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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | oppglemOLD 19301 | Obsolete version of setsplusg 19300 as of 18-Oct-2024. Lemma for oppgbas 19302. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ π = (oppgβπ ) & β’ πΈ = Slot π & β’ π β β & β’ π β 2 β β’ (πΈβπ ) = (πΈβπ) | ||
Theorem | oppgbas 19302 | Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
β’ π = (oppgβπ ) & β’ π΅ = (Baseβπ ) β β’ π΅ = (Baseβπ) | ||
Theorem | oppgbasOLD 19303 | Obsolete version of oppgbas 19302 as of 18-Oct-2024. Base set of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ π = (oppgβπ ) & β’ π΅ = (Baseβπ ) β β’ π΅ = (Baseβπ) | ||
Theorem | oppgtset 19304 | Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
β’ π = (oppgβπ ) & β’ π½ = (TopSetβπ ) β β’ π½ = (TopSetβπ) | ||
Theorem | oppgtsetOLD 19305 | Obsolete version of oppgtset 19304 as of 18-Oct-2024. Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ π = (oppgβπ ) & β’ π½ = (TopSetβπ ) β β’ π½ = (TopSetβπ) | ||
Theorem | oppgtopn 19306 | Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
β’ π = (oppgβπ ) & β’ π½ = (TopOpenβπ ) β β’ π½ = (TopOpenβπ) | ||
Theorem | oppgmnd 19307 | The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) |
β’ π = (oppgβπ ) β β’ (π β Mnd β π β Mnd) | ||
Theorem | oppgmndb 19308 | Bidirectional form of oppgmnd 19307. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
β’ π = (oppgβπ ) β β’ (π β Mnd β π β Mnd) | ||
Theorem | oppgid 19309 | Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) |
β’ π = (oppgβπ ) & β’ 0 = (0gβπ ) β β’ 0 = (0gβπ) | ||
Theorem | oppggrp 19310 | The opposite of a group is a group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
β’ π = (oppgβπ ) β β’ (π β Grp β π β Grp) | ||
Theorem | oppggrpb 19311 | Bidirectional form of oppggrp 19310. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
β’ π = (oppgβπ ) β β’ (π β Grp β π β Grp) | ||
Theorem | oppginv 19312 | Inverses in a group are a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
β’ π = (oppgβπ ) & β’ πΌ = (invgβπ ) β β’ (π β Grp β πΌ = (invgβπ)) | ||
Theorem | invoppggim 19313 | The inverse is an antiautomorphism on any group. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
β’ π = (oppgβπΊ) & β’ πΌ = (invgβπΊ) β β’ (πΊ β Grp β πΌ β (πΊ GrpIso π)) | ||
Theorem | oppggic 19314 | Every group is (naturally) isomorphic to its opposite. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
β’ π = (oppgβπΊ) β β’ (πΊ β Grp β πΊ βπ π) | ||
Theorem | oppgsubm 19315 | Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
β’ π = (oppgβπΊ) β β’ (SubMndβπΊ) = (SubMndβπ) | ||
Theorem | oppgsubg 19316 | Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
β’ π = (oppgβπΊ) β β’ (SubGrpβπΊ) = (SubGrpβπ) | ||
Theorem | oppgcntz 19317 | A centralizer in a group is the same as the centralizer in the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
β’ π = (oppgβπΊ) & β’ π = (CntzβπΊ) β β’ (πβπ΄) = ((Cntzβπ)βπ΄) | ||
Theorem | oppgcntr 19318 | The center of a group is the same as the center of the opposite group. (Contributed by Mario Carneiro, 21-Apr-2016.) |
β’ π = (oppgβπΊ) & β’ π = (CntrβπΊ) β β’ π = (Cntrβπ) | ||
Theorem | gsumwrev 19319 | 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.) |
β’ π΅ = (Baseβπ) & β’ π = (oppgβπ) β β’ ((π β Mnd β§ π β Word π΅) β (π Ξ£g π) = (π Ξ£g (reverseβπ))) | ||
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 π΄ as the set of one-to-one onto functions
from π΄ to itself under function composition, see df-symg 19321. However, the
set is allowed to be empty, see symgbas0 19342. Hint: The symmetric groups
should not be confused with "symmetry groups" which is a different topic in
group theory.
