The Primitive Idempotents in

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1 Iteratoal Joural of Algebra, Vol, 00, o 5, 3 - The Prmtve Idempotets FC - I Kulvr gh Departmet of Mathematcs, H College r Jwa Nagar (rsa)-5075, Ida kulvrsheora@yahoocom K Arora Departmet of Mathematcs, Maharsh Dayaad Uversty Rohtak-00, Ida Abstract Let F be the sem smple group algebra of the cyclc group of order ( 3) over the fte feld F( = F(q) ) of prme power order q (odd),where q s quadratc resdue modulo The q = 8k (k s eve or odd) I case k s odd the F has () prmtve dempotets, the explct expressos for these prmtve dempotets are obtaed Mathematcs ubect Classfcato: 9B5, 63, 0C05 Keywords : roup algebra; prmtve dempotets ; cyclotomc cosets Itroducto Let = C m = < g > be a fte cyclc group of order m ad F( = F (q)) be a feld of order q, a power of ts prme (odd) characterstc ρ, (say) Let t be the multplcatve order of q modulo m The, t φ(m) I case m = p ad t = φ ( p ) for some odd prme p ad teger, the explct expresso (wthout proof) for the prmtve dempotets FC p are gve by

2 3 K gh ad K Arora [8] ad are completely dscussed by Pruth ad Arora [] Arora ad Pruth [3] also obtaed the complete set of prmtve dempotets case m = p ( ) ad t = φ ( p ) Further Batra ad Arora [], [6] ad [7] whle descrbg mmal cyclc codes of legth p ad p φ case t = ( p ) φ ad ( p ) respectvely obtaed the complete set of prmtve dempotets these cases φ( ) The case whe m =, ad t = s dscussed depedetly by Pruth [0], Arora et al [], [5] ad Baksh et al [] Here we assume that q s quadratc resdue modulo By Theorem 9 [ 9, p 0 ], q (mod 8) ad α0 α coversely e q = 8k (k = α p p r ; α 0 ) Observe that the order of q modulo s dfferet case whe k s eve or odd ad the descrpto of prmtve dempotets for both the cases make the paper too legthy whch s ot sutable for compact dscussos Therefore, we preset the result two papers, wth the vew pot that few basc results proved ths paper wll frequetly be used to prove the results part - II, wherever ad wheever eeded I ths paper we cosder the case whe α 0 = 0 e k s odd By Lemma, order of q modulo φ ( 3 ) s ( ) Let be the set { 0,,, } For a, b, the relato a bq (mod ), parttos to ( ) dsot q-cyclotomc cosets, as obtaed Theorem Therefore, FC has () prmtve dempotets The explct expressos for these prmtve dempotets are obtaed Theorem r Cyclotomc Cosets Lemma Let q = 8k ( k s odd ) The, φ( ) s the order of q modulo Proof Trval Theorem uppose q = 8k (k s odd) The ( ), q-cyclotomc cosets modulo are gve by : Ω 0 = {0}, Ω (), = { }, Ω (), = {3 }, Ω (), = { } ; ad for, ( ) Ω (), = { (8λ ) : 0 λ },

3 Prmtve dempotets 33 ( ) Ω (), = { (8λ 3) : 0 λ }, ( ) Ω (3), = { (8λ 5) : 0 λ }, ( ) Ω (), = { (8λ 7) : 0 λ } Proof Follows by Lemma Notatos 3 () For ad β, deote by ( β), the elemets s g s Ω( β ), FC () For, ( lm, ), = ( ( l), ( m), ), where l, m ad l m 3 ( ) () For 0, let X ( ) x x x x = l = x l = 0 et X = X( g) Further for, let Y = X X ad Y0 = X0 The, for 0, X = ( ), ( (), (), ) β (), = β= ad for, Y = ( ), ( (), (), ) β (), ( β), = β= β= (v) The suffx δ> ( δ ), s cosdered reduced modulo Lemma k () uppose 0 The for ay k, g X = X () uppose The X X = X () uppose 0, The Y f =, Y Y = o otherwse Proof () By our choce o(g) = It the follows easly, by otato 3() that

