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Theorem List for Metamath Proof Explorer - 17501-17600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoppgtopn 17501 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝑅)    &   𝐽 = (TopOpen‘𝑅)       𝐽 = (TopOpen‘𝑂)
 
Theoremoppgmnd 17502 The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Mnd → 𝑂 ∈ Mnd)
 
Theoremoppgmndb 17503 Bidirectional form of oppgmnd 17502. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)
 
Theoremoppgid 17504 Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
𝑂 = (oppg𝑅)    &    0 = (0g𝑅)        0 = (0g𝑂)
 
Theoremoppggrp 17505 The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Grp → 𝑂 ∈ Grp)
 
Theoremoppggrpb 17506 Bidirectional form of oppggrp 17505. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp)
 
Theoremoppginv 17507 Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)    &   𝐼 = (invg𝑅)       (𝑅 ∈ Grp → 𝐼 = (invg𝑂))
 
Theoreminvoppggim 17508 The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝐺)    &   𝐼 = (invg𝐺)       (𝐺 ∈ Grp → 𝐼 ∈ (𝐺 GrpIso 𝑂))
 
Theoremoppggic 17509 Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝐺)       (𝐺 ∈ Grp → 𝐺𝑔 𝑂)
 
Theoremoppgsubm 17510 Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (SubMnd‘𝐺) = (SubMnd‘𝑂)
 
Theoremoppgsubg 17511 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (SubGrp‘𝐺) = (SubGrp‘𝑂)
 
Theoremoppgcntz 17512 A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑂 = (oppg𝐺)    &   𝑍 = (Cntz‘𝐺)       (𝑍𝐴) = ((Cntz‘𝑂)‘𝐴)
 
Theoremoppgcntr 17513 The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑂 = (oppg𝐺)    &   𝑍 = (Cntr‘𝐺)       𝑍 = (Cntr‘𝑂)
 
Theoremgsumwrev 17514 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 = (oppg𝑀)       ((𝑀 ∈ Mnd ∧ 𝑊 ∈ Word 𝐵) → (𝑂 Σg 𝑊) = (𝑀 Σg (reverse‘𝑊)))
 
10.2.9  Symmetric groups
 
10.2.9.1  Definition and basic properties

According to Wikipedia ("Symmetric group", 09-Mar-2019, https://en.wikipedia.org/wiki/symmetric_group) "In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions." and according to Encyclopedia of Mathematics ("Symmetric group", 09-Mar-2019, https://www.encyclopediaofmath.org/index.php/Symmetric_group) "The group of all permutations (self-bijections) of a set with the operation of composition (see Permutation group).". In [Rotman] p. 27 "If X is a nonempty set, a permutation of X is a function a : X -> X that is a one-to-one correspondence." and "If X is a nonempty set, the symmetric group on X, denoted SX, is the group whose elements are the permutations of X and whose binary operation is composition of functions.". Therefore, we define the symmetric group on a set 𝐴 as the set of one-to-one onto functions from 𝐴 to itself under function composition, see df-symg 17516. However, the set is allowed to be empty, see symgbas0 17532. Hint: The symmetric groups should not be confused with "symmetry groups" which is a different topic in group theory.

In this context, the one-to-one onto functions are called permutations for short. Since the base set of symmetric groups on a set 𝐴 is the set of all permutations of 𝐴 (see symgbas 17518), we can formally say 𝑃 ∈ (SymGrp‘𝐴) expressing "𝑃 is a permutation of 𝐴" if we are not interested in the group (or topology) structure.

In general, a permutation group "... is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself)." (see Wikipedia "Permutation group", 17-Mar-2019, https://en.wikipedia.org/wiki/Permutation_group). This means that a symmetric group is a permutation group, and each permutation group is a subgroup of a symmetric group (see pgrpsubgsymgbi 17545 and pgrpsubgsymg 17546). For example, the structure with the singleton containing only the identity function restricted to a set as base set and the function composition as group operation is a permutation group (group consisting of permutations), see idrespermg 17549, which is a (proper) subgroup of a symmetric group, see idressubgsymg 17548.

As in [Rotman] p. 28 "Let 𝑥𝑋 and 𝑝 ∈ SymGrp(𝑋); we say 𝑝 fixes 𝑥 if (𝑝𝑥) = 𝑥; otherwise 𝑝 moves 𝑥.". The theorems starting with symgfix2 17554 are about fixed/moved elements.