| ||
Syntax | csymg 19320 | Extend class notation to include the class of symmetric groups. |
class SymGrp | ||
Definition | df-symg 19321* | Define the symmetric group on set π₯. We represent the group as the set of one-to-one onto functions from π₯ to itself under function composition, and topologize it as a function space assuming the set is discrete. This definition is based on the fact that a symmetric group is a restriction of the monoid of endofunctions. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by AV, 28-Mar-2024.) |
β’ SymGrp = (π₯ β V β¦ ((EndoFMndβπ₯) βΎs {β β£ β:π₯β1-1-ontoβπ₯})) | ||
Theorem | symgval 19322* | The value of the symmetric group function at π΄. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 28-Mar-2024.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = {π₯ β£ π₯:π΄β1-1-ontoβπ΄} β β’ πΊ = ((EndoFMndβπ΄) βΎs π΅) | ||
Theorem | permsetexOLD 19323* | Obsolete version of f1osetex 8871 as of 8-Aug-2024. (Contributed by AV, 30-Mar-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ (π΄ β π β {π β£ π:π΄β1-1-ontoβπ΄} β V) | ||
Theorem | symgbas 19324* | The base set of the symmetric group. (Contributed by Mario Carneiro, 12-Jan-2015.) (Proof shortened by AV, 29-Mar-2024.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ π΅ = {π₯ β£ π₯:π΄β1-1-ontoβπ΄} | ||
Theorem | symgbasexOLD 19325 | Obsolete as of 8-Aug-2024. π΅ β V follows immediatly from fvex 6903. (Contributed by AV, 30-Mar-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (π΄ β π β π΅ β V) | ||
Theorem | elsymgbas2 19326 | Two ways of saying a function is a 1-1-onto mapping of A to itself. (Contributed by Mario Carneiro, 28-Jan-2015.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (πΉ β π β (πΉ β π΅ β πΉ:π΄β1-1-ontoβπ΄)) | ||
Theorem | elsymgbas 19327 | 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.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (π΄ β π β (πΉ β π΅ β πΉ:π΄β1-1-ontoβπ΄)) | ||
Theorem | symgbasf1o 19328 | Elements in the symmetric group are 1-1 onto functions. (Contributed by SO, 9-Jul-2018.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (πΉ β π΅ β πΉ:π΄β1-1-ontoβπ΄) | ||
Theorem | symgbasf 19329 | A permutation (element of the symmetric group) is a function from a set into itself. (Contributed by AV, 1-Jan-2019.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (πΉ β π΅ β πΉ:π΄βΆπ΄) | ||
Theorem | symgbasmap 19330 | A permutation (element of the symmetric group) is a mapping (or set exponentiation) from a set into itself. (Contributed by AV, 30-Mar-2024.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (πΉ β π΅ β πΉ β (π΄ βm π΄)) | ||
Theorem | symghash 19331 | The symmetric group on π objects has cardinality π!. (Contributed by Mario Carneiro, 22-Jan-2015.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (π΄ β Fin β (β―βπ΅) = (!β(β―βπ΄))) | ||
Theorem | symgbasfi 19332 | The symmetric group on a finite index set is finite. (Contributed by SO, 9-Jul-2018.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (π΄ β Fin β π΅ β Fin) | ||
Theorem | symgfv 19333 | The function value of a permutation. (Contributed by AV, 1-Jan-2019.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ ((πΉ β π΅ β§ π β π΄) β (πΉβπ) β π΄) | ||
