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Theorem List for Metamath Proof Explorer - 17901-18000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsymgfixelsi 17901* The restriction of a permutation fixing an element to the set with this element removed is an element of the restricted symmetric group. (Contributed by AV, 4-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐷 = (𝑁 ∖ {𝐾})       ((𝐾𝑁𝐹𝑄) → (𝐹𝐷) ∈ 𝑆)
 
Theoremsymgfixf 17902* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a function. (Contributed by AV, 4-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))       (𝐾𝑁𝐻:𝑄𝑆)
 
Theoremsymgfixf1 17903* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a 1-1 function. (Contributed by AV, 4-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))       (𝐾𝑁𝐻:𝑄1-1𝑆)
 
Theoremsymgfixfolem1 17904* Lemma 1 for symgfixfo 17905. (Contributed by AV, 7-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝑁𝑉𝐾𝑁𝑍𝑆) → 𝐸𝑄)
 
Theoremsymgfixfo 17905* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is an onto function. (Contributed by AV, 7-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))       ((𝑁𝑉𝐾𝑁) → 𝐻:𝑄onto𝑆)
 
Theoremsymgfixf1o 17906* The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a bijection. (Contributed by AV, 7-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))       ((𝑁𝑉𝐾𝑁) → 𝐻:𝑄1-1-onto𝑆)
 
10.2.9.4  Transpositions in the symmetric group

Transpositions are special cases of "cycles" as defined in [Rotman] p. 28: "Let i1 , i2 , ... , ir be distinct integers between 1 and n. If α in Sn fixes the other integers and α(i1) = i2, α(i2) = i3, ..., α(ir-1 ) = ir, α(ir) = i1, then α is an r-cycle. We also say that α is a cycle of length r." and in [Rotman] p. 31: "A 2-cycle is also called transposition.".

We (currently) do not have/need a definition for cycles, so transpositions are explicitly defined in df-pmtr 17908.

 
Syntaxcpmtr 17907 Syntax for the transposition generator function.
class pmTrsp
 
Definitiondf-pmtr 17908* Define a function that generates the transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑𝑦 ≈ 2𝑜} ↦ (𝑧𝑑 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
 
Theoremf1omvdmvd 17909 A permutation of any class moves a point which is moved to a different point which is moved. (Contributed by Stefan O'Rear, 22-Aug-2015.)
((𝐹:𝐴1-1-onto𝐴𝑋 ∈ dom (𝐹 ∖ I )) → (𝐹𝑋) ∈ (dom (𝐹 ∖ I ) ∖ {𝑋}))
 
Theoremf1omvdcnv 17910 A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
(𝐹:𝐴1-1-onto𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ I ))
 
Theoremmvdco 17911 Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.)
dom ((𝐹𝐺) ∖ I ) ⊆ (dom (𝐹 ∖ I ) ∪ dom (𝐺 ∖ I ))
 
Theoremf1omvdconj 17912 Conjugation of a permutation takes the image of the moved subclass. (Contributed by Stefan O'Rear, 22-Aug-2015.)
((𝐹:𝐴𝐴𝐺:𝐴1-1-onto𝐴) → dom (((𝐺𝐹) ∘ 𝐺) ∖ I ) = (𝐺 “ dom (𝐹 ∖ I )))
 
Theoremf1otrspeq 17913 A transposition is characterized by the points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
(((𝐹:𝐴1-1-onto𝐴𝐺:𝐴1-1-onto𝐴) ∧ (dom (𝐹 ∖ I ) ≈ 2𝑜 ∧ dom (𝐺 ∖ I ) = dom (𝐹 ∖ I ))) → 𝐹 = 𝐺)
 
Theoremf1omvdco2 17914 If exactly one of two permutations is limited to a set of points, then the composition will not be. (Contributed by Stefan O'Rear, 23-Aug-2015.)
((𝐹:𝐴1-1-onto𝐴𝐺:𝐴1-1-onto𝐴 ∧ (dom (𝐹 ∖ I ) ⊆ 𝑋 ⊻ dom (𝐺 ∖ I ) ⊆ 𝑋)) → ¬ dom ((𝐹𝐺) ∖ I ) ⊆ 𝑋)
 