 
Syntaxcsymg 17515 Extend class notation to include the class of symmetric groups.
class SymGrp
 
Definitiondf-symg 17516* 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. (Contributed by Paul Chapman, 25-Feb-2008.)
SymGrp = (𝑥 ∈ V ↦ {:𝑥1-1-onto𝑥} / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓𝑏, 𝑔𝑏 ↦ (𝑓𝑔))⟩, ⟨(TopSet‘ndx), (∏t‘(𝑥 × {𝒫 𝑥}))⟩})
 
Theoremsymgval 17517* The value of the symmetric group function at 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}    &    + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))    &   𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))       (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
 
Theoremsymgbas 17518* The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
 
Theoremelsymgbas2 17519 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)       (𝐹𝑉 → (𝐹𝐵𝐹:𝐴1-1-onto𝐴))
 
Theoremelsymgbas 17520 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)       (𝐴𝑉 → (𝐹𝐵𝐹:𝐴1-1-onto𝐴))
 
Theoremsymgbasf1o 17521 Elements in the symmetric group are 1-1 onto functions. (Contributed by SO, 9-Jul-2018.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐹𝐵𝐹:𝐴1-1-onto𝐴)
 
Theoremsymgbasf 17522 A permutation (element of the symmetric group) is a function of a set into itself. (Contributed by AV, 1-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐹𝐵𝐹:𝐴𝐴)
 
Theoremsymghash 17523 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)))
 
Theoremsymgbasfi 17524 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)
 
Theoremsymgfv 17525 The function value of a permutation. (Contributed by AV, 1-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐹𝐵𝑋𝐴) → (𝐹𝑋) ∈ 𝐴)
 
Theoremsymgfvne 17526 The function values of a permutation for different arguments are different. (Contributed by AV, 8-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐹𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = 𝑍 → (𝑌𝑋 → (𝐹𝑌) ≠ 𝑍)))
 
Theoremsymgplusg 17527* 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.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)        + = (𝑓𝐵, 𝑔𝐵 ↦ (𝑓𝑔))
 
Theoremsymgov 17528 The value of the group operation of the symmetric group on 𝐴. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑋𝑌))
 
Theoremsymgcl 17529 The group operation of the symmetric group on 𝐴 is closed, i.e. a magma. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by Mario Carneiro, 28-Jan-2015.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &    + = (+g𝐺)       ((𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
 
Theoremsymgmov1 17530* For a permutation of a set, each element of the set replaces an(other) element of the set. (Contributed by AV, 2-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))       (𝑄𝑃 → ∀𝑛𝑁𝑘𝑁 (𝑄𝑛) = 𝑘)
 
Theoremsymgmov2 17531* For a permutation of a set, each element of the set is replaced by an(other) element of the set. (Contributed by AV, 2-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))       (𝑄𝑃 → ∀𝑛𝑁𝑘𝑁 (𝑄𝑘) = 𝑛)
 
Theoremsymgbas0 17532 The base set of the symmetric group on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Feb-2019.)
(Base‘(SymGrp‘∅)) = {∅}
 
Theoremsymg1hash 17533 The symmetric group on a singleton has cardinality 1. (Contributed by AV, 9-Dec-2018.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼}       (𝐼𝑉 → (#‘𝐵) = 1)
 
Theoremsymg1bas 17534 The symmetric group on a singleton is the symmetric group S1 consisting of the identity only. (Contributed by AV, 9-Dec-2018.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼}       (𝐼𝑉𝐵 = {{⟨𝐼, 𝐼⟩}})
 
Theoremsymg2hash 17535 The symmetric group on a (proper) pair has cardinality 2. (Contributed by AV, 9-Dec-2018.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼, 𝐽}       ((𝐼𝑉𝐽𝑊𝐼𝐽) → (#‘𝐵) = 2)
 
Theoremsymg2bas 17536 The symmetric group on a pair is the symmetric group S2 consisting of the identity and the transposition. This theorem is also valid if the elements are identical: then it collapses to theorem symg1bas 17534. (Contributed by AV, 9-Dec-2018.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐴 = {𝐼, 𝐽}       ((𝐼𝑉𝐽𝑊) → 𝐵 = {{⟨𝐼, 𝐼⟩, ⟨𝐽, 𝐽⟩}, {⟨𝐼, 𝐽⟩, ⟨𝐽, 𝐼⟩}})
 
Theoremsymgtset 17537 The topology of the symmetric group on 𝐴. This component is defined on a larger set than the true base - the product topology is defined on the set of all functions, not just bijections - but the definition of TopOpen ensures that it is trimmed down before it gets use. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉 → (∏t‘(𝐴 × {𝒫 𝐴})) = (TopSet‘𝐺))
 
Theoremsymggrp 17538 The symmetric group on a set 𝐴 is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉𝐺 ∈ Grp)
 
