CYCLIC HYPERGROUPS WHICH ARE INDUCED BY THE CHARACTER OF SOME FINITE GROUPS

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1 italia joural of pure ad applied atheatics CYCLIC HYPERGROUPS WHICH ARE INDUCED BY THE CHARACTER OF SOME FINITE GROUPS Sara Sehavatizadeh Departet of Matheatics Tarbiat Modares Uiversity Tehra Ira Mohaad Mehdi Zahedi Departet of Matheatics Shahid Bahoar Uiversity Kera Ira Ali Iraaesh Departet of Matheatics Tarbiat Modares Uiversity Tehra Ira Abstract Let G be a fiite group ad Ĝ be the set of all irreducible characters of G I this paper, the hypergroups obtaied fro the character table Ĝ are cosidered Moreover, we show that Ŝ for 3 ad  for 4 are sigle-power cyclic hypergroups ad D ˆ is cyclic with fiite period Keywords: polygroup, cyclic hypergroup, character AMS subject classificatio: 0N0, 0C, 03 Itroductio Hypergroups have bee studied by ay researchers i various fields for a log tie; for exaples, see [], [] ad [4] Cyclic hypergroups already cosidered at the begiig of the theory s history by [3] have bee later o studied i depth by Vougioulis [] ad afterwards by Leoreau [8] The hypergroup H will be called cyclic with fiite period with respect to h H, if there exists a positive iteger s Z +, such that where h t = hhh }{{} t ties H = h h h s,

2 4 s sehavatizadeh, zahedi a iraaesh The iiu such a s will be called period of the geerator h If there exists h H ad s Z +, the iiu oe, such that H = h s, the H will be called sigle -power cyclic ad h is a geerator with sigle- power period s Quasicaoical hypergroups were itroduced by P Corsii ad later were studied by P Boasiga ad Ch Massouros They satisfy all the coditios of caoical hypergroups, except the coutativity Later, S D Coer i [] itroduced this class of hypergroups idepedetly, usig the ae of polygroups A polygroup is a syste P =< P,, e, >, where e P, is a uitary operatio o P, aps P P ito the o-epty subset of P, ad the followig axios hold for all x, y, z P : xyz = xyz; ex = xe = x; 3 x yz iplies y xz ad z y x Roth i [0] showed that for a fiite group G, there exists a polygroup syste Ĝ,, χ, where Ĝ is the set of all irreducible characters [6] of G Later o, Ĝ have bee studied i various fields McMulle i [9] proved that CĜ is seisiple ad Coer i [] showed that a atural hypergroup is associated with every character algebra ad also showed certai edge colorig of graphs give raise to hypergroups with special properties Syetry groups have bee widely applied i cheistry [3] ad crystallography [7] May of these applicatios, have ivolved coset decopositio, decopositios ito cojugacy classes ad group characters Let S be syetric group o letters ad A be alteratig group o letters ad D be dihedral group I this paper, we will show that the hypergroups which obtaied fro character tables of S for 3 ad A for 4 are siglepower cyclic hypergroup I cotiue, we will show the hypergroup D ˆ is cyclic hypergroup with fiite period Preliiaries I this sectio, we etio soe fudaetal otios ad facts of character of fiite groups ad character hypergroups, referrig to Issacs s boo [6] ad Roth s paper [0] Let G be a fiite group ad F be a field Also, let V be a fiite diesioal vector space o F A represetatio of G over V is a hooorphis T : G GLV, T xy = T xt y; x, y G A represetatio T of G is called irreducible, if V is a irreducible F G od Let the diesio of V over F be The GLV = GL, F where GL, F is the set of all square ivertible atrixes Let T be a represetatio of G The the character χ of G afforded by T is the fuctio give by χg = trt g ad χ

3 cyclic hypergroups which are iduced by the character 5 is a irreducible character if the represetatio T is irreducible For a character χ, the erel of χ is defied by er χ = {g G : χg = χe} If er χ = {e}, the χ is called a faithful character We assue that the field F is equal to coplex uber If χ ad ψ are ay two coplex characters of G, the χ, ψ deotes the usual ier product: χ, ψ = χgψg Let IrrG = {χ, χ,, χ }, where χ i for i are irreducible characters of G Sice we eed soe well ow results relate to character theory we brig the i follow: Theore [6]Orthogoality Relatios Let χ i, χ j ad χ be coplex characters of G ad g, h G The χ i gχ j g = δ ij, χ IrrG χgχh = 0 Theore [6] The character table of D for eve iteger =, = e πi ad j is as follow: Table I D a a r r b ab cg i 4 4 χ χ χ 3 r χ 4 r ψ j j jr + jr 0 0 Theore 3 [6] The character table of D for odd iteger, = e πi j is as follow: Table II ad D a r r b cg i χ χ ψ j jr + jr 0

