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Theorem List for Metamath Proof Explorer - 18501-18600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-pmtr 18501* 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 18502 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 18503 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 18504 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 18505 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 18506 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 18507 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 18508 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 18509* 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 18510* 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 18511 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 18512 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 18513 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 18514 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃):𝐷𝐷)
 
Theorempmtrmvd 18515 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 18516 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 18517 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 18518 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 18519* 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 18520 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 18521 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 18522 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 18523 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅𝐹 = 𝐹)
 
Theorempmtrfb 18524 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 18525 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 18526* 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 18527* 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 18528 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 18529* 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 18530 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 18531 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 18532 Lemma 1 for pmtr3ncom 18534. (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 18533 Lemma 2 for pmtr3ncom 18534. (Contributed by AV, 17-Mar-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝐹 = (𝑇‘{𝑋, 𝑌})    &   𝐺 = (𝑇‘{𝑌, 𝑍})       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (𝐺𝐹) ≠ (𝐹𝐺))
 
Theorempmtr3ncom 18534* 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 18535 Lemma 1 for pmtrdifel 18539. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇𝑆𝑅)
 
Theorempmtrdifellem2 18536 Lemma 2 for pmtrdifel 18539. (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 18537* Lemma 3 for pmtrdifel 18539. (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 18538 Lemma 4 for pmtrdifel 18539. (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 18539* 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 18540* Lemma 1 for pmtrdifwrdel 18544. (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 18541* Lemma 2 for pmtrdifwrdel 18544. (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 18542* Lemma 3 for pmtrdifwrdel 18544. (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 18543* Lemma 1 for pmtrdifwrdel2 18545. (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 18544* 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 18545* 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 18546* 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 18547 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⟩}}
 
10.2.10.5  The sign of a permutation
 
Syntaxcpsgn 18548 Syntax for the sign of a permutation.
class pmSgn
 
Syntaxcevpm 18549 Syntax for even permutations.
class pmEven
 
Definitiondf-psgn 18550* 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 18551 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 18552* Lemma for psgnuni 18558. 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 18553* Lemma for psgnuni 18558. 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.) (Proof shortened by AV, 12-Oct-2022.)
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 18554* Lemma for psgnuni 18558. 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 18555* Lemma for psgnuni 18558. 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 18556 Lemma for psgnuni 18558. 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 18557 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 18558 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 18559* 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 18560* 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 18561 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 18562 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 18563* 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 18564 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 18565* 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 18566* 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 18567* 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 18568 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 18569 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 18570 The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.)
((pmSgn‘∅)‘∅) = 1
 
Theoremsygbasnfpfi 18571 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 18572* 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 18573* 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 18574 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 18575 The group sum of transpositions of a finite set is a permutation, see also psgneldm2i 18564. (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 18576* 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 18577* 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 18578 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 18579 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 18580* 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 18581 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 18582 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
 
10.2.11  p-Groups and Sylow groups; Sylow's theorems
 
Syntaxcod 18583 Extend class notation to include the order function on the elements of a group.
class od
 
Syntaxcgex 18584 Extend class notation to include the order function on the elements of a group.
class gEx
 
Syntaxcpgp 18585 Extend class notation to include the class of all p-groups.
class pGrp
 
Syntaxcslw 18586 Extend class notation to include the class of all Sylow p-subgroups of a group.
class pSyl
 
Definitiondf-od 18587* 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 18588* 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 18589* 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 18590* 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 18591* Value of the order function. For a shorter proof using ax-rep 5182, see odfvalALT 18592. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) Remove depedency on ax-rep 5182. (Revised by Rohan Ridenour, 17-Aug-2023.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)       𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
 
TheoremodfvalALT 18592* Shorter proof of odfval 18591 using ax-rep 5182. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑋 = (Base‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   𝑂 = (od‘𝐺)       𝑂 = (𝑥𝑋{𝑦 ∈ ℕ ∣ (𝑦 · 𝑥) = 0 } / 𝑖if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
 
Theoremodval 18593* 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 18594* 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))
 
Theoremodcl 18595 The order of a group element is always a nonnegative integer. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       (𝐴𝑋 → (𝑂𝐴) ∈ ℕ0)
 
Theoremodf 18596 Functionality of the group element order. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)       𝑂:𝑋⟶ℕ0
 
Theoremodid 18597 Any element to the power of its order is the identity. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (𝐴𝑋 → ((𝑂𝐴) · 𝐴) = 0 )
 
Theoremodlem2 18598 Any positive annihilator of a group element is an upper bound on the (positive) order of the element. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 5-Oct-2020.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐴𝑋𝑁 ∈ ℕ ∧ (𝑁 · 𝐴) = 0 ) → (𝑂𝐴) ∈ (1...𝑁))
 
Theoremodmodnn0 18599 Reduce the argument of a group multiple by modding out the order of the element. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (((𝐺 ∈ Mnd ∧ 𝐴𝑋𝑁 ∈ ℕ0) ∧ (𝑂𝐴) ∈ ℕ) → ((𝑁 mod (𝑂𝐴)) · 𝐴) = (𝑁 · 𝐴))
 
Theoremmndodconglem 18600 Lemma for mndodcong 18601. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Mnd)    &   (𝜑𝐴𝑋)    &   (𝜑 → (𝑂𝐴) ∈ ℕ)    &   (𝜑𝑀 ∈ ℕ0)    &   (𝜑𝑁 ∈ ℕ0)    &   (𝜑𝑀 < (𝑂𝐴))    &   (𝜑𝑁 < (𝑂𝐴))    &   (𝜑 → (𝑀 · 𝐴) = (𝑁 · 𝐴))       ((𝜑𝑀𝑁) → 𝑀 = 𝑁)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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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 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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