Theoremsymgid 17539 The group identity element of the symmetric group on a set 𝐴. (Contributed by Paul Chapman, 25-Jul-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉 → ( I ↾ 𝐴) = (0g𝐺))
 
Theoremsymginv 17540 The group inverse in the symmetric group corresponds to the functional inverse. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝑁 = (invg𝐺)       (𝐹𝐵 → (𝑁𝐹) = 𝐹)
 
Theoremgalactghm 17541* The currying of a group action is a group homomorphism between the group 𝐺 and the symmetric group (SymGrp‘𝑌). (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = (SymGrp‘𝑌)    &   𝐹 = (𝑥𝑋 ↦ (𝑦𝑌 ↦ (𝑥 𝑦)))       ( ∈ (𝐺 GrpAct 𝑌) → 𝐹 ∈ (𝐺 GrpHom 𝐻))
 
Theoremlactghmga 17542* The converse of galactghm 17541. The uncurrying of a homomorphism into (SymGrp‘𝑌) is a group action. Thus, group actions and group homomorphisms into a symmetric group are essentially equivalent notions. (Contributed by Mario Carneiro, 15-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = (SymGrp‘𝑌)    &    = (𝑥𝑋, 𝑦𝑌 ↦ ((𝐹𝑥)‘𝑦))       (𝐹 ∈ (𝐺 GrpHom 𝐻) → ∈ (𝐺 GrpAct 𝑌))
 
Theoremsymgtopn 17543 The topology of the symmetric group on 𝐴. (Contributed by Mario Carneiro, 29-Aug-2015.)
𝐺 = (SymGrp‘𝑋)    &   𝐵 = (Base‘𝐺)       (𝑋𝑉 → ((∏t‘(𝑋 × {𝒫 𝑋})) ↾t 𝐵) = (TopOpen‘𝐺))
 
Theoremsymgga 17544* The symmetric group induces a group action on its base set. (Contributed by Mario Carneiro, 24-Jan-2015.)
𝐺 = (SymGrp‘𝑋)    &   𝐵 = (Base‘𝐺)    &   𝐹 = (𝑓𝐵, 𝑥𝑋 ↦ (𝑓𝑥))       (𝑋𝑉𝐹 ∈ (𝐺 GrpAct 𝑋))
 
Theorempgrpsubgsymgbi 17545 Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       (𝐴𝑉 → (𝑃 ∈ (SubGrp‘𝐺) ↔ (𝑃𝐵 ∧ (𝐺s 𝑃) ∈ Grp)))
 
Theorempgrpsubgsymg 17546* Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)    &   𝐹 = (Base‘𝑃)       (𝐴𝑉 → ((𝑃 ∈ Grp ∧ 𝐹𝐵 ∧ (+g𝑃) = (𝑓𝐹, 𝑔𝐹 ↦ (𝑓𝑔))) → 𝐹 ∈ (SubGrp‘𝐺)))
 
Theoremidresperm 17547 The identity function restricted to a set is a permutation of this set. (Contributed by AV, 17-Mar-2019.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉 → ( I ↾ 𝐴) ∈ (Base‘𝐺))
 
Theoremidressubgsymg 17548 The singleton containing only the identity function restricted to a set is a subgroup of the symmetric group of this set. (Contributed by AV, 17-Mar-2019.)
𝐺 = (SymGrp‘𝐴)       (𝐴𝑉 → {( I ↾ 𝐴)} ∈ (SubGrp‘𝐺))
 
Theoremidrespermg 17549 The structure with the singleton containing only the identity function restricted to a set as base set and the function composition as group operation (constructed by (structure) restricting the symmetric group to that singleton) is a permutation group (group consisting of permutations). (Contributed by AV, 17-Mar-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐸 = (𝐺s {( I ↾ 𝐴)})       (𝐴𝑉 → (𝐸 ∈ Grp ∧ (Base‘𝐸) ⊆ (Base‘𝐺)))
 
10.2.9.2  Cayley's theorem
 
Theoremcayleylem1 17550* Lemma for cayley 17552. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐻 = (SymGrp‘𝑋)    &   𝑆 = (Base‘𝐻)    &   𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))       (𝐺 ∈ Grp → 𝐹 ∈ (𝐺 GrpHom 𝐻))
 
Theoremcayleylem2 17551* Lemma for cayley 17552. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &    0 = (0g𝐺)    &   𝐻 = (SymGrp‘𝑋)    &   𝑆 = (Base‘𝐻)    &   𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))       (𝐺 ∈ Grp → 𝐹:𝑋1-1𝑆)
 