4 3 K gh ad K Arora k g X = X Part () ad () follows from part () 3 ome famles the roup Algebra ad ther propertes Lemma 3 I uppose ad k 0 be ay teger The for 3, k g ( β), = ( β), II If =, the for, ( ), f,,6, k β k = g ( β), = ( β ), f k =,3,5, III () If =, the for, ( ), f,5,9, k β k = g ( β), = ( β 3), f k = 3,7,, (), f ad,3,5,7 k β= k= () g ( β), = (), f β= ad k=,3,5,7 Proof I ce, for, k Ω (β), Ω (β), (mod ), therefore, k k s s ( β), = = = ( β), s Ω( β ), s Ω(β), g g g II Let k = t for some postve teger t The, k Ω ( β ), = {t r : r β (mod 8)} = { r : r β (mod 8)} = Ω ( β ), Now, let k = t, for some postve teger t The, k Ω ( β ), = {(t ) r : r β (mod 8)} = { r : r β 3 (mod 8)} = Ω ( β ),, provg II III Follows o smlar les as above Lemma 3 uppose ξ β Ω ( β ), The for, I ξ β Ω ( β ), = Ω ( β ), Ω ( β ), II () ξ Ω ( ), = ξ 3 Ω ( ),

5 Prmtve dempotets 35 = Ω ( ), Ω ( ), Ω ( 3 ), Ω ( ), () ξ Ω ( 3 ), = Ω ( ), Ω ( ), () ξ Ω ( ), = Ω ( ), Ω ( 3 ), (v) ξ Ω ( ), = ξ Ω ( 3 ), = (Ω ( ), 3 Ω ( ), 3 ) (Ω ( ), Ω ( ), ) (Ω ( ), Ω ( ), ) Ω ( ), Ω 0 Proof I By defto, Ω ( ), = { r : r ( mod 8 )} The ξ Ω ( ), = { ( r s ) : r = 8m, s = 8 ; m, 0 } = { ( 8t ) : t 0} = { ( t ) : t 0} = { r : r or 5 ( mod 8 ) = Ω ( ), Ω ( 3 ), mlarly, for β, ξ β Ω ( β ), = Ω ( β ), Ω ( β ), II Follows o smlar les as above Lemma 33 uppose The I ( β), ( β), = ( ( β ), ( β ), ) II () (), (), = (3), (), = ( (), (), ) () (), (3), = ( (), (), ) () = ( ) (), (), (), (3), (v) (), (), = (), (3), (), 3 (), 3 (), = [( ) ] Proof The result follows by Lemma 3 Lemma 3 For <, ( γ ), f = ad β=,3 ( 3), f ad, γ = β= ( β), ( γ), = ( γ ), f =, for all β ( γ), f 3, for all β Proof ce Ω = -, therefore result follows by lemma 3 (β ), Lemma 35 uppose The () X (,3), = 0 () X (,), = 0 () X (,), = 0 (v) X (3,), = 0 Proof () Assume that = The, by defto of X, ( l, m ), ad

6 36 K gh ad K Arora by Lemma 3, we get that : X (,3), = ( ) [ ( (), (3), ) ( (), (3), )( (), (3), ) ( (), (), )( (), (3), ) ( (), (), )( (), (3), ) ] ( ) ( ) = [ ( (), (3), ) ( (3), (), ) ] = 0 Next, assume > The, X (,3), = ( (), (3), ) X s t = g g X s ( mod 8) t 5( mod 8) ce ad Ω (), = Ω (3),, t the follows from Lemma, that Ω(), X (,3), = ( X X ) = 0 (), () ad (v) follows smlarly by usg Lemmas ad 3 Lemma 36 (,3), f =, () Y (,3), = 0 otherwse () Y (,), (,), f =, = 0 otherwse () (v) ( (,), (3,), ) f =, Y (,), = ( (,), (3,) ) f =, 0 otherwse ( (,), (3,), ) f =, Y (3,), = ( (,), (3,), ) f =, 0 otherwse

