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Theorem List for Metamath Proof Explorer - 17601-17700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremoppgplusfval 17601 Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
+ = (+g𝑅)    &   𝑂 = (oppg𝑅)    &    = (+g𝑂)        = tpos +
 
Theoremoppgplus 17602 Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
+ = (+g𝑅)    &   𝑂 = (oppg𝑅)    &    = (+g𝑂)       (𝑋 𝑌) = (𝑌 + 𝑋)
 
Theoremoppglem 17603 Lemma for oppgbas 17604. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)    &   𝐸 = Slot 𝑁    &   𝑁 ∈ ℕ    &   𝑁 ≠ 2       (𝐸𝑅) = (𝐸𝑂)
 
Theoremoppgbas 17604 Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)    &   𝐵 = (Base‘𝑅)       𝐵 = (Base‘𝑂)
 
Theoremoppgtset 17605 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝑅)    &   𝐽 = (TopSet‘𝑅)       𝐽 = (TopSet‘𝑂)
 
Theoremoppgtopn 17606 Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝑅)    &   𝐽 = (TopOpen‘𝑅)       𝐽 = (TopOpen‘𝑂)
 
Theoremoppgmnd 17607 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 17608 Bidirectional form of oppgmnd 17607. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Mnd ↔ 𝑂 ∈ Mnd)
 
Theoremoppgid 17609 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 17610 The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Grp → 𝑂 ∈ Grp)
 
Theoremoppggrpb 17611 Bidirectional form of oppggrp 17610. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝑅)       (𝑅 ∈ Grp ↔ 𝑂 ∈ Grp)
 
Theoremoppginv 17612 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 17613 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 17614 Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.)
𝑂 = (oppg𝐺)       (𝐺 ∈ Grp → 𝐺𝑔 𝑂)
 
Theoremoppgsubm 17615 Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (SubMnd‘𝐺) = (SubMnd‘𝑂)
 
Theoremoppgsubg 17616 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.)
𝑂 = (oppg𝐺)       (SubGrp‘𝐺) = (SubGrp‘𝑂)
 
Theoremoppgcntz 17617 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 17618 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 17619 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 17621. However, the set is allowed to be empty, see symgbas0 17637. 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 17623), 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 17650 and pgrpsubgsymg 17651). 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 17654, which is a (proper) subgroup of a symmetric group, see idressubgsymg 17653.

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 17659 are about fixed/moved elements.

 
Syntaxcsymg 17620 Extend class notation to include the class of symmetric groups.
class SymGrp
 
Definitiondf-symg 17621* 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 17622* 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 17623* 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 17624 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 17625 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 17626 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 17627 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 17628 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 17629 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 17630 The function value of a permutation. (Contributed by AV, 1-Jan-2019.)
𝐺 = (SymGrp‘𝐴)    &   𝐵 = (Base‘𝐺)       ((𝐹𝐵𝑋𝐴) → (𝐹𝑋) ∈ 𝐴)
 
Theoremsymgfvne 17631 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 17632* 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 17633 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 17634 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 17635* 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 17636* 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 17637 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 17638 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 17639 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 17640 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 17641 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 17639. (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 17642 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 17643 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 17644 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 17645 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 17646* 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 17647* The converse of galactghm 17646. 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 17648 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 17649* 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 17650 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 17651* 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 17652 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 17653 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 17654 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 17655* Lemma for cayley 17657. (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 17656* Lemma for cayley 17657. (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 17657* 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 17658* 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 17659* 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 17660* 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 17661* 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 17662* 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 17663* Lemma for symgextf1 17664. (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 17664* 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 17665* 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 17666* 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 17667* 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 17668* 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 17669 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 17670* Lemma 1 for gsmsymgrfix 17671. (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 17671* 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 17672* 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 17673* Lemma 1 for gsmsymgreq 17675. (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 17674* Lemma 2 for gsmsymgreq 17675. (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 17675* 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 17676* 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 17677* 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 17678* 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 17679* 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 17680* 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 17681* Lemma 1 for symgfixfo 17682. (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 17682* 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 17683* 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 17685.

 
Syntaxcpmtr 17684 Syntax for the transposition generator function.
class pmTrsp
 
Definitiondf-pmtr 17685* 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 17686 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 17687 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 17688 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 17689 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 17690 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 17691 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 17692 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 17693* 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 17694* 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 17695 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 17696 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 17697 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 17698 Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
𝑇 = (pmTrsp‘𝐷)       ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃):𝐷𝐷)
 
Theorempmtrmvd 17699 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 17700 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)
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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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