Aled Mathematcal Sceces, Vol., 7, o. 9, 95-9 HIKARI Ltd, www.m-hkar.com htts://do.or/.9/ams.7.7576 O the Ratoal Valued Characters Table of the Grou (Q m C Whe m s a Eve Number Raaa Hassa Abass Deartmet of Mathematcs, Faculty of Educato for Grls The Uversty of Kufa, Iraq Coyrht 7 Raaa Hassa Abass. Ths artcle s dstrbuted uder the Creatve Commos Attrbuto Lcese, whch ermts urestrcted use, dstrbuto, ad reroducto ay medum, rovded the oral work s roerly cted. Abstract The ma urose of ths aer s to rove that the eeral form of a ratoal valued characters table of the rou (Qm C such that Qm be the Quatero rou of order m whe m s a eve umber ad C be the cyclc rou of order ad (Qm C be drect roduct of Qm ad C such that Qm C = {(a,b:a Qm,b C} ad Qm C = Qm. C =m.=6m. Ths table deeds o a ratoal valued characters table of a quatero rou of order m whe m s a eve umber ad symbolzed by (Qm C. Mathematcs subect Classfcato: 6G, 7B Keywords: Grou, Quatero, Cyclc, a ratoal valued, a ratoal characters table Itroducto Let G be a fte rou, the two elemets of G are sad to be г-couate f the cyclc subrous they eerate are couate G. Ths rocess defes a equvalece relato o G; ts classes are called г-classes. Let * (G deotes the r r matrx whch the rows corresods to the θ's ad the colums corresod to the г-classes of G. The matrx exress R(G bass terms of the cf(g,z bass s * (G.I 959, M.J. Hall [ s foud" The ratoal valued characters table of fte rou". I 9, M.S. Krdar [5 s foud "The ratoal valued characters table of the cyclc rou C". I 995, N.R. Mahamood
96 Raaa Hassa Abass [6 studed "The ratoal valued characters of Qm". I ths work we have foud the ratoal valued characters table of the rou (Qm C. Prelmares I ths secto we reset some mortat deftos ad basc cocets of a ratoal valued characters ad a ratoal valued characters table. Defto (.: [ A ratoal valued character of G s a character whose values are Z, whch s ( Z, for all G Corollary (.: [ The ratoal valued characters = ( Gal ( Q ( / Q form the bass for R (G, where are the rreducble characters of G ad ther umbers are equal to the umber of couacy classes of cyclc subrou of G. Theorem (.:[ Let T : G GL(,F ad T : G GL(m,F be two rreducble reresetatos of the rous G ad G wth characters ad resectvely the : T T s rreducble reresetato of the rou G G wth the character. Proosto (.: [ The umber of all ratoal valued characters of a fte rou G s equal to the umber of all dstct - classes o G. Defto (.5: [ the comlete formato about ratoal valued characters of a fte rou G s dslayed a table called ratoal valued characters table of G. We refer to t by (G whch s matrx whose colums are -classes ad rows whch are the values of all ratoal valued characters of G, where s the umber of -classes. Proosto (.6:[5 The ratoal valued characters table of cyclc rou C s of rak s+ where s rme umber whch symbolzed by ( (C s by: - classes [ [r s P (- s [r s [r s [r [r, s ve s - (- P (- s - P s (- P (- (- s - s P P [r
O the ratoal valued characters table of the rou (Qm C 97 s s s P(- P(- P(- P(- P(- - P - - - - - - - Table ( where ts rak s+ reresets the umber of all dstct -classes. Remark(.7:[5 I eeral, for =... m m where.c.d (, =, f ad 's are rmes umbers, the we have the follow formula : (C = (C (C (C Pm m Proosto (.:[6 The eeral form of the ratoal valued characters table of the Quatero rou Qm whe m= h, h s ostve teer ad t s ve by: ( Q m ( Q. h Γ-classes [ [r h [r h [r h [r [s [rs [r θ θ θ h - h h- h- - h- h- h- h- - h- θ l - θ l - - θ l - - θ l - - θ l Table ( where l s the umber of Γ-classes of Cm. Proosto (.9: [6 The ratoal valued characters table of the Quatero rou Qm whe m s a eve umber s ve by:
9 Raaa Hassa Abass ( Q m Γ-classes θ θ Γ-classes of Cm [r k ( Cm [s - - θ l θ l θ l θ l - - - - Table ( [rs Where θ l (r k = θ (rk= l f k eve umber f k odd umber l s umber of Γ-classes of Cm. ad θ l (r k = θ l (r k = The ma results I ths secto we fd the ratoal valued character table of the rou (Qm C whe m s a eve umber. Proosto(.: The ratoal valued characters table of the rou (Qm C whe m s a eve umber s equal to the tesor roduct of the ratoal valued characters table of Qm whe m s a eve umber ad the ratoal valued characters table of C that s: ( Qm C ( Qm ( C. Proof :- C = {I, c, c,c } Sce ( C h h h - - Table(
O the ratoal valued characters table of the rou (Qm C 99 Where h ={(I}, ={c }, ={c,c } the, h h = = = =- = = = = = = = = - = = = = = = From the defto of Qm C, ad Theorem(. we have (Qm C = ( Qm ( C Each elemet Qm C h = h h h Qm, h C,,,,,m, {I,c,c, c }, each rreducble character of Qm C s, =. where s a rreducble character of Qm ad s the rreducble character of C,the (, ( h ( h ( h ( h ( h f f f f ad { I, c } f ad { c, c } f ad ad { I} ad { c } ad { c, c } C From Corollary (., = (, Gal ( Q (, / Q where, s the ratoal valued character of Qm C The, ( = (, h, Gal ( Q (, / Q
9 Raaa Hassa Abass (I (a If = ad, (. h {I} ( = ( = = = h ( ( h ( =. ( h (b If = ad {c } ( = (, h = - ( Gal ( Q( / Q I =. - = (. ( h h (c f = ad {c, } ( = ( = (, h = Gal ( Q( / Q I Gal ( Q( / Q where s the ratoal valued character of Qm. (II (a If = ad {I,c }, = ( = (b If = ad {c,c }, = ( = = (. -=.-=.. (III If = ad C = ( = From [I, [II ad [III we have.., = h, Gal ( Q ( / Q, Gal ( Q ( / Q Gal ( Q ( / Q Gal ( Q ( / Q Gal ( Q ( / Q The * ( Qm C = * ( Qm * ( C. Gal ( Q ( / Q I. =. = Gal ( Q ( / Q. =.. Examle(.: To fd * (Q6 C by us the Proosto (. we et the follow table: * (Q C= Gal ( Q ( / Q (.
O the ratoal valued characters table of the rou (Qm C 9 Γ- classes [,I [r,i [r,i [r,i [r, I [s, I [rs,i [,c [r,c [r,c 6 - -6 6 (, 6 - - - (, - - - - (, - - - - - - - (, - - - - - - - (5, - - - - - - - (6, - - - - - - - (7, - - - (, - - - - (, - - - - - (, - - - - - - - - - (, - - - - - - - - - (5, - - - - - - - - - (6, - - - - - - - (7, - - - (, - - - (, - - - (, - - - - - - (, - - - - - - (5, - - - - - - (6, (7, [r,c Table(5 [r,c [s, c [rs,c [, c [r,c [r,c [r,c [r,c [s, c [rs,c Examle (.: To fd * (Q C by us the Proosto (. we et the follow table: * (Q 5 C=
9 Raaa Hassa Abass 6 6 6 6 Refereces [ C.W. Curts ad I. Reer, Reresetato Theory of Fte Grous ad Assocatve Alebra, AMS Chelsea Publsh, 96 [ J.P. Serre, Lear Reresetato of Fte Grous, Srer-Verlae, 977. htts://do.or/.7/97--6-95-7 [ K. Sekvch, Extesos ad the Irreducbltes of the Iduced Characters of Cyclc -Grou, Hroshma Math. Joural, (, 65-7. [ M. J. Hall, The Theory of Grou, Macmlla, New York, 959. [5 M. S. Krdar, The Factor Grou of the Z-Valued Class Fucto Modulo the
O the ratoal valued characters table of the rou (Qm C 9 Grou of the Geeralzed Characters, Ph.D. Thess, Uversty of Brmham, 9. [6 N.R. Mahamood, The Cyclc Decomosto of the Factor Grou cf(qm,z/ R (Qm, M.Sc. Thess, Uversty of Techoloy, 995. Receved: Jue, 7; Publshed: July, 7