Home Metamath Proof ExplorerTheorem List (p. 177 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27159) Hilbert Space Explorer (27160-28684) Users' Mathboxes (28685-42360)

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)

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
 Copyright terms: Public domain < Previous  Next >