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Theorem List for Metamath Proof Explorer - 18501-18600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1omvdcnv 18501 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 18502 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 18503 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 18504 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 18505 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 18506 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 18507* 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 18508* 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 18509 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 18510 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 18511 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 18512 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃):𝐷𝐷)
 
Theorempmtrmvd 18513 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 18514 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 18515 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 18516 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 18517* 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 18518 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 18519 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 18520 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 18521 A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)    &   𝑅 = ran 𝑇       (𝐹𝑅𝐹 = 𝐹)
 
Theorempmtrfb 18522 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 18523 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 18524* 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 18525* 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 18526 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 18527* 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 18528 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 18529 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 18530 Lemma 1 for pmtr3ncom 18532. (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 18531 Lemma 2 for pmtr3ncom 18532. (Contributed by AV, 17-Mar-2018.)
𝑇 = (pmTrsp‘𝐷)    &   𝐹 = (𝑇‘{𝑋, 𝑌})    &   𝐺 = (𝑇‘{𝑌, 𝑍})       ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑍𝐷) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (𝐺𝐹) ≠ (𝐹𝐺))
 
Theorempmtr3ncom 18532* 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 18533 Lemma 1 for pmtrdifel 18537. (Contributed by AV, 15-Jan-2019.)
𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))    &   𝑅 = ran (pmTrsp‘𝑁)    &   𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))       (𝑄𝑇𝑆𝑅)
 
Theorempmtrdifellem2 18534 Lemma 2 for pmtrdifel 18537. (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 18535* Lemma 3 for pmtrdifel 18537. (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 18536 Lemma 4 for pmtrdifel 18537. (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 18537* 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 18538* Lemma 1 for pmtrdifwrdel 18542. (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 18539* Lemma 2 for pmtrdifwrdel 18542. (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 18540* Lemma 3 for pmtrdifwrdel 18542. (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 18541* Lemma 1 for pmtrdifwrdel2 18543. (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 18542* 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 18543* 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 18544* 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 18545 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 18546 Syntax for the sign of a permutation.
class pmSgn
 
Syntaxcevpm 18547 Syntax for even permutations.
class pmEven
 
Definitiondf-psgn 18548* 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 18549 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 18550* Lemma for psgnuni 18556. 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 18551* Lemma for psgnuni 18556. 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 18552* Lemma for psgnuni 18556. 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 18553* Lemma for psgnuni 18556. 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 18554 Lemma for psgnuni 18556. 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 18555 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 18556 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 18557* 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 18558* 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 18559 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 18560 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 18561* 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 18562 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 18563* 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 18564* 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 18565* 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 18566 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 18567 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 18568 The permutation sign function for an empty set at an empty set is 1. (Contributed by AV, 27-Feb-2019.)
((pmSgn‘∅)‘∅) = 1
 
Theoremsygbasnfpfi 18569 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 18570* 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 18571* 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 18572 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 18573 The group sum of transpositions of a finite set is a permutation, see also psgneldm2i 18562. (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 18574* 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 18575* 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 18576 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 18577 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 18578* 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 18579 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 18580 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 18581 Extend class notation to include the order function on the elements of a group.
class od
 
Syntaxcgex 18582 Extend class notation to include the order function on the elements of a group.
class gEx
 
Syntaxcpgp 18583 Extend class notation to include the class of all p-groups.
class pGrp
 
Syntaxcslw 18584 Extend class notation to include the class of all Sylow p-subgroups of a group.
class pSyl
 
Definitiondf-od 18585* 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 18586* 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 18587* 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 18588* 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 18589* Value of the order function. For a shorter proof using ax-rep 5181, see odfvalALT 18590. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by AV, 5-Oct-2020.) Remove depedency on ax-rep 5181. (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 18590* Shorter proof of odfval 18589 using ax-rep 5181. (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 18591* 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 18592* 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 18593 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 18594 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 18595 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 18596 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 18597 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 18598 Lemma for mndodcong 18599. (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)
 
Theoremmndodcong 18599 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       (((𝐺 ∈ Mnd ∧ 𝐴𝑋) ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0) ∧ (𝑂𝐴) ∈ ℕ) → ((𝑂𝐴) ∥ (𝑀𝑁) ↔ (𝑀 · 𝐴) = (𝑁 · 𝐴)))
 
Theoremmndodcongi 18600 If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. For monoids, the reverse implication is false for elements with infinite order. For example, the powers of 2 mod 10 are 1,2,4,8,6,2,4,8,6,... so that the identity 1 never repeats, which is infinite order by our definition, yet other numbers like 6 appear many times in the sequence. (Contributed by Mario Carneiro, 23-Sep-2015.)
𝑋 = (Base‘𝐺)    &   𝑂 = (od‘𝐺)    &    · = (.g𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Mnd ∧ 𝐴𝑋 ∧ (𝑀 ∈ ℕ0𝑁 ∈ ℕ0)) → ((𝑂𝐴) ∥ (𝑀𝑁) → (𝑀 · 𝐴) = (𝑁 · 𝐴)))
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