Theoremf1omvdco3 17915 If a point is moved by exactly one of two permutations, then it will be moved by their composite. (Contributed by Stefan O'Rear, 23-Aug-2015.)
((𝐹:𝐴1-1-onto𝐴𝐺:𝐴1-1-onto𝐴 ∧ (𝑋 ∈ dom (𝐹 ∖ I ) ⊻ 𝑋 ∈ dom (𝐺 ∖ I ))) → 𝑋 ∈ dom ((𝐹𝐺) ∖ I ))
 
Theorempmtrfval 17916* The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       (𝐷𝑉𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷𝑦 ≈ 2𝑜} ↦ (𝑧𝐷 ↦ if(𝑧𝑝, (𝑝 ∖ {𝑧}), 𝑧))))
 
Theorempmtrval 17917* A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
 
Theorempmtrfv 17918 General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
 
Theorempmtrprfv 17919 In a transposition of two given points, each maps to the other. (Contributed by Stefan O'Rear, 25-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑋𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑋) = 𝑌)
 
Theorempmtrprfv3 17920 In a transposition of two given points, all other points are mapped to themselves. (Contributed by AV, 17-Mar-2019.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → ((𝑇‘{𝑋, 𝑌})‘𝑍) = 𝑍)
 
Theorempmtrf 17921 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃):𝐷𝐷)
 
Theorempmtrmvd 17922 A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → dom ((𝑇𝑃) ∖ I ) = 𝑃)
 
Theorempmtrrn 17923 Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) ∈ 𝑅)
 
Theorempmtrfrn 17924 A transposition (as a kind of function) is the function transposing the two points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇    &   𝑃 = dom (𝐹 ∖ I )       (𝐹𝑅 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃)))
 
Theorempmtrffv 17925 Mapping of a point under a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇    &   𝑃 = dom (𝐹 ∖ I )       ((𝐹𝑅𝑍𝐷) → (𝐹𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
 
Theorempmtrrn2 17926* For any transposition there are two points it is transposing. (Contributed by SO, 15-Jul-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 → ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝐹 = (𝑇‘{𝑥, 𝑦})))
 
Theorempmtrfinv 17927 A transposition function is an involution. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 → (𝐹𝐹) = ( I ↾ 𝐷))
 
Theorempmtrfmvdn0 17928 A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 → dom (𝐹 ∖ I ) ≠ ∅)
 
Theorempmtrff1o 17929 A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅𝐹:𝐷1-1-onto𝐷)
 
Theorempmtrfcnv 17930 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅𝐹 = 𝐹)
 
Theorempmtrfb 17931 An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅 ↔ (𝐷 ∈ V ∧ 𝐹:𝐷1-1-onto𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2𝑜))
 
Theorempmtrfconj 17932 Any conjugate of a transposition is a transposition. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       ((𝐹𝑅𝐺:𝐷1-1-onto𝐷) → ((𝐺𝐹) ∘ 𝐺) ∈ 𝑅)
 
Theoremsymgsssg 17933* The symmetric group has subgroups restricting the set of non-fixed points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐷𝑉 → {𝑥𝐵 ∣ dom (𝑥 ∖ I ) ⊆ 𝑋} ∈ (SubGrp‘𝐺))
 
Theoremsymgfisg 17934* The symmetric group has a subgroup of permutations that move finitely many points. (Contributed by Stefan O'Rear, 24-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝐷𝑉 → {𝑥𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin} ∈ (SubGrp‘𝐺))
 
Theoremsymgtrf 17935 Transpositions are elements of the symmetric group. (Contributed by Stefan O'Rear, 23-Aug-2015.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       𝑇𝐵
 
Theoremsymggen 17936* The span of the transpositions is the subgroup that moves finitely many points. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝐾 = (mrCls‘(SubMnd‘𝐺))       (𝐷𝑉 → (𝐾𝑇) = {𝑥𝐵 ∣ dom (𝑥 ∖ I ) ∈ Fin})
 
Theoremsymggen2 17937 A finite permutation group is generated by the transpositions, see also Theorem 3.4 in [Rotman] p. 31. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝐾 = (mrCls‘(SubMnd‘𝐺))       (𝐷 ∈ Fin → (𝐾𝑇) = 𝐵)
 