Theoremcayley 17552* Cayley's Theorem (constructive version): given group 𝐺, 𝐹 is an isomorphism between 𝐺 and the subgroup 𝑆 of the symmetric group 𝐻 on the underlying set 𝑋 of 𝐺. See also Theorem 3.15 in [Rotman] p. 42. (Contributed by Paul Chapman, 3-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = (SymGrp‘𝑋)    &    + = (+g𝐺)    &   𝐹 = (𝑔𝑋 ↦ (𝑎𝑋 ↦ (𝑔 + 𝑎)))    &   𝑆 = ran 𝐹       (𝐺 ∈ Grp → (𝑆 ∈ (SubGrp‘𝐻) ∧ 𝐹 ∈ (𝐺 GrpHom (𝐻s 𝑆)) ∧ 𝐹:𝑋1-1-onto𝑆))
 
Theoremcayleyth 17553* Cayley's Theorem (existence version): every group 𝐺 is isomorphic to a subgroup of the symmetric group on the underlying set of 𝐺. (For any group 𝐺 there exists an isomorphism 𝑓 between 𝐺 and a subgroup of the symmetric group on the underlying set of 𝐺.) See also Theorem 3.15 in [Rotman] p. 42. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.)
𝑋 = (Base‘𝐺)    &   𝐻 = (SymGrp‘𝑋)       (𝐺 ∈ Grp → ∃𝑠 ∈ (SubGrp‘𝐻)∃𝑓 ∈ (𝐺 GrpHom (𝐻s 𝑠))𝑓:𝑋1-1-onto𝑠)
 
10.2.9.3  Permutations fixing one element
 
Theoremsymgfix2 17554* If a permutation does not move a certain element of a set to a second element, there is a third element which is moved to the second element. (Contributed by AV, 2-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))       (𝐿𝑁 → (𝑄 ∈ (𝑃 ∖ {𝑞𝑃 ∣ (𝑞𝐾) = 𝐿}) → ∃𝑘 ∈ (𝑁 ∖ {𝐾})(𝑄𝑘) = 𝐿))
 
Theoremsymgextf 17555* The extension of a permutation, fixing the additional element, is a function. (Contributed by AV, 6-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁𝑁)
 
Theoremsymgextfv 17556* The function value of the extension of a permutation, fixing the additional element, for elements in the original domain. (Contributed by AV, 6-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸𝑋) = (𝑍𝑋)))
 
Theoremsymgextfve 17557* The function value of the extension of a permutation, fixing the additional element, for the additional element. (Contributed by AV, 6-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       (𝐾𝑁 → (𝑋 = 𝐾 → (𝐸𝑋) = 𝐾))
 
Theoremsymgextf1lem 17558* Lemma for symgextf1 17559. (Contributed by AV, 6-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → ((𝑋 ∈ (𝑁 ∖ {𝐾}) ∧ 𝑌 ∈ {𝐾}) → (𝐸𝑋) ≠ (𝐸𝑌)))
 
Theoremsymgextf1 17559* The extension of a permutation, fixing the additional element, is a 1-1 function. (Contributed by AV, 6-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁1-1𝑁)
 
Theoremsymgextfo 17560* The extension of a permutation, fixing the additional element, is an onto function. (Contributed by AV, 7-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁onto𝑁)
 
Theoremsymgextf1o 17561* The extension of a permutation, fixing the additional element, is a bijection. (Contributed by AV, 7-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → 𝐸:𝑁1-1-onto𝑁)
 
Theoremsymgextsymg 17562* The extension of a permutation is an element of the extended symmetric group. (Contributed by AV, 9-Mar-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝑁𝑉𝐾𝑁𝑍𝑆) → 𝐸 ∈ (Base‘(SymGrp‘𝑁)))
 
Theoremsymgextres 17563* The restriction of the extension of a permutation, fixing the additional element, to the original domain. (Contributed by AV, 6-Jan-2019.)
𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))       ((𝐾𝑁𝑍𝑆) → (𝐸 ↾ (𝑁 ∖ {𝐾})) = 𝑍)
 
Theoremgsumccatsymgsn 17564 Homomorphic property of composites of permutations with a singleton. (Contributed by AV, 20-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐴𝑉𝑊 ∈ Word 𝐵𝑍𝐵) → (𝐺 Σg (𝑊 ++ ⟨“𝑍”⟩)) = ((𝐺 Σg 𝑊) ∘ 𝑍))
 