Theorem | symgfvne 19334 | The function values of a permutation for different arguments are different. (Contributed by AV, 8-Jan-2019.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ ((πΉ β π΅ β§ π β π΄ β§ π β π΄) β ((πΉβπ) = π β (π β π β (πΉβπ) β π))) | ||
Theorem | symgressbas 19335 | The symmetric group on π΄ characterized as structure restriction of the monoid of endofunctions on π΄ to its base set. (Contributed by AV, 30-Mar-2024.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) & β’ π = (EndoFMndβπ΄) β β’ πΊ = (π βΎs π΅) | ||
Theorem | symgplusg 19336* | 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.) (Proof shortened by AV, 19-Feb-2024.) (Revised by AV, 29-Mar-2024.) (Proof shortened by AV, 14-Aug-2024.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (π΄ βm π΄) & β’ + = (+gβπΊ) β β’ + = (π β π΅, π β π΅ β¦ (π β π)) | ||
Theorem | symgov 19337 | The value of the group operation of the symmetric group on π΄. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 28-Jan-2015.) (Revised by AV, 30-Mar-2024.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) β β’ ((π β π΅ β§ π β π΅) β (π + π) = (π β π)) | ||
Theorem | symgcl 19338 | The group operation of the symmetric group on π΄ is closed, i.e. a magma. (Contributed by Mario Carneiro, 12-Jan-2015.) (Revised by Mario Carneiro, 28-Jan-2015.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) & β’ + = (+gβπΊ) β β’ ((π β π΅ β§ π β π΅) β (π + π) β π΅) | ||
Theorem | idresperm 19339 | The identity function restricted to a set is a permutation of this set. (Contributed by AV, 17-Mar-2019.) |
β’ πΊ = (SymGrpβπ΄) β β’ (π΄ β π β ( I βΎ π΄) β (BaseβπΊ)) | ||
Theorem | symgmov1 19340* | For a permutation of a set, each element of the set replaces an(other) element of the set. (Contributed by AV, 2-Jan-2019.) |
β’ π = (Baseβ(SymGrpβπ)) β β’ (π β π β βπ β π βπ β π (πβπ) = π) | ||
Theorem | symgmov2 19341* | 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.) |
β’ π = (Baseβ(SymGrpβπ)) β β’ (π β π β βπ β π βπ β π (πβπ) = π) | ||
Theorem | symgbas0 19342 | The base set of the symmetric group on the empty set is the singleton containing the empty set. (Contributed by AV, 27-Feb-2019.) |
β’ (Baseβ(SymGrpββ )) = {β } | ||
Theorem | symg1hash 19343 | The symmetric group on a singleton has cardinality 1. (Contributed by AV, 9-Dec-2018.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) & β’ π΄ = {πΌ} β β’ (πΌ β π β (β―βπ΅) = 1) | ||
Theorem | symg1bas 19344 | The symmetric group on a singleton is the symmetric group S1 consisting of the identity only. (Contributed by AV, 9-Dec-2018.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) & β’ π΄ = {πΌ} β β’ (πΌ β π β π΅ = {{β¨πΌ, πΌβ©}}) | ||
Theorem | symg2hash 19345 | The symmetric group on a (proper) pair has cardinality 2. (Contributed by AV, 9-Dec-2018.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) & β’ π΄ = {πΌ, π½} β β’ ((πΌ β π β§ π½ β π β§ πΌ β π½) β (β―βπ΅) = 2) | ||
Theorem | symg2bas 19346 | The symmetric group on a pair is the symmetric group S2 consisting of the identity and the transposition. Notice that this statement is valid for proper pairs only. In the case that both elements are identical, i.e., the pairs are actually singletons, this theorem would be about S1, see Theorem symg1bas 19344. (Contributed by AV, 9-Dec-2018.) (Proof shortened by AV, 16-Jun-2022.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) & β’ π΄ = {πΌ, π½} β β’ ((πΌ β π β§ π½ β π) β π΅ = {{β¨πΌ, πΌβ©, β¨π½, π½β©}, {β¨πΌ, π½β©, β¨π½, πΌβ©}}) | ||
Theorem | 0symgefmndeq 19347 | The symmetric group on the empty set is identical with the monoid of endofunctions on the empty set. (Contributed by AV, 30-Mar-2024.) |
β’ (EndoFMndββ ) = (SymGrpββ ) | ||
Theorem | snsymgefmndeq 19348 | The symmetric group on a singleton π΄ is identical with the monoid of endofunctions on π΄. (Contributed by AV, 31-Mar-2024.) |
β’ (π΄ = {π} β (EndoFMndβπ΄) = (SymGrpβπ΄)) | ||
Theorem | symgpssefmnd 19349 | For a set π΄ with more than one element, the symmetric group on π΄ is a proper subset of the monoid of endofunctions on π΄. (Contributed by AV, 31-Mar-2024.) |
β’ π = (EndoFMndβπ΄) & β’ πΊ = (SymGrpβπ΄) β β’ ((π΄ β π β§ 1 < (β―βπ΄)) β (BaseβπΊ) β (Baseβπ)) | ||
Theorem | symgvalstruct 19350* | The value of the symmetric group function at π΄ represented as extensible structure with three slots. This corresponds to the former definition of SymGrp. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 31-Mar-2024.) (Proof shortened by AV, 6-Nov-2024.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = {π₯ β£ π₯:π΄β1-1-ontoβπ΄} & β’ π = (π΄ βm π΄) & β’ + = (π β π, π β π β¦ (π β π)) & β’ π½ = (βtβ(π΄ Γ {π« π΄})) β β’ (π΄ β π β πΊ = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(TopSetβndx), π½β©}) | ||
Theorem | symgvalstructOLD 19351* | Obsolete proof of symgvalstruct 19350 as of 6-Nov-2024. The value of the symmetric group function at π΄ represented as extensible structure with three slots. This corresponds to the former definition of SymGrp. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 31-Mar-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = {π₯ β£ π₯:π΄β1-1-ontoβπ΄} & β’ π = (π΄ βm π΄) & β’ + = (π β π, π β π β¦ (π β π)) & β’ π½ = (βtβ(π΄ Γ {π« π΄})) β β’ (π΄ β π β πΊ = {β¨(Baseβndx), π΅β©, β¨(+gβndx), + β©, β¨(TopSetβndx), π½β©}) | ||
Theorem | symgsubmefmnd 19352 | The symmetric group on a set π΄ is a submonoid of the monoid of endofunctions on π΄. (Contributed by AV, 18-Feb-2024.) |
β’ π = (EndoFMndβπ΄) & β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (π΄ β π β π΅ β (SubMndβπ)) | ||
Theorem | symgtset 19353 | The topology of the symmetric group on π΄. 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.) (Proof revised by AV, 30-Mar-2024.) |
β’ πΊ = (SymGrpβπ΄) β β’ (π΄ β π β (βtβ(π΄ Γ {π« π΄})) = (TopSetβπΊ)) | ||
Theorem | symggrp 19354 | The symmetric group on a set π΄ is a group. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 28-Jan-2024.) |
β’ πΊ = (SymGrpβπ΄) β β’ (π΄ β π β πΊ β Grp) | ||
Theorem | symgid 19355 | The group identity element of the symmetric group on a set π΄. (Contributed by Paul Chapman, 25-Jul-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) (Proof shortened by AV, 1-Apr-2024.) |
β’ πΊ = (SymGrpβπ΄) β β’ (π΄ β π β ( I βΎ π΄) = (0gβπΊ)) | ||
Theorem | symginv 19356 | 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.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) & β’ π = (invgβπΊ) β β’ (πΉ β π΅ β (πβπΉ) = β‘πΉ) | ||
Theorem | symgsubmefmndALT 19357 | The symmetric group on a set π΄ is a submonoid of the monoid of endofunctions on π΄. Alternate proof based on issubmndb 18756 and not on injsubmefmnd 18848 and sursubmefmnd 18847. (Contributed by AV, 18-Feb-2024.) (Revised by AV, 30-Mar-2024.) (New usage is discouraged.) (Proof modification is discouraged.) |
β’ π = (EndoFMndβπ΄) & β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (π΄ β π β π΅ β (SubMndβπ)) | ||
Theorem | galactghm 19358* | The currying of a group action is a group homomorphism between the group πΊ and the symmetric group (SymGrpβπ). (Contributed by FL, 17-May-2010.) (Proof shortened by Mario Carneiro, 13-Jan-2015.) |
β’ π = (BaseβπΊ) & β’ π» = (SymGrpβπ) & β’ πΉ = (π₯ β π β¦ (π¦ β π β¦ (π₯ β π¦))) β β’ ( β β (πΊ GrpAct π) β πΉ β (πΊ GrpHom π»)) | ||
Theorem | lactghmga 19359* | The converse of galactghm 19358. The uncurrying of a homomorphism into (SymGrpβπ) 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.) |
β’ π = (BaseβπΊ) & β’ π» = (SymGrpβπ) & β’ β = (π₯ β π, π¦ β π β¦ ((πΉβπ₯)βπ¦)) β β’ (πΉ β (πΊ GrpHom π») β β β (πΊ GrpAct π)) | ||
Theorem | symgtopn 19360 | The topology of the symmetric group on π΄. (Contributed by Mario Carneiro, 29-Aug-2015.) |
β’ πΊ = (SymGrpβπ) & β’ π΅ = (BaseβπΊ) β β’ (π β π β ((βtβ(π Γ {π« π})) βΎt π΅) = (TopOpenβπΊ)) | ||
Theorem | symgga 19361* | The symmetric group induces a group action on its base set. (Contributed by Mario Carneiro, 24-Jan-2015.) |
β’ πΊ = (SymGrpβπ) & β’ π΅ = (BaseβπΊ) & β’ πΉ = (π β π΅, π₯ β π β¦ (πβπ₯)) β β’ (π β π β πΉ β (πΊ GrpAct π)) | ||
Theorem | pgrpsubgsymgbi 19362 | Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ (π΄ β π β (π β (SubGrpβπΊ) β (π β π΅ β§ (πΊ βΎs π) β Grp))) | ||
Theorem | pgrpsubgsymg 19363* | Every permutation group is a subgroup of the corresponding symmetric group. (Contributed by AV, 14-Mar-2019.) (Revised by AV, 30-Mar-2024.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) & β’ πΉ = (Baseβπ) β β’ (π΄ β π β ((π β Grp β§ πΉ β π΅ β§ (+gβπ) = (π β πΉ, π β πΉ β¦ (π β π))) β πΉ β (SubGrpβπΊ))) | ||
Theorem | idressubgsymg 19364 | 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.) |
β’ πΊ = (SymGrpβπ΄) β β’ (π΄ β π β {( I βΎ π΄)} β (SubGrpβπΊ)) | ||
Theorem | idrespermg 19365 | 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.) |
β’ πΊ = (SymGrpβπ΄) & β’ πΈ = (πΊ βΎs {( I βΎ π΄)}) β β’ (π΄ β π β (πΈ β Grp β§ (BaseβπΈ) β (BaseβπΊ))) | ||
Theorem | cayleylem1 19366* | Lemma for cayley 19368. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) |
β’ π = (BaseβπΊ) & β’ + = (+gβπΊ) & β’ 0 = (0gβπΊ) & β’ π» = (SymGrpβπ) & β’ π = (Baseβπ») & β’ πΉ = (π β π β¦ (π β π β¦ (π + π))) β β’ (πΊ β Grp β πΉ β (πΊ GrpHom π»)) | ||
Theorem | cayleylem2 19367* | Lemma for cayley 19368. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) |
β’ π = (BaseβπΊ) & β’ + = (+gβπΊ) & β’ 0 = (0gβπΊ) & β’ π» = (SymGrpβπ) & β’ π = (Baseβπ») & β’ πΉ = (π β π β¦ (π β π β¦ (π + π))) β β’ (πΊ β Grp β πΉ:πβ1-1βπ) | ||
Theorem | cayley 19368* | Cayley's Theorem (constructive version): given group πΊ, πΉ is an isomorphism between πΊ and the subgroup π of the symmetric group π» on the underlying set π of πΊ. See also Theorem 3.15 in [Rotman] p. 42. (Contributed by Paul Chapman, 3-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Jan-2015.) |
β’ π = (BaseβπΊ) & β’ π» = (SymGrpβπ) & β’ + = (+gβπΊ) & β’ πΉ = (π β π β¦ (π β π β¦ (π + π))) & β’ π = ran πΉ β β’ (πΊ β Grp β (π β (SubGrpβπ») β§ πΉ β (πΊ GrpHom (π» βΎs π)) β§ πΉ:πβ1-1-ontoβπ)) | ||
Theorem | cayleyth 19369* | Cayley's Theorem (existence version): every group πΊ is isomorphic to a subgroup of the symmetric group on the underlying set of πΊ. (For any group πΊ there exists an isomorphism π between πΊ and a subgroup β of the symmetric group on the underlying set of πΊ.) See also Theorem 3.15 in [Rotman] p. 42. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by Mario Carneiro, 13-Jan-2015.) |
β’ π = (BaseβπΊ) & β’ π» = (SymGrpβπ) β β’ (πΊ β Grp β βπ β (SubGrpβπ»)βπ β (πΊ GrpHom (π» βΎs π ))π:πβ1-1-ontoβπ ) | ||
Theorem | symgfix2 19370* | 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.) |
β’ π = (Baseβ(SymGrpβπ)) β β’ (πΏ β π β (π β (π β {π β π β£ (πβπΎ) = πΏ}) β βπ β (π β {πΎ})(πβπ) = πΏ)) | ||
Theorem | symgextf 19371* | The extension of a permutation, fixing the additional element, is a function. (Contributed by AV, 6-Jan-2019.) |
β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ πΈ = (π₯ β π β¦ if(π₯ = πΎ, πΎ, (πβπ₯))) β β’ ((πΎ β π β§ π β π) β πΈ:πβΆπ) | ||
Theorem | symgextfv 19372* | 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.) |
β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ πΈ = (π₯ β π β¦ if(π₯ = πΎ, πΎ, (πβπ₯))) β β’ ((πΎ β π β§ π β π) β (π β (π β {πΎ}) β (πΈβπ) = (πβπ))) | ||
Theorem | symgextfve 19373* | The function value of the extension of a permutation, fixing the additional element, for the additional element. (Contributed by AV, 6-Jan-2019.) |
β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ πΈ = (π₯ β π β¦ if(π₯ = πΎ, πΎ, (πβπ₯))) β β’ (πΎ β π β (π = πΎ β (πΈβπ) = πΎ)) | ||
Theorem | symgextf1lem 19374* | Lemma for symgextf1 19375. (Contributed by AV, 6-Jan-2019.) |
β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ πΈ = (π₯ β π β¦ if(π₯ = πΎ, πΎ, (πβπ₯))) β β’ ((πΎ β π β§ π β π) β ((π β (π β {πΎ}) β§ π β {πΎ}) β (πΈβπ) β (πΈβπ))) | ||
Theorem | symgextf1 19375* | The extension of a permutation, fixing the additional element, is a 1-1 function. (Contributed by AV, 6-Jan-2019.) |
β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ πΈ = (π₯ β π β¦ if(π₯ = πΎ, πΎ, (πβπ₯))) β β’ ((πΎ β π β§ π β π) β πΈ:πβ1-1βπ) | ||
Theorem | symgextfo 19376* | The extension of a permutation, fixing the additional element, is an onto function. (Contributed by AV, 7-Jan-2019.) |
β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ πΈ = (π₯ β π β¦ if(π₯ = πΎ, πΎ, (πβπ₯))) β β’ ((πΎ β π β§ π β π) β πΈ:πβontoβπ) | ||
Theorem | symgextf1o 19377* | The extension of a permutation, fixing the additional element, is a bijection. (Contributed by AV, 7-Jan-2019.) |
β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ πΈ = (π₯ β π β¦ if(π₯ = πΎ, πΎ, (πβπ₯))) β β’ ((πΎ β π β§ π β π) β πΈ:πβ1-1-ontoβπ) | ||
Theorem | symgextsymg 19378* | The extension of a permutation is an element of the extended symmetric group. (Contributed by AV, 9-Mar-2019.) |
β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ πΈ = (π₯ β π β¦ if(π₯ = πΎ, πΎ, (πβπ₯))) β β’ ((π β π β§ πΎ β π β§ π β π) β πΈ β (Baseβ(SymGrpβπ))) | ||
Theorem | symgextres 19379* | The restriction of the extension of a permutation, fixing the additional element, to the original domain. (Contributed by AV, 6-Jan-2019.) |