7 Prmtve dempotets 37 Proof () Assume = The, by defto of Y, ( l, m ), ad Lemma 3, we get that : Y (,3), = [ ( (), (3), ) ( (3), (), )] = ( (), (3), ) = (,3), Next, assume If, the by Lemma 3(I), trvally Y (,3), = 0 If =, the by Lemmas 35 () ad 3, Y = X (,3), (,3), 3 3 = [ ( (3), (), ) ( (), (3), )] = 0 Further, f, aga by Lemma 35(), Y = X X = 0 Ths proves () (,3), (,3), (,3), (), () ad (v) follows smlarly by usg Lemmas 3, 3, ad 35 Lemma 37 () (,3), (,3), ( (,), (3,), ) f =, = 8 0 otherwse () () (,), (,), (,), (,3), ( (,), (3,), ) f =, = 8 0 otherwse Y f =, = 6 0 otherwse Proof Follows by Lemmas 33 ad 3 Lemma 38 () Y ( (,), (3,), ) f =, 3 6 (,), (,), = (,3), f =, 0 otherwse

8 38 K gh ad K Arora () () (v) Y ( (,), (3,), ) f =, 3 6 (,), (,3), = (,), f =, 0 otherwse Y ( (,), (3,), ) f =, 3 6 (3,), (,), = (,3), f =, 0 otherwse Y ( (,), (3,), ) f =, 3 6 (3,), (,3), = (,), f =, 0 otherwse Proof () Assume = By Lemma 33, (,), (,), = [( (), (), ) {( (), 3 (), 3 ) (), } ( ) ( )] (), (), (), (3), = [{( (), (), ) ( (3), (), )} {( (), 3 (), 3 ) (), ( (), (), )}] = ( (, ), (3, ), ) Y 6 3 Next assume The, by repeated applcato of Lemma 3, we get the followg: If =, the, (, ), (, ), = 3 (), (), (), (), (), (), (), (), [ ] = ( (3), (), ) = (,3), If =, the,

9 Prmtve dempotets 39 (, ), (, ), = [ (), (), (), (), (), (), (), (), ] = 0 If 3, the aga t follows o smlar les that (,), (,), = 0 Further f, aga the repeated applcato of Lemma 3 yelds that (,), (,), = 0 Ths complete the proof of () The proof of (), () ad (v) follows o smlar les as dscussed above Lemma 39 () () [( (), (), ) ( (), (), ) f, = (,), (,), = ( (,3), (,), ) f =, 8 0 otherwse [( (), (), ) ( (), (), ) f, = (3,), (3,), = ( (,3), (,), ) f =, 8 0 otherwse [( (), (), ) {( (), 3 (), 3 ) (), }] f =, () (,), (3,), = ( (,3), (,), ) f =, 8 ( (,3), (,), ) f =, 8 0 otherwse Proof Follows by Lemmas 33 ad 3

10 0 K gh ad K Arora The Ma Result Theorem FC has ( ) prmtve dempotets gve by : e = Y ; e(), = Y 0 0 e(), = [ Y ( (,), (3,), )] θ, e(), = [ Y ( (,), (3,), )] θ, ad for, e(), = [ Y θ ( (,), (3,), ) θ ( (,), θ (,3), )], e(), = [ Y θ ( (,), (3,), ) θ ( θ (,), (,3), )], e(3), = [ Y θ ( (,), (3,), ) θ ( (,), θ (,3), )], e(), = [ Y θ ( (,), (3,), ) θ ( θ (,), (,3), )], where θ = ad θ F(ρ) or F(ρ ) (ρ beg the characterstc of F) Proof Ths ca be proved by meas of Lemma ad Lemmas 36 to 39 Ackowledgemet Research supported by: CIR,( FNo9/38(0), k3-emr-i), New Delh, Ida Refereces [] Arora, K, Batra, ad Cohe, D ; The prmtve dempotets of a cyclc group Algebra II, outheast Asa Bullet of Mathematcs, 9(005), [] Arora, K, Batra,, Cohe, D ad Pruth, M ; The prmtve dempotets of a cyclc group Algebra, outheast Asa Bullet of Mathematcs, 6(003), [3] Arora, K ad Pruth, M ; Mmal cyclc codes of legth p, Fte Felds ad Ther Applcatos, 5(999), [] Baksh, K, Dumr, VC ad Raka M ; Mmal cyclc codes of legth m, Rach Uversty, Math Joural, 33(00), -8

11 Prmtve dempotets [5] Batra, ad Arora, K ; Mmal quadratc resdue cyclc codes of legth, J App Math & Computg, 8(-)(005), 5-3 [6] Batra ad Arora K ; Mmal Quadratc Resdue Cyclc