Theoremgsmsymgrfixlem1 17565* Lemma 1 for gsmsymgrfix 17566. (Contributed by AV, 20-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)       (((𝑊 ∈ Word 𝐵𝑃𝐵) ∧ (𝑁 ∈ Fin ∧ 𝐾𝑁) ∧ (∀𝑖 ∈ (0..^(#‘𝑊))((𝑊𝑖)‘𝐾) = 𝐾 → ((𝑆 Σg 𝑊)‘𝐾) = 𝐾)) → (∀𝑖 ∈ (0..^((#‘𝑊) + 1))(((𝑊 ++ ⟨“𝑃”⟩)‘𝑖)‘𝐾) = 𝐾 → ((𝑆 Σg (𝑊 ++ ⟨“𝑃”⟩))‘𝐾) = 𝐾))
 
Theoremgsmsymgrfix 17566* The composition of permutations fixing one element also fixes this element. (Contributed by AV, 20-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)       ((𝑁 ∈ Fin ∧ 𝐾𝑁𝑊 ∈ Word 𝐵) → (∀𝑖 ∈ (0..^(#‘𝑊))((𝑊𝑖)‘𝐾) = 𝐾 → ((𝑆 Σg 𝑊)‘𝐾) = 𝐾))
 
Theoremfvcosymgeq 17567* The values of two compositions of permutations are equal if the values of the composed permutations are pairwise equal. (Contributed by AV, 26-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (SymGrp‘𝑀)    &   𝑃 = (Base‘𝑍)    &   𝐼 = (𝑁𝑀)       ((𝐺𝐵𝐾𝑃) → ((𝑋𝐼 ∧ (𝐺𝑋) = (𝐾𝑋) ∧ ∀𝑛𝐼 (𝐹𝑛) = (𝐻𝑛)) → ((𝐹𝐺)‘𝑋) = ((𝐻𝐾)‘𝑋)))
 
Theoremgsmsymgreqlem1 17568* Lemma 1 for gsmsymgreq 17570. (Contributed by AV, 26-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (SymGrp‘𝑀)    &   𝑃 = (Base‘𝑍)    &   𝐼 = (𝑁𝑀)       (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝐽𝐼) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ((∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ∧ (𝐶𝐽) = (𝑅𝐽)) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝐽) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝐽)))
 
Theoremgsmsymgreqlem2 17569* Lemma 2 for gsmsymgreq 17570. (Contributed by AV, 26-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (SymGrp‘𝑀)    &   𝑃 = (Base‘𝑍)    &   𝐼 = (𝑁𝑀)       (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (#‘𝑋) = (#‘𝑌))) → ((∀𝑖 ∈ (0..^(#‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(#‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
 
Theoremgsmsymgreq 17570* Two combination of permutations moves an element of the intersection of the base sets of the permutations to the same element if each pair of corresponding permutations moves such an element to the same element. (Contributed by AV, 20-Jan-2019.)
𝑆 = (SymGrp‘𝑁)    &   𝐵 = (Base‘𝑆)    &   𝑍 = (SymGrp‘𝑀)    &   𝑃 = (Base‘𝑍)    &   𝐼 = (𝑁𝑀)       (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝑊 ∈ Word 𝐵𝑈 ∈ Word 𝑃 ∧ (#‘𝑊) = (#‘𝑈))) → (∀𝑖 ∈ (0..^(#‘𝑊))∀𝑛𝐼 ((𝑊𝑖)‘𝑛) = ((𝑈𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑊)‘𝑛) = ((𝑍 Σg 𝑈)‘𝑛)))
 
Theoremsymgfixelq 17571* A permutation of a set fixing an element of the set. (Contributed by AV, 4-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}       (𝐹𝑉 → (𝐹𝑄 ↔ (𝐹:𝑁1-1-onto𝑁 ∧ (𝐹𝐾) = 𝐾)))
 
Theoremsymgfixels 17572* The restriction of a permutation to a set with one element removed is an element of the restricted symmetric group if the restriction is a 1-1 onto function. (Contributed by AV, 4-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}    &   𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))    &   𝐷 = (𝑁 ∖ {𝐾})       (𝐹𝑉 → ((𝐹𝐷) ∈ 𝑆 ↔ (𝐹𝐷):𝐷1-1-onto𝐷))
 
Theoremsymgfixelsi 17573* 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 17574* 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 17575* 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 17576* Lemma 1 for symgfixfo 17577. (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 17577* 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 17578* 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 17580.

 
Syntaxcpmtr 17579 Syntax for the transposition generator function.
class pmTrsp
 
Definitiondf-pmtr 17580* 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 17581 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 17582 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 17583 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 17584 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 17585 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 17586 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 17587 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 17588* 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 17589* 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 17590 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 17591 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 17592 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 17593 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃):𝐷𝐷)
 
Theorempmtrmvd 17594 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 17595 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 17596 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 17597 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 17598* 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 17599 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 17600 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 ) ≠ ∅)
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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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