β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ πΈ = (π₯ β π β¦ if(π₯ = πΎ, πΎ, (πβπ₯))) β β’ ((πΎ β π β§ π β π) β (πΈ βΎ (π β {πΎ})) = π) | ||
Theorem | gsumccatsymgsn 19380 | Homomorphic property of composites of permutations with a singleton. (Contributed by AV, 20-Jan-2019.) |
β’ πΊ = (SymGrpβπ΄) & β’ π΅ = (BaseβπΊ) β β’ ((π΄ β π β§ π β Word π΅ β§ π β π΅) β (πΊ Ξ£g (π ++ β¨βπββ©)) = ((πΊ Ξ£g π) β π)) | ||
Theorem | gsmsymgrfixlem1 19381* | Lemma 1 for gsmsymgrfix 19382. (Contributed by AV, 20-Jan-2019.) |
β’ π = (SymGrpβπ) & β’ π΅ = (Baseβπ) β β’ (((π β Word π΅ β§ π β π΅) β§ (π β Fin β§ πΎ β π) β§ (βπ β (0..^(β―βπ))((πβπ)βπΎ) = πΎ β ((π Ξ£g π)βπΎ) = πΎ)) β (βπ β (0..^((β―βπ) + 1))(((π ++ β¨βπββ©)βπ)βπΎ) = πΎ β ((π Ξ£g (π ++ β¨βπββ©))βπΎ) = πΎ)) | ||
Theorem | gsmsymgrfix 19382* | The composition of permutations fixing one element also fixes this element. (Contributed by AV, 20-Jan-2019.) |
β’ π = (SymGrpβπ) & β’ π΅ = (Baseβπ) β β’ ((π β Fin β§ πΎ β π β§ π β Word π΅) β (βπ β (0..^(β―βπ))((πβπ)βπΎ) = πΎ β ((π Ξ£g π)βπΎ) = πΎ)) | ||
Theorem | fvcosymgeq 19383* | 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.) |
β’ π = (SymGrpβπ) & β’ π΅ = (Baseβπ) & β’ π = (SymGrpβπ) & β’ π = (Baseβπ) & β’ πΌ = (π β© π) β β’ ((πΊ β π΅ β§ πΎ β π) β ((π β πΌ β§ (πΊβπ) = (πΎβπ) β§ βπ β πΌ (πΉβπ) = (π»βπ)) β ((πΉ β πΊ)βπ) = ((π» β πΎ)βπ))) | ||
Theorem | gsmsymgreqlem1 19384* | Lemma 1 for gsmsymgreq 19386. (Contributed by AV, 26-Jan-2019.) |
β’ π = (SymGrpβπ) & β’ π΅ = (Baseβπ) & β’ π = (SymGrpβπ) & β’ π = (Baseβπ) & β’ πΌ = (π β© π) β β’ (((π β Fin β§ π β Fin β§ π½ β πΌ) β§ ((π β Word π΅ β§ πΆ β π΅) β§ (π β Word π β§ π β π) β§ (β―βπ) = (β―βπ))) β ((βπ β πΌ ((π Ξ£g π)βπ) = ((π Ξ£g π)βπ) β§ (πΆβπ½) = (π βπ½)) β ((π Ξ£g (π ++ β¨βπΆββ©))βπ½) = ((π Ξ£g (π ++ β¨βπ ββ©))βπ½))) | ||
Theorem | gsmsymgreqlem2 19385* | Lemma 2 for gsmsymgreq 19386. (Contributed by AV, 26-Jan-2019.) |
β’ π = (SymGrpβπ) & β’ π΅ = (Baseβπ) & β’ π = (SymGrpβπ) & β’ π = (Baseβπ) & β’ πΌ = (π β© π) β β’ (((π β Fin β§ π β Fin) β§ ((π β Word π΅ β§ πΆ β π΅) β§ (π β Word π β§ π β π) β§ (β―βπ) = (β―βπ))) β ((βπ β (0..^(β―βπ))βπ β πΌ ((πβπ)βπ) = ((πβπ)βπ) β βπ β πΌ ((π Ξ£g π)βπ) = ((π Ξ£g π)βπ)) β (βπ β (0..^(β―β(π ++ β¨βπΆββ©)))βπ β πΌ (((π ++ β¨βπΆββ©)βπ)βπ) = (((π ++ β¨βπ ββ©)βπ)βπ) β βπ β πΌ ((π Ξ£g (π ++ β¨βπΆββ©))βπ) = ((π Ξ£g (π ++ β¨βπ ββ©))βπ)))) | ||
Theorem | gsmsymgreq 19386* | 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.) |
β’ π = (SymGrpβπ) & β’ π΅ = (Baseβπ) & β’ π = (SymGrpβπ) & β’ π = (Baseβπ) & β’ πΌ = (π β© π) β β’ (((π β Fin β§ π β Fin) β§ (π β Word π΅ β§ π β Word π β§ (β―βπ) = (β―βπ))) β (βπ β (0..^(β―βπ))βπ β πΌ ((πβπ)βπ) = ((πβπ)βπ) β βπ β πΌ ((π Ξ£g π)βπ) = ((π Ξ£g π)βπ))) | ||
Theorem | symgfixelq 19387* | A permutation of a set fixing an element of the set. (Contributed by AV, 4-Jan-2019.) |
β’ π = (Baseβ(SymGrpβπ)) & β’ π = {π β π β£ (πβπΎ) = πΎ} β β’ (πΉ β π β (πΉ β π β (πΉ:πβ1-1-ontoβπ β§ (πΉβπΎ) = πΎ))) | ||
Theorem | symgfixels 19388* | 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.) |
β’ π = (Baseβ(SymGrpβπ)) & β’ π = {π β π β£ (πβπΎ) = πΎ} & β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ π· = (π β {πΎ}) β β’ (πΉ β π β ((πΉ βΎ π·) β π β (πΉ βΎ π·):π·β1-1-ontoβπ·)) | ||
