![]() |
Metamath
Proof Explorer Theorem List (p. 193 of 479) | < 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: | ![]() (1-30166) |
![]() (30167-31689) |
![]() (31690-47842) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cntzsubm 19201 | Centralizers in a monoid are submonoids. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.) |
β’ π΅ = (Baseβπ) & β’ π = (Cntzβπ) β β’ ((π β Mnd β§ π β π΅) β (πβπ) β (SubMndβπ)) | ||
Theorem | cntzsubg 19202 | Centralizers in a group are subgroups. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
β’ π΅ = (Baseβπ) & β’ π = (Cntzβπ) β β’ ((π β Grp β§ π β π΅) β (πβπ) β (SubGrpβπ)) | ||
Theorem | cntzidss 19203 | If the elements of π commute, the elements of a subset π also commute. (Contributed by Mario Carneiro, 25-Apr-2016.) |
β’ π = (CntzβπΊ) β β’ ((π β (πβπ) β§ π β π) β π β (πβπ)) | ||
Theorem | cntzmhm 19204 | Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.) |
β’ π = (CntzβπΊ) & β’ π = (Cntzβπ») β β’ ((πΉ β (πΊ MndHom π») β§ π΄ β (πβπ)) β (πΉβπ΄) β (πβ(πΉ β π))) | ||
Theorem | cntzmhm2 19205 | Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.) |
β’ π = (CntzβπΊ) & β’ π = (Cntzβπ») β β’ ((πΉ β (πΊ MndHom π») β§ π β (πβπ)) β (πΉ β π) β (πβ(πΉ β π))) | ||
Theorem | cntrsubgnsg 19206 | A central subgroup is normal. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
β’ π = (Cntrβπ) β β’ ((π β (SubGrpβπ) β§ π β π) β π β (NrmSGrpβπ)) | ||
Theorem | cntrnsg 19207 | The center of a group is a normal subgroup. (Contributed by Stefan O'Rear, 6-Sep-2015.) |
β’ π = (Cntrβπ) β β’ (π β Grp β π β (NrmSGrpβπ)) | ||
Syntax | coppg 19208 | The opposite group operation. |
class oppg | ||
Definition | df-oppg 19209 | Define an opposite group, which is the same as the original group but with addition written the other way around. df-oppr 20149 does the same thing for multiplication. (Contributed by Stefan O'Rear, 25-Aug-2015.) |
β’ oppg = (π€ β V β¦ (π€ sSet β¨(+gβndx), tpos (+gβπ€)β©)) | ||
Theorem | oppgval 19210 | Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
β’ + = (+gβπ ) & β’ π = (oppgβπ ) β β’ π = (π sSet β¨(+gβndx), tpos + β©) | ||
Theorem | oppgplusfval 19211 | Value of the addition operation of an opposite group. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
β’ + = (+gβπ ) & β’ π = (oppgβπ ) & β’ β = (+gβπ) β β’ β = tpos + | ||
Theorem | oppgplus 19212 | Value of the addition operation of an opposite ring. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Fan Zheng, 26-Jun-2016.) |
β’ + = (+gβπ ) & β’ π = (oppgβπ ) & β’ β = (+gβπ) β β’ (π β π) = (π + π) | ||
Theorem | setsplusg 19213 | The other components of an extensible structure remain unchanged if the +g component is set/substituted. (Contributed by Stefan O'Rear, 26-Aug-2015.) Generalisation of the former oppglem and mgplem. (Revised by AV, 18-Oct-2024.) |
β’ π = (π sSet β¨(+gβndx), πβ©) & β’ πΈ = Slot (πΈβndx) & β’ (πΈβndx) β (+gβndx) β β’ (πΈβπ ) = (πΈβπ) | ||
Theorem | oppglemOLD 19214 | Obsolete version of setsplusg 19213 as of 18-Oct-2024. Lemma for oppgbas 19215. (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 19215 | 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 19216 | Obsolete version of oppgbas 19215 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 19217 | Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
β’ π = (oppgβπ ) & β’ π½ = (TopSetβπ ) β β’ π½ = (TopSetβπ) | ||
Theorem | oppgtsetOLD 19218 | Obsolete version of oppgtset 19217 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 19219 | Topology of an opposite group. (Contributed by Mario Carneiro, 17-Sep-2015.) |
β’ π = (oppgβπ ) & β’ π½ = (TopOpenβπ ) β β’ π½ = (TopOpenβπ) | ||
Theorem | oppgmnd 19220 | 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 19221 | Bidirectional form of oppgmnd 19220. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
β’ π = (oppgβπ ) β β’ (π β Mnd β π β Mnd) | ||
Theorem | oppgid 19222 | 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 19223 | 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 19224 | Bidirectional form of oppggrp 19223. (Contributed by Stefan O'Rear, 26-Aug-2015.) |
β’ π = (oppgβπ ) β β’ (π β Grp β π β Grp) | ||
Theorem | oppginv 19225 | 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 19226 | 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 19227 | 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 19228 | Being a submonoid is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
β’ π = (oppgβπΊ) β β’ (SubMndβπΊ) = (SubMndβπ) | ||
Theorem | oppgsubg 19229 | Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 17-Sep-2015.) |
β’ π = (oppgβπΊ) β β’ (SubGrpβπΊ) = (SubGrpβπ) | ||
Theorem | oppgcntz 19230 | 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 19231 | 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 19232 | 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 19234. However, the
set is allowed to be empty, see symgbas0 19255. Hint: The symmetric groups
should not be confused with "symmetry groups" which is a different topic in
group theory.
| ||
Syntax | csymg 19233 | Extend class notation to include the class of symmetric groups. |
class SymGrp | ||
Definition | df-symg 19234* | 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 19235* | 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 19236* | Obsolete version of f1osetex 8852 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 19237* | 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 19238 | Obsolete as of 8-Aug-2024. π΅ β V follows immediatly from fvex 6904. (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 19239 | 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 19240 | 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 19241 | 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 19242 | 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 19243 | 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 19244 | 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 19245 | 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 19246 | 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 19247 | 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 19248 | 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 19249* | 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 19250 | 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 19251 | 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 19252 | 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 19253* | 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 19254* | 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 19255 | 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 19256 | 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 19257 | 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 19258 | 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 19259 | 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 19257. (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 19260 | 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 19261 | 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 19262 | 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 19263* | 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 19264* | Obsolete proof of symgvalstruct 19263 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 19265 | 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 19266 | 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 19267 | 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 19268 | 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 19269 | 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 19270 | The symmetric group on a set π΄ is a submonoid of the monoid of endofunctions on π΄. Alternate proof based on issubmndb 18685 and not on injsubmefmnd 18777 and sursubmefmnd 18776. (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 19271* | 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 19272* | The converse of galactghm 19271. 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 19273 | 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 19274* | 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 19275 | 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 19276* | 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 19277 | 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 19278 | 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 19279* | Lemma for cayley 19281. (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 19280* | Lemma for cayley 19281. (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 19281* | 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 19282* | 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 19283* | 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 19284* | 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 19285* | 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 19286* | 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 19287* | Lemma for symgextf1 19288. (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 19288* | 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 19289* | 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 19290* | 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 19291* | 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 19292* | 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 19293 | 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 19294* | Lemma 1 for gsmsymgrfix 19295. (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 19295* | 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 19296* | 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 19297* | Lemma 1 for gsmsymgreq 19299. (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 19298* | Lemma 2 for gsmsymgreq 19299. (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 19299* | 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 19300* | 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))) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |