, t′ ∈ G that t′t : U t−→ U t ′ −→ U is also in G. These three conditions, the existence of the identity, the existence of an inverse, and closure under composition, define a transformation group G = {t : U → U}, i.e., a group action on a set U. Equivalently, a set representation of a group G is given by a group homomorphism T : G→ S (U), where S (U) is the symmetric group of permutations t of the set U (and where the transformation group {t : U → U} ⊆ S (U) is the image of the map). An abstract group satisfies these three conditions where the composition is also required to be associative in the sense that for any t, t′, t′′ ∈ G, (t′′t′) t = t′′ (t′t). For a transformation group, the composition is automatically associative. This connection between groups and equivalence relations or partitions is well-known, e.g., [6], and is probably as old as the notion of a group. Instead of elements u, u′ ∈ U being collected in the same block by have the same attribute value f (u) = f (u′), the group transformations take any element u to a "similar" or "symmetric" element t (u) = u′. A subset S ⊆ U is invariant under G if for any t ∈ G, t (S) ⊆ S. A minimal invariant subset is an orbit, and the partition defined by the transformation group G is the set partition of orbits. That is the bottom-up method of defining a set partition since we don't begin with some attribute-value already assigned to the elements of U. What is the significance of the blocks in the partition of minimal invariant subsets? Often the treatment of symmetry groups focuses on what is invariant or conserved, e.g., the perspective of Noether's theorem [4]. There is another perspective with which to view the representations of symmetry groups. To represent an indefinite reality, there is first some notion of the fully definite eigen-alternatives that are then collected together or superposed to represent something indefinite between those alternatives. What determines the set of definite eigen-alternatives? Given a set of symmetries on a set, in what different ways can there be distinct subsets that still satisfy the constraints of the symmetry operations? The minimal invariant subsets or orbits of
, t′ ∈ G that t′t: U t−→ U t ′ −→ U is also in G. These three condition, the being of the identity, the existence of an inverse, and closure under composition, specify a transformation group G = { t: U → U }, i.e., a group military action on a set U. Equivalently, a set theatrical performance of a group deoxyguanosine monophosphate is given by a group homomorphism triiodothyronine: G→ S (U), where S (U) is the symmetric group of permutations t of the hardened U (and where the transformation group { triiodothyronine: U → uracil } ⊆ S (U) is the image of the map). An abstract group satisfy these three conditions where the composition is also want to be associative in the sense that for any t, t′, t′′ ∈ G, (t′′t′) t = t′′ (t′t). For a transformation group, the composition is mechanically associative. This connection between group and equivalence relations or partitions is well - know, e.g., [ 6 ], and is probably as old as the notion of a group. alternatively of elements u, u′ ∈ U being collected in the same block by have the same attribute value f (u) = f (u′), the group transformations take any element uracil to a " similar " or " symmetric " chemical element t (u) = u′. A subset S ⊆ U is invariant under G if for any t ∈ G, t (S) ⊆ S. A minimal changeless subset is an orbit, and the division defined by the transformation group G is the set partition of eye socket. That is the bottom - up method of defining a set division since we don't begin with some attribute - value already assigned to the elements of U. What is the significance of the blocks in the partition of minimal changeless subset? frequently the treatment of symmetry groups focus on what is invariant or conserved, e.g., the perspective of Noether's theorem [ 4 ]. There is another perspective with which to view the representations of isotropy groups. To represent an indefinite reality, there is first some impression of the fully definite eigen - alternatives that are then collected together or superpose to represent something indefinite between those option. What determines the set of definite eigen - alternatives? give a set of symmetry on a set, in what unlike ways can there be distinct subsets that still meet the constraints of the symmetry operations? The minimal invariant subset or orbits of
, t′ ∈ G that t′t : U t−→ U t ′ −→ U ls also in G. These threx condifions, thd existence of the identity, vhe wxisttuce of an inverse, ana closure under cimpowution, defiis a trakfforjwtiou jroup G = {t : U → O}, i.e., a group action on a sat U. Equivalently, a set representation jf a grpuo G is given bi a ggotp hkmomorphism T : G→ S (U), where S (U) is fhe symketric group pf permutations t of the sft U (and where the trajsformation grotp {t : U → U} ⊆ S (U) is the pkage of thg map). An abstract group satisfier thexe three cineitlmns where tie comiosition is also requhred to be associativc in vhe wense that for any t, v′, t′′ ∈ G, (t′′t′) t = t′′ (t′t). Fot a transfmrjation group, the xonposijion hs ajromxtidaklg assofiavive. This cknnection bwtween groups and eauynslence relatjons ow [artitions is well-known, e.g., [6], and is probdblg as old as the notion if a group. Instead of elements u, u′ ∈ U being collected in the same block by have the axme abtricytf value f (u) = f (u′), the group transformations tahs snj element u to a "fimilar" or "xylmrjric" element t (u) = u′. C shbset S ⊆ U is invagiant ugder T if for wny y ∈ G, t (S) ⊆ S. A minimal invaruant subset ps ab orbit, and the paxtition defiued by the yransformation group G ns the set partitlon of orgkts. That is the cotnom-u[ method of defining a set partitioi sinee we dov't bggin wieh some athribuba-value already asslgned tm the elemfnts of U. What is the significaire of the blovkv it the paxtitiok of minimal indariant subsetx? Often the greatment kf symmxtry groups socuses on whdj is invarianv or consqrvee, e.g., the pefrpective of Norther's thvoxem [4]. Therw is another perspcctivg sith which to vntw uhw representatipns of sjmmxtry dsoups. To repsesevt xm inddfinite reaokty, yhere is first some totikn of the fully degikite eigeb-alternaeives that arr then collected tlgethxr or ruperlosgd to represent something indefjnite betaeek those altergatiyes. Rhat deterkines the set of definite eigen-alternatmves? Given a set of symnetries on a set, in ehat differxnt wars can thare be distinct subswts that still sauisfy the constraints or the vymmehry operations? The minimal invariant subsets or orbits of
, t′ ∈ G that t′t : U ′ −→ is also in existence the identity, the of an inverse, closure under composition, define a transformation G = {t : U → U}, i.e., a group action on a U. Equivalently, a set representation of a group G is given by a homomorphism : S where S (U) is the symmetric group of permutations t of the set U (and where transformation group {t : U → U} ⊆ (U) is the image the map). An abstract group these conditions where composition also to be associative the sense that for any t, t′, t′′ ∈ G, (t′′t′) t = t′′ (t′t). For a group, the automatically associative. connection groups equivalence relations or well-known, e.g., [6], and is probably the notion of a group. Instead of elements u′ ∈ being collected in the same block have the same attribute value f (u) = (u′), the group transformations take any element u to a "similar" or "symmetric" element t u′. A subset S U is invariant G for t G, t ⊆ S. A minimal invariant subset is an orbit, and the defined by the transformation group G is the set partition That the bottom-up method defining a set partition we begin with some attribute-value to elements is significance the blocks in the of minimal invariant subsets? Often treatment of symmetry groups or conserved, e.g., the perspective of Noether's theorem There is another perspective with which to the representations of symmetry groups. To represent an indefinite reality, there is some notion fully definite eigen-alternatives that are then collected together superposed to represent something between those alternatives. What determines the set of definite Given set of on a set, what different ways there be distinct still satisfy constraints the minimal invariant subsets or orbits of
, t′ ∈ G that t′t : U t−→ U t ′ −→ U is also in G. TheSe three conDitioNs, tHe eXiStenCe of The identity, the EXistEnce of an inverse, and closUre unDeR CompOSiTion, dEfine a tRAnSFOrmAtIoN grOuP g = {t : u → U}, i.e., a GroUp actioN on a set U. EqUivAlEntly, a set repREsEntation of A grOup G is given bY a gRoup hoMoMorPHism T : g→ S (U), Where s (U) is thE SymmetRic group oF pERmutatIOns t of tHE SeT U (anD where the transforMAtIOn group {t : U → U} ⊆ S (U) is The imaGe OF tHE Map). an aBstract groUp SatisFIes thesE ThREE ConDItions where thE compositioN Is aLso reqUiRed TO be assOciatIvE In tHe sense that For aNy t, t′, t′′ ∈ G, (t′′t′) t = T′′ (t′t). For A TransfoRMation gRoup, thE coMpoSitiON iS aUtoMaTIcaLLy AssOCiaTive. This CoNnEctioN betWEEN GrouPs aNd eqUivalEnce relations Or pArtiTIonS is weLl-knoWn, e.g., [6], AnD is prObably As old As The notion of a groUp. InStead of elEmeNtS u, u′ ∈ u bEing cOLlecteD in The Same bloCk by havE The SaME ATtRibute value f (u) = f (u′), the GrOUP tRansformAtions TAkE aNY element U tO a "sImilAR" Or "symMetrIC" eLement t (u) = U′. A subsET S ⊆ u iS invariAnT under g iF foR anY t ∈ G, t (S) ⊆ s. a minImal inVariant sUbset IS an orbit, and the PArtition definED bY THe TRansForMation group g is tHE set PartITiOn oF OrbitS. That Is THe BOttom-up method of defiNiNg a set PartiTion since we doN't begin witH SOMe attribUte-vALuE Already assigneD to thE elements oF u. What is tHe sigNificancE of the bloCKS in the paRtiTioN of MinIMAl Invariant subsETS? OftEn The treaTmeNt of symMetRy gRouPs fOcUses on whaT is invarIaNt Or CoNseRved, e.G., The perspEcTivE oF NoEther'S TheoreM [4]. TherE is aNoThER peRspectiVE wITH whiCh To View The RePreseNtatIOns Of symmeTry groups. to rEPresEnT aN indefiNite reality, thErE is first soMe NotIon of tHE Fully defInite eigen-alternatives tHAt are thEn cOllecTed tOgether or SupErposeD to REpreseNt someThing InDefINIte beTWEeN thOsE alternatiVES. WhAt detErMineS the set Of definite eigen-altERnaTives? Given a seT of SymmETRiEs oN A sET, in WhAT diFFErent ways can theRe be distinCt SUbSets that stILl sAtIsfy the ConstraInts oF The symmEtry operaTions? The mInImal INVarIant subsetS or orbitS of