Theorem | symgfixelsi 19389* | 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.) |
β’ π = (Baseβ(SymGrpβπ)) & β’ π = {π β π β£ (πβπΎ) = πΎ} & β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ π· = (π β {πΎ}) β β’ ((πΎ β π β§ πΉ β π) β (πΉ βΎ π·) β π) | ||
Theorem | symgfixf 19390* | 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.) |
β’ π = (Baseβ(SymGrpβπ)) & β’ π = {π β π β£ (πβπΎ) = πΎ} & β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ π» = (π β π β¦ (π βΎ (π β {πΎ}))) β β’ (πΎ β π β π»:πβΆπ) | ||
Theorem | symgfixf1 19391* | 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.) |
β’ π = (Baseβ(SymGrpβπ)) & β’ π = {π β π β£ (πβπΎ) = πΎ} & β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ π» = (π β π β¦ (π βΎ (π β {πΎ}))) β β’ (πΎ β π β π»:πβ1-1βπ) | ||
Theorem | symgfixfolem1 19392* | Lemma 1 for symgfixfo 19393. (Contributed by AV, 7-Jan-2019.) |
β’ π = (Baseβ(SymGrpβπ)) & β’ π = {π β π β£ (πβπΎ) = πΎ} & β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ π» = (π β π β¦ (π βΎ (π β {πΎ}))) & β’ πΈ = (π₯ β π β¦ if(π₯ = πΎ, πΎ, (πβπ₯))) β β’ ((π β π β§ πΎ β π β§ π β π) β πΈ β π) | ||
Theorem | symgfixfo 19393* | 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.) |
β’ π = (Baseβ(SymGrpβπ)) & β’ π = {π β π β£ (πβπΎ) = πΎ} & β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ π» = (π β π β¦ (π βΎ (π β {πΎ}))) β β’ ((π β π β§ πΎ β π) β π»:πβontoβπ) | ||
Theorem | symgfixf1o 19394* | 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.) |
β’ π = (Baseβ(SymGrpβπ)) & β’ π = {π β π β£ (πβπΎ) = πΎ} & β’ π = (Baseβ(SymGrpβ(π β {πΎ}))) & β’ π» = (π β π β¦ (π βΎ (π β {πΎ}))) β β’ ((π β π β§ πΎ β π) β π»:πβ1-1-ontoβπ) | ||
Transpositions are special cases of "cycles" as defined in [Rotman] p. 28: "Let
i1 , i2 , ... , ir be distinct integers
between 1 and n. If Ξ± in Sn fixes the other integers and
Ξ±(i1) = i2, Ξ±(i2) = i3,
..., Ξ±(ir-1 ) = ir, Ξ±(ir) =
i1, then Ξ± is an r-cycle. We also say that Ξ± is a
cycle of length r." and in [Rotman] p. 31: "A 2-cycle is also called
transposition.".
| ||
Syntax | cpmtr 19395 | Syntax for the transposition generator function. |
class pmTrsp | ||
Definition | df-pmtr 19396* | Define a function that generates the transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
β’ pmTrsp = (π β V β¦ (π β {π¦ β π« π β£ π¦ β 2o} β¦ (π§ β π β¦ if(π§ β π, βͺ (π β {π§}), π§)))) | ||
Theorem | f1omvdmvd 19397 | 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.) |
β’ ((πΉ:π΄β1-1-ontoβπ΄ β§ π β dom (πΉ β I )) β (πΉβπ) β (dom (πΉ β I ) β {π})) | ||
Theorem | f1omvdcnv 19398 | A permutation and its inverse move the same points. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
β’ (πΉ:π΄β1-1-ontoβπ΄ β dom (β‘πΉ β I ) = dom (πΉ β I )) | ||
Theorem | mvdco 19399 | Composing two permutations moves at most the union of the points. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
β’ dom ((πΉ β πΊ) β I ) β (dom (πΉ β I ) βͺ dom (πΊ β I )) | ||
Theorem | f1omvdconj 19400 | Conjugation of a permutation takes the image of the moved subclass. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
β’ ((πΉ:π΄βΆπ΄ β§ πΊ:π΄β1-1-ontoβπ΄) β dom (((πΊ β πΉ) β β‘πΊ) β I ) = (πΊ β dom (πΉ β I ))) |
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