Theoremsymgtrinv 17938 To invert a permutation represented as a sequence of transpositions, reverse the sequence. (Contributed by Stefan O'Rear, 27-Aug-2015.)
𝑇 = ran (pmTrsp‘𝐷)    &   𝐺 = (SymGrp‘𝐷)    &   𝐼 = (invg𝐺)       ((𝐷𝑉𝑊 ∈ Word 𝑇) → (𝐼‘(𝐺 Σg 𝑊)) = (𝐺 Σg (reverse‘𝑊)))
 
Theorempmtr3ncomlem1 17939 Lemma 1 for pmtr3ncom 17941. (Contributed by AV, 17-Mar-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝐹 = (𝑇‘{𝑋, 𝑌})    &   𝐺 = (𝑇‘{𝑌, 𝑍})       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → ((𝐺𝐹)‘𝑋) ≠ ((𝐹𝐺)‘𝑋))
 
Theorempmtr3ncomlem2 17940 Lemma 2 for pmtr3ncom 17941. (Contributed by AV, 17-Mar-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝐹 = (𝑇‘{𝑋, 𝑌})    &   𝐺 = (𝑇‘{𝑌, 𝑍})       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (𝐺𝐹) ≠ (𝐹𝐺))
 
Theorempmtr3ncom 17941* Transpositions over sets with at least 3 elements are not commutative, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉 ∧ 3 ≤ (#‘𝐷)) → ∃𝑓 ∈ ran 𝑇𝑔 ∈ ran 𝑇(𝑔𝑓) ≠ (𝑓𝑔))
 
Theorempmtrdifellem1 17942 Lemma 1 for pmtrdifel 17946. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇𝑆𝑅)
 
Theorempmtrdifellem2 17943 Lemma 2 for pmtrdifel 17946. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
 
Theorempmtrdifellem3 17944* Lemma 3 for pmtrdifel 17946. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑄𝑥) = (𝑆𝑥))
 
Theorempmtrdifellem4 17945 Lemma 4 for pmtrdifel 17946. (Contributed by AV, 28-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       ((𝑄𝑇𝐾𝑁) → (𝑆𝐾) = 𝐾)
 
Theorempmtrdifel 17946* A transposition of elements of a set without a special element corresponds to a transposition of elements of the set. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)       𝑡𝑇𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥)
 
Theorempmtrdifwrdellem1 17947* Lemma 1 for pmtrdifwrdel 17951. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(#‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       (𝑊 ∈ Word 𝑇𝑈 ∈ Word 𝑅)
 
Theorempmtrdifwrdellem2 17948* Lemma 2 for pmtrdifwrdel 17951. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(#‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       (𝑊 ∈ Word 𝑇 → (#‘𝑊) = (#‘𝑈))
 
Theorempmtrdifwrdellem3 17949* Lemma 3 for pmtrdifwrdel 17951. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(#‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       (𝑊 ∈ Word 𝑇 → ∀𝑖 ∈ (0..^(#‘𝑊))∀𝑛 ∈ (𝑁 ∖ {𝐾})((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛))
 
Theorempmtrdifwrdel2lem1 17950* Lemma 1 for pmtrdifwrdel2 17952. (Contributed by AV, 31-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑈 = (𝑥 ∈ (0..^(#‘𝑊)) ↦ ((pmTrsp‘𝑁)‘dom ((𝑊𝑥) ∖ I )))       ((𝑊 ∈ Word 𝑇𝐾𝑁) → ∀𝑖 ∈ (0..^(#‘𝑊))((𝑈𝑖)‘𝐾) = 𝐾)
 
Theorempmtrdifwrdel 17951* A sequence of transpositions of elements of a set without a special element corresponds to a sequence of transpositions of elements of the set. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)       𝑤 ∈ Word 𝑇𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥))
 
Theorempmtrdifwrdel2 17952* A sequence of transpositions of elements of a set without a special element corresponds to a sequence of transpositions of elements of the set not moving the special element. (Contributed by AV, 31-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)       (𝐾𝑁 → ∀𝑤 ∈ Word 𝑇𝑢 ∈ Word 𝑅((#‘𝑤) = (#‘𝑢) ∧ ∀𝑖 ∈ (0..^(#‘𝑤))(((𝑢𝑖)‘𝐾) = 𝐾 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})((𝑤𝑖)‘𝑥) = ((𝑢𝑖)‘𝑥))))
 
Theorempmtrprfval 17953* The transpositions on a pair. (Contributed by AV, 9-Dec-2018.)
(pmTrsp‘{1, 2}) = (𝑝 ∈ {{1, 2}} ↦ (𝑧 ∈ {1, 2} ↦ if(𝑧 = 1, 2, 1)))
 