4 6 s sehavatizadeh, zahedi a iraaesh Suppose that the group G acts o a set Ω ad g G The we defie the set of fixed poits of g by fixg = {α Ω α g = α} Theore 4 [6] I syetric group S, χg = fixg is a faithful irreducible character Theore 5 [6] I alteratig group A, χg A character is a faithful irreducible Theore 6 Cauchy-Frobeius Lea [5] Let G be a fiite group actig o a fiite set Ω The G has orbits o Ω where = f ixg Let G be a fiite group with Ĝ = {χ, χ,, χ } Roth i [0], itroduced the character polygroup < Ĝ,, χ, > where the product χ i χ j is the set of those irreducible copoets which appear i the eleet wise product χ i χ j Further, χ, the coplex cojugate of χ, is the iverse of χ If θ χ ψ, the θ, χψ > 0, hece θ χ, ψ > 0 ad ψ θ χ Lea 7 [0] Let G be a fiite abelia group The Ĝ is isoorphic to G A ey theore i the study of the character hypergroup theore of Burside: Ĝ is the classical Theore 8 Burside [6] Let χ be a faithful character of G ad suppose χg taes o exactly differet values for g G The every ψ IrrG is a costituet of oe of the characters χ j for 0 j < 3 Mai results I this sectio, we will obtai the ai results related to the character table of syetric group S, A ad D I fact, we give Theores 34, 35, 38 ad 3 as the ai results For a irreducible character χ i, we let χ t i = χ i χ i χ }{{} i, t ties where the hyper operatio is as above Lea 3 I the syetric group S for 3, χ taes o exactly differet values for ay g S Proof We ow that S has cojugacy classes of the fore i i where 0 i ad i Moreover, i each of classes we have χg = i Lea 3 I the alteratig group A for 4, χ A taes o exactly differet values for g A

5 cyclic hypergroups which are iduced by the character 7 Proof A has cojugacy classes of the fore i i where i is a odd iteger such that 0 i ad i ad also has cojugacy classes of the fore i i ad ij ij for eve iteger i, j Let Ω be a fiite set ad for ay positive iteger t, Ω t = Ω } Ω {{ Ω } t ties The we give a corollary of Cauchy-Frobeius Lea as follow: Corollary 33 Let Ω be a fiite set ad G be a fiite group actig o Ω t The G has orbits o Ω t where fixg t = G Proof Cosider the set F = {ω, ω,, ω t, g Ω Ω Ω G ω, ω,, ω t g = ω, ω,, ω t } we shall cout the uber of F i two ways First, suppose that the orbits of G are Ω, Ω,, Ω The, usig the orbit-stabilizer property, we have F = i= ω,ω,,ω t Ω i Ω i = = i= Secod, F = f ixg The result follows Theore 34 For 3, Ŝ is a sigle-power cyclic polygroup with respect to geerator χg = fixg I fact Ŝ = χ Proof By the Burside theore 8, we have Ŝ = χ 0 χ χ We ust prove that χ χ We ow that for Ω = {,,, }, S acts o Ω, Ω ad Ω 3 The actio o Ω has two orbits A = {i, i i Ω} ad A = {i, j i, j Ω, i j} Siilarly the actio o Ω 3 has five orbits Now, put F g = fixg The, by the Cauchy-Frobeius Lea 6 ad the previous lea: Hece, F g =, χ, χ = F g =, F g 3 = 5 χg 3 = F g 3 = G Therefore, χ χ Cosequetly, χ χ 3 χ

6 8 s sehavatizadeh, zahedi a iraaesh Hece, Ŝ = χ 0 χ χ = χ Sice the alteratig group A is a iportat siple group, we would lie to give soe results as above o it Theore 35 For 4,  is a sigle-power cyclic polygroup with respect to geerator χ A I fact,  = χ A Proof By the Burside theore 8, we have  = χ A 0 χ A χ A Sice χ, χ 0, we have χ A, χ A 0 Hece, χ A χ A Therefore, χ A χ A 3 χ A Sice dihedral groups are faous betwee of all o-abelia groups, we show that Dˆ has a cyclic hypergroup structure Lea 36 Cosider the dihedral group D for eve iteger = Let be a eve iteger The: Case a For a eve iteger j, ψ j ψ Also the ultiplicities of ψ j i ψ j Case b For a odd iteger j, ψ j ψ is j Proof Case a Let j be a eve iteger The, we have Also the ultiplicities of ψ j i ψ ψ, ψ j = j + j j + + j + +4+j + = = + + +j + ++j + + +j + +j + +4+j j + ++j j + + j + + j j+j + +3+j j + + +j 3+ j + j + + j +4 j j +3 j +j j++j + +j + +j is + +j j + j + j + +j+j I this equality, we cosider the colu A, A, B ad B for all 0 j, A is equal to the colu as follows:

7 cyclic hypergroups which are iduced by the character 9 B is equal to the colu ++j + +4+j j j, + j + +4 j + ++ j+j ad for all j <, A is equal to the colu ++j + +4+j j j, ad B is equal to colu + j + +4 j + ++ j+j Now by usig the orthogoality relatios for A, A, B ad B ad by soe aipulatios we get that A + B =, A + B =, j Ad, for = j, we have for each copoet of A ad B is equal to oe Hece, but ψ, ψ j = + + j So ψ, ψ j = j ad hece the proof of Case a is copleted Case b The proof is siilar to Case a = + = = + = j + j

8 30 s sehavatizadeh, zahedi a iraaesh Lea 37 Cosider the dihedral group D for eve iteger = Let be a odd iteger The: Case a For a eve iteger j, ψ j ψ ad the ultiplicities of ψ j i ψ is j Case b For a odd iteger j, ψ j ψ ad the ultiplicities of ψ j i ψ is j Proof The proof is siilar to Lea 36 Theore 38 Cosider the dihedral group D ad = The cyclic hypergroup with geerator ψ I fact, Dˆ = ψ ψ ˆ D is a Proof First let be a eve iteger By Lea 36 it is eough to show that for i 4, χ i are i ψ Sice is a eve iteger ad by the character value uber of χ, χ i the character table of D, we have: χ, χ ψ Now, for χ 3 we have: ψ, χ 3 = = = I this equality, we cosider the colu A ad B as follows: for all 0, A is equal to the colu ad, for all <, B its equal to colu

9 cyclic hypergroups which are iduced by the character Now, by usig the orthogoality relatios for A ad B ad soe aipulatios, we get that A + B = for =, we have that each copoet of A is equal to oe Hece, ψ, χ 3 = + = + But So = = ψ, χ 3 = 0 Therefore, χ 3 ψ, ad siilarly χ 4 ψ Hece the proof is copleted Now, let be a odd iteger The proof i this case is siilar to the above Lea 39 Cosider the dihedral group D for odd iteger Put = ad let be a eve iteger The: Case a For a eve iteger j, ψ j ψ ad the ultiplicities of ψ j i ψ is j Case b For a odd iteger j, ψ j ψ is j Proof The proof is siilar to Lea 36 ad the ultiplicities of ψ j i ψ Lea 30 Cosider the dihedral group D for odd iteger Put = ad let be a odd iteger The: Case a For a eve iteger j, ψ j ψ ad the ultiplicities of ψ j i ψ is j Case b For a odd iteger j, ψ j ψ ad the ultiplicities of ψ j i ψ is j

10 3 s sehavatizadeh, zahedi a iraaesh Proof The proof is siilar to Lea 36 Theore 3 Cosider the dihedral group D for odd iteger > 3 The Dˆ is a cyclic with fiite period hypergroup with geerator ψ where = I fact, Dˆ = ψ ψ Proof By Leas 39 ad 30, it is eough to show that χ ad χ are i ψ ψ By the character value uber of χ ad χ i the character table of D, we have χ, χ ψ if is a eve iteger ad χ, χ ψ if is a odd iteger This copletes the proof 4 Coclusio I this paper, a relatio betwee character theory ad polygroup theory has obtaied I fact, we could give a structure of hypergroup by character tables ad usig a special hyperactio o the Now there is a questio, ca we exteded this idea to a arbitrary fiite group i which its character tables is ow? Refereces [] Coer, SD, Hyperstructures associated with character algebra ad color schees, New Frotiers i Hyperstructures, Hadoric Press, 996, [] Corsii, P, Prolegoea of Hypergroup Theory, Secod editio, Aviai Editore, 933 [3] Cotto, FA, Cheical Applicatios of Group Theory, J Wiley ad Sos, 990 [4] Davvaz, B, Polygroup Theory ad related systes, World Scietific Publishig Co Pte Ltd, 03 [5] Dixo, JD, Mortier, B, Perutatio Groups, Graduate Texts i Matheatics, vol 63, Spriger-Verlag, New Yor, 99 [6] Isaacs, IM, Character Theory of Fiite Groups, Acadeic Press, New Yor, 976 [7] Jaovec, V, Dvoraova, E, Wie, TR, Litvi, DB, The coset ad double coset decopositios of the 3 crystallographic poit groups, Acta Cryst, sect A, , [8] Leoreau, V, About the siplifiable cyclic seihypergroups, Italia J Pure Appl Math, 7 000, [9] McMulle, JR, Price, JF, Reversible Hypergroups, Cofereza teuta il 5 e 6 aggio, 977 [0] Roth, RL, Character ad cojugacy class hypergroups of fiite group, A Math Pura Appl, , 95-3 [] Vougioulis, T, Cyclicity i a special class of hypergroups, Acta Uiv Caroliae-Math et Physica, 98, 3-6 [] Vougioulis, T, Hyperstructures ad therir Represetatios, Hadroic Press, Ic, 5, Pal Harber, USA, 994 [3] Wall, HS, Hypergroups, Aer J Math, , Accepted: 08004