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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | ghmquskerlem1 19301* | Lemma for ghmqusker 19305. (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 19302* | In the case of theorem ghmqusker 19305, 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 19303* | Lemma for ghmqusker 19305. (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 19304* | 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 19305* | 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 19306 | The image 𝐻 of a group homomorphism 𝐹 is isomorphic with the quotient group 𝑄 over 𝐹's kernel 𝐾. Together with ghmker 19260 and ghmima 19255, 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) | ||
| Syntax | cga 19307 | Extend class definition to include the class of group actions. |
| class GrpAct | ||
| Definition | df-ga 19308* | 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 19309* | 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 19324). 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 19310 | 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 19311 | The right argument of a group action is a set. (Contributed by Mario Carneiro, 30-Apr-2015.) |
| ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝑌 ∈ V) | ||
| Theorem | gagrpid 19312 | 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 19313 | 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 19314 | 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 19315 | 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 19316 | 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 19317 | 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 19318* | 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 19319* | 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 19320 | 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 19321* | 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 19322 | 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 19323 | 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 19324* | 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 19325* | 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 19326* | 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 19327* | 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 19328* | 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 19329* | Existence and uniqueness for the function of orbsta 19331. (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 19330* | 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 19331* | 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 19332* | 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))) | ||
| Syntax | ccntz 19333 | Syntax for the centralizer of a set in a monoid. |
| class Cntz | ||
| Syntax | ccntr 19334 | Syntax for the centralizer of a monoid. |
| class Cntr | ||
| Definition | df-cntz 19335* | 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 19336 | 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 19337 | 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 19338* | 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 19339* | 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 19340* | 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 19341* | 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 19342* | 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 19343 | 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 19344* | 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 19345 | 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 19346 | 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 19347 | 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 19348* | 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 19349 | The center is a subset of the base field. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
| ⊢ 𝐵 = (Base‘𝑀) ⇒ ⊢ (Cntr‘𝑀) ⊆ 𝐵 | ||
| Theorem | cntri 19350 | 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 19351 | 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 19352* | 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 19353 | 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 19354 | 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 19355* | 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 19356 | 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 19357 | 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 19358 | 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 19359 | 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 19360 | 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 19361 | 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 19362 | 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)) | ||
| Syntax | coppg 19363 | The opposite group operation. |
| class oppg | ||
| Definition | df-oppg 19364 | Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 20334 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 19365 | 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 19366 | 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 19367 | 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 19368 | 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 | oppglemOLD 19369 | Obsolete version of setsplusg 19368 as of 27-Oct-2024. Lemma for oppgbas 19370. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
| ⊢ 𝑂 = (oppg‘𝑅) & ⊢ 𝐸 = Slot 𝑁 & ⊢ 𝑁 ∈ ℕ & ⊢ 𝑁 ≠ 2 ⇒ ⊢ (𝐸‘𝑅) = (𝐸‘𝑂) | ||
| Theorem | oppgbas 19370 | 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 19371 | Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝑂 = (oppg‘𝑅) & ⊢ 𝐽 = (TopSet‘𝑅) ⇒ ⊢ 𝐽 = (TopSet‘𝑂) | ||
| Theorem | oppgtopn 19372 | Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝑂 = (oppg‘𝑅) & ⊢ 𝐽 = (TopOpen‘𝑅) ⇒ ⊢ 𝐽 = (TopOpen‘𝑂) | ||
| Theorem | oppgmnd 19373 | 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 19374 | Bidirectional form of oppgmnd 19373. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
| ⊢ 𝑂 = (oppg‘𝑅) ⇒ ⊢ (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd) | ||
| Theorem | oppgid 19375 | 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 19376 | 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 19377 | Bidirectional form of oppggrp 19376. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
| ⊢ 𝑂 = (oppg‘𝑅) ⇒ ⊢ (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp) | ||
| Theorem | oppginv 19378 | 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 19379 | 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 19380 | 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 19381 | Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝑂 = (oppg‘𝐺) ⇒ ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂) | ||
| Theorem | oppgsubg 19382 | Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
| ⊢ 𝑂 = (oppg‘𝐺) ⇒ ⊢ (SubGrp‘𝐺) = (SubGrp‘𝑂) | ||
| Theorem | oppgcntz 19383 | 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 19384 | 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 19385 | 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))) | ||
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 19387. However, the
set is allowed to be empty, see symgbas0 19406. Hint: The symmetric groups
should not be confused with "symmetry groups" which is a different topic in
group theory.
| ||
| Syntax | csymg 19386 | Extend class notation to include the class of symmetric groups. |
| class SymGrp | ||
| Definition | df-symg 19387* | 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 19388* | 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 19389* | 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 19390 | 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 19391 | 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 19392 | 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 19393 | 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 19394 | 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 19395 | 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 19396 | 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 19397 | 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 19398 | 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 19399 | 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 19400* | 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)) | ||
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