Theorempmtrprfvalrn 17954 The range of the transpositions on a pair is actually a singleton: the transposition of the two elements of the pair. (Contributed by AV, 9-Dec-2018.)
ran (pmTrsp‘{1, 2}) = {{⟨1, 2⟩, ⟨2, 1⟩}}
 
10.2.9.5  The sign of a permutation
 
Syntaxcpsgn 17955 Syntax for the sign of a permutation.
class pmSgn
 
Syntaxcevpm 17956 Syntax for even permutations.
class pmEven
 
Definitiondf-psgn 17957* Define a function which takes the value 1 for even permutations and -1 for odd. (Contributed by Stefan O'Rear, 28-Aug-2015.)
pmSgn = (𝑑 ∈ V ↦ (𝑥 ∈ {𝑝 ∈ (Base‘(SymGrp‘𝑑)) ∣ dom (𝑝 ∖ I ) ∈ Fin} ↦ (℩𝑠𝑤 ∈ Word ran (pmTrsp‘𝑑)(𝑥 = ((SymGrp‘𝑑) Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))))
 
Definitiondf-evpm 17958 Define the set of even permutations on a given set. (Contributed by Stefan O'Rear, 9-Jul-2018.)
pmEven = (𝑑 ∈ V ↦ ((pmSgn‘𝑑) “ {1}))
 
Theorempsgnunilem1 17959* Lemma for psgnuni 17965. Given two consequtive transpositions in a representation of a permutation, either they are equal and therefore equivalent to the identity, or they are not and it is possible to commute them such that a chosen point in the left transposition is preserved in the right. By repeating this process, a point can be removed from a representation of the identity. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑃𝑇)    &   (𝜑𝑄𝑇)    &   (𝜑𝐴 ∈ dom (𝑃 ∖ I ))       (𝜑 → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
 
Theorempsgnunilem5 17960* Lemma for psgnuni 17965. It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving 𝐴 in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   (𝜑 → (#‘𝑊) = 𝐿)    &   (𝜑𝐼 ∈ (0..^𝐿))    &   (𝜑𝐴 ∈ dom ((𝑊𝐼) ∖ I ))    &   (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ))       (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))
 
Theorempsgnunilem2 17961* Lemma for psgnuni 17965. Induction step for moving a transposition as far to the right as possible. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   (𝜑 → (#‘𝑊) = 𝐿)    &   (𝜑𝐼 ∈ (0..^𝐿))    &   (𝜑𝐴 ∈ dom ((𝑊𝐼) ∖ I ))    &   (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ))    &   (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))       (𝜑 → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (#‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ))))
 
Theorempsgnunilem3 17962* Lemma for psgnuni 17965. Any nonempty representation of the identity can be incrementally transformed into a representation two shorter. (Contributed by Stefan O'Rear, 25-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (#‘𝑊) = 𝐿)    &   (𝜑 → (#‘𝑊) ∈ ℕ)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))    &   (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))        ¬ 𝜑
 
Theorempsgnunilem4 17963 Lemma for psgnuni 17965. An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))       (𝜑 → (-1↑(#‘𝑊)) = 1)
 
Theoremm1expaddsub 17964 Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)
((𝑋 ∈ ℤ ∧ 𝑌 ∈ ℤ) → (-1↑(𝑋𝑌)) = (-1↑(𝑋 + 𝑌)))
 
Theorempsgnuni 17965 If the same permutation can be written in more than one way as a product of transpositions, the parity of those products must agree; otherwise the product of one with the inverse of the other would be an odd representation of the identity. (Contributed by Stefan O'Rear, 27-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   (𝜑𝐷𝑉)    &   (𝜑𝑊 ∈ Word 𝑇)    &   (𝜑𝑋 ∈ Word 𝑇)    &   (𝜑 → (𝐺 Σg 𝑊) = (𝐺 Σg 𝑋))       (𝜑 → (-1↑(#‘𝑊)) = (-1↑(#‘𝑋)))
 
Theorempsgnfval 17966* Function definition of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝐹 = {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       𝑁 = (𝑥𝐹 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
 