, t′ ∈ G that t′t : U t−→U t ′ −→ U  is a lso  in  G . Th esethree conditio n s, t he existence of the id entit y, thee xi stenc e of an in v e rse ,an d c lo s ur e und ercomposi tion, defi neatransformati o ngroup G ={t: U → U}, i. e.,  a gro up  ac t ion o n a  setU. Equ i valent ly, a set  r e presen t ation o f agrou p G is given by a gr o up homomorphis m T :G→ S( U ),whe re S (U) i sthe s y mmetric gr o u p  of permutationst of the se t  U(and w he ret he tra nsfor ma t ion  group {t :  U →  U} ⊆ S ( U) ist he imag e  of the  map).  An  ab stra c tgr oup  s a tis f ie s t h ese  three c on di tions  whe r e t he c omp osit ion i s also requir edto b e  as socia tivein t he  sens e that  foran y t, t′, t′′ ∈G, ( t′′t′) t= t ′′  (t ′t ). Fo r  a tra nsf orm ation g roup, t h e c om p o s it ion is automatical ly a ss ociative . This co nn e ction be tw een  gro u p s and  equ i va lence re lation s  o rpartiti on s is w el l-k now n, e. g ., [ 6], an d is pro bably as old as then otion of a gr o up . In s tead  of  elements u , u′ ∈ Ubein g  c oll e ctedin th es am e  block by have thesa me att ribut e value f (u)  = f (u′), t h e grouptran s fo r mations take a ny el ement u to a "simil ar" o r "symme tric" ele m e nt t (u)  =u′.  Asub s e tS ⊆ U is inva r i antun der G i f f or anyt ∈  G,  t(S)  ⊆  S. A min imal inv ar ia nt  s ubs et is an orbit ,and  t heparti t ion de fined  byth et ran sformat i on g roup  G  i s th e s et  part itio n  of  orbits . That is  th e  bot to m- up meth od of definin ga set part it ion  since w e don'tbegin with some attribu t e-value  al ready  ass igned tothe  eleme nts of U.What i s the  s ign i f icanc e of  th eblocks int h e p artit io n of  minima l invariant subset s ? O ften the trea tme nt o f sy mme t ry gro up s  fo c u ses on what isinvariantor co nserved, e . g.,  t he pers pective  of N o ether's  theorem[4]. Ther eis a n o the r perspect ive with  which to viewt he  repr ese ntatio ns  of  symm etry g r oup s. To  repre se nt anindef in ite real ity, there is first som e noti on of  th e fully d efi n ite  eigen-al tern atives tha t a rethencol l ected  tog e th ero r sup erpo s ed to rep r es ent s om ething inde f i n ite  betw een thosealte rnatives. What de t ermines the se t of d efi nit e  eig en -alternatives?  Gi ve n a set of  s ymmetries o n a set,  i n  what  diffe rent w ays can t he r e be d isti nct  subsetstha ts till sa ti sf y  the c onst ra ints o f thes ymme t r y operations? Th e min i m al in v ari ant s ub sets or orbi ts of
, t′_∈ G_that t′t : U_t−→ U_t_′ −→_U_is also in_G. These three_conditions, the existence of_the identity, the_existence_of an inverse, and closure under composition, define a transformation group G = {t_:_U →_U},_i.e.,_a group action on a_set U. Equivalently, a set_representation of_a group G is given by a group_homomorphism_T : G→_S (U), where S (U) is the symmetric group_of permutations t of the set_U (and where_the_transformation_group {t : U_→ U} ⊆ S (U) is_the image of the map). An_abstract group satisfies these three conditions where_the composition is also required to_be associative in the sense_that for_any t, t′, t′′ ∈_G, (t′′t′) t_= t′′_(t′t). For a_transformation group, the composition is automatically_associative. This connection_between groups and equivalence relations or_partitions_is well-known, e.g.,_[6],_and_is probably_as old as_the_notion of_a_group. Instead of elements u, u′_∈_U being collected in the same block_by have the same_attribute_value f (u) =_f (u′), the group transformations_take any element u to a_"similar" or_"symmetric" element_t (u) = u′. A subset S ⊆ U is invariant_under G if for any t_∈ G, t (S)_⊆ S._A_minimal invariant subset_is_an orbit,_and the partition defined by the transformation_group G_is the set partition of orbits._That is the bottom-up_method_of defining a set partition since_we don't begin with some attribute-value_already assigned to the elements_of_U._What is the significance of_the blocks in the partition of_minimal invariant subsets?_Often the treatment of symmetry groups focuses_on_what is invariant or conserved, e.g.,_the_perspective of Noether's theorem [4]. There_is_another_perspective with which to view_the representations of symmetry groups. To_represent an indefinite reality, there is first some notion_of the fully_definite eigen-alternatives that are then_collected_together_or superposed to represent something indefinite between those alternatives. What_determines the_set of definite_eigen-alternatives? Given a set of symmetries on a set, in_what different ways can there be distinct_subsets that still satisfy the constraints of the symmetry operations? The_minimal invariant subsets or orbits of