19488. (Contributed by Stefan O'Rear, 26-Aug-2015.)
⊢ 𝑂 = (oppg‘𝑅)    ⇒   ⊢ (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp)
Theorem	oppginv 19490	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 19491	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 19492	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 19493	Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂)
Theorem	oppgsubg 19494	Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
⊢ 𝑂 = (oppg‘𝐺)    ⇒   ⊢ (SubGrp‘𝐺) = (SubGrp‘𝑂)
Theorem	oppgcntz 19495	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 19496	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 19497	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 19498	less-than relation of an opposite group. (Contributed by Thierry Arnoux, 13-Apr-2018.)
⊢ 𝑂 = (oppg‘𝑅)    &   ⊢ ≤ = (le‘𝑅)    ⇒   ⊢ ≤ = (le‘𝑂)
Theorem	oppglt 19499	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 19501. However, the set is allowed to be empty, see symgbas0 19520. 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 19503), 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 19503). This means that a symmetric group is a permutation group, and each permutation group is a subgroup of a symmetric group (see pgrpsubgsymgbi 19539 and pgrpsubgsymg 19540). 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 19542, which is a (proper) subgroup of a symmetric group, see idressubgsymg 19541. 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 19547 are about fixed/moved elements.
Syntax	csymg 19500	Extend class notation to include the class of symmetric groups.
class SymGrp