Theorempsgnfn 17967* Functionality and domain of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝐹 = {𝑝𝐵 ∣ dom (𝑝 ∖ I ) ∈ Fin}    &   𝑁 = (pmSgn‘𝐷)       𝑁 Fn 𝐹
 
Theorempsgndmsubg 17968 The finitary permutations are a subgroup. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝐷𝑉 → dom 𝑁 ∈ (SubGrp‘𝐺))
 
Theorempsgneldm 17969 Property of being a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)    &   𝐵 = (Base‘𝐺)       (𝑃 ∈ dom 𝑁 ↔ (𝑃𝐵 ∧ dom (𝑃 ∖ I ) ∈ Fin))
 
Theorempsgneldm2 17970* The finitary permutations are the span of the transpositions. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝐷𝑉 → (𝑃 ∈ dom 𝑁 ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σg 𝑤)))
 
Theorempsgneldm2i 17971 A sequence of transpositions describes a finitary permutation. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       ((𝐷𝑉𝑊 ∈ Word 𝑇) → (𝐺 Σg 𝑊) ∈ dom 𝑁)
 
Theorempsgneu 17972* A finitary permutation has exactly one parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝑃 ∈ dom 𝑁 → ∃!𝑠𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))
 
Theorempsgnval 17973* Value of the permutation sign function. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝑃 ∈ dom 𝑁 → (𝑁𝑃) = (℩𝑠𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
 
Theorempsgnvali 17974* A finitary permutation has at least one representation for its parity. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝑃 ∈ dom 𝑁 → ∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ (𝑁𝑃) = (-1↑(#‘𝑤))))
 
Theorempsgnvalii 17975 Any representation of a permutation is length matching the permutation sign. (Contributed by Stefan O'Rear, 28-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       ((𝐷𝑉𝑊 ∈ Word 𝑇) → (𝑁‘(𝐺 Σg 𝑊)) = (-1↑(#‘𝑊)))
 
Theorempsgnpmtr 17976 All transpositions are odd. (Contributed by Stefan O'Rear, 29-Aug-2015.)
𝐺 = (SymGrp‘𝐷)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝑃𝑇 → (𝑁𝑃) = -1)
 
Theorempsgn0fv0 17977 The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.)
((pmSgn‘∅)‘∅) = 1
 
Theoremsygbasnfpfi 17978 The class of non-fixed points of a permutation of a finite set is finite. (Contributed by AV, 13-Jan-2019.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)       ((𝐷 ∈ Fin ∧ 𝑃𝐵) → dom (𝑃 ∖ I ) ∈ Fin)
 
Theorempsgnfvalfi 17979* Function definition of the permutation sign function for permutations of finite sets. (Contributed by AV, 13-Jan-2019.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝐷 ∈ Fin → 𝑁 = (𝑥𝐵 ↦ (℩𝑠𝑤 ∈ Word 𝑇(𝑥 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))))
 
Theorempsgnvalfi 17980* Value of the permutation sign function for permutations of finite sets. (Contributed by AV, 13-Jan-2019.)
𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       ((𝐷 ∈ Fin ∧ 𝑃𝐵) → (𝑁𝑃) = (℩𝑠𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
 
Theorempsgnran 17981 The range of the permutation sign function for finite permutations. (Contributed by AV, 1-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)       ((𝑁 ∈ Fin ∧ 𝑄𝑃) → (𝑆𝑄) ∈ {1, -1})
 
Theoremgsmtrcl 17982 The group sum of transpositions of a finite set is a permutation, see also psgneldm2i 17971. (Contributed by AV, 19-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)    &   𝑇 = ran (pmTrsp‘𝑁)       ((𝑁 ∈ Fin ∧ 𝑊 ∈ Word 𝑇) → (𝑆 Σg 𝑊) ∈ 𝐵)
 
Theorempsgnfitr 17983* A permutation of a finite set is generated by transpositions. (Contributed by AV, 13-Jan-2019.)
𝐺 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝐺)    &   𝑇 = ran (pmTrsp‘𝑁)       (𝑁 ∈ Fin → (𝑄𝐵 ↔ ∃𝑤 ∈ Word 𝑇𝑄 = (𝐺 Σg 𝑤)))
 
Theorempsgnfieu 17984* A permutation of a finite set has exactly one parity. (Contributed by AV, 13-Jan-2019.)
𝐺 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝐺)    &   𝑇 = ran (pmTrsp‘𝑁)       ((𝑁 ∈ Fin ∧ 𝑄𝐵) → ∃!𝑠𝑤 ∈ Word 𝑇(𝑄 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤))))
 
Theorempmtrsn 17985 The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)
(pmTrsp‘{𝐴}) = ∅
 
Theorempsgnsn 17986 The permutation sign function for a singleton. (Contributed by AV, 6-Aug-2019.)
𝐷 = {𝐴}    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑁 = (pmSgn‘𝐷)       ((𝐴𝑉𝑋𝐵) → (𝑁𝑋) = 1)
 
Theorempsgnprfval 17987* The permutation sign function for a pair. (Contributed by AV, 10-Dec-2018.)
𝐷 = {1, 2}    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝑋𝐵 → (𝑁𝑋) = (℩𝑠𝑤 ∈ Word 𝑇(𝑋 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(#‘𝑤)))))
 
Theorempsgnprfval1 17988 The permutation sign of the identity for a pair. (Contributed by AV, 11-Dec-2018.)
𝐷 = {1, 2}    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝑁‘{⟨1, 1⟩, ⟨2, 2⟩}) = 1
 
Theorempsgnprfval2 17989 The permutation sign of the transposition for a pair. (Contributed by AV, 10-Dec-2018.)
𝐷 = {1, 2}    &   𝐺 = (SymGrp‘𝐷)    &   𝐵 = (Base‘𝐺)    &   𝑇 = ran (pmTrsp‘𝐷)    &   𝑁 = (pmSgn‘𝐷)       (𝑁‘{⟨1, 2⟩, ⟨2, 1⟩}) = -1
 
10.2.10  p-Groups and Sylow groups; Sylow's theorems
 
Syntaxcod 17990 Extend class notation to include the order function on the elements of a group.
class od
 
Syntaxcgex 17991 Extend class notation to include the order function on the elements of a group.
class gEx
 
Syntaxcpgp 17992 Extend class notation to include the class of all p-groups.
class pGrp
 
Syntaxcslw 17993 Extend class notation to include the class of all Sylow p-subgroups of a group.
class pSyl
 
Definitiondf-od 17994* Define the order of an element in a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.) (Revised by AV, 5-Oct-2020.)
od = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔) ↦ {𝑛 ∈ ℕ ∣ (𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < ))))
 
Definitiondf-gex 17995* Define the exponent of a group. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 4-Sep-2015.) (Revised by AV, 26-Sep-2020.)
gEx = (𝑔 ∈ V ↦ {𝑛 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑛(.g𝑔)𝑥) = (0g𝑔)} / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
 
Definitiondf-pgp 17996* Define the set of p-groups, which are groups such that every element has a power of 𝑝 as its order. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by AV, 5-Oct-2020.)
pGrp = {⟨𝑝, 𝑔⟩ ∣ ((𝑝 ∈ ℙ ∧ 𝑔 ∈ Grp) ∧ ∀𝑥 ∈ (Base‘𝑔)∃𝑛 ∈ ℕ0 ((od‘𝑔)‘𝑥) = (𝑝𝑛))}
 
Definitiondf-slw 17997* Define the set of Sylow p-subgroups of a group 𝑔. A Sylow p-subgroup is a p-group that is not a subgroup of any other p-groups in 𝑔. (Contributed by Mario Carneiro, 16-Jan-2015.)
pSyl = (𝑝 ∈ ℙ, 𝑔 ∈ Grp ↦ { ∈ (SubGrp‘𝑔) ∣ ∀𝑘 ∈ (SubGrp‘𝑔)((𝑘𝑝 pGrp (𝑔s 𝑘)) ↔ = 𝑘)})
 
Theoremodfval 17998* Value of the order function. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)       𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
 
Theoremodval 17999* Second substitution for the group order definition. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 5-Sep-2015.) (Revised by AV, 5-Oct-2020.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }       (𝐴𝑋 → (𝑂𝐴) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
 
Theoremodlem1 18000* The group element order is either zero or a nonzero multiplier that annihilates the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) (Revised by AV, 5-Oct-2020.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)    &   𝐼 = {𝑦 ∈ ℕ ∣ (𝑦 · 𝐴) = 0 }       (𝐴𝑋 → (((𝑂𝐴) = 0 ∧ 𝐼 = ∅) ∨ (𝑂𝐴) ∈ 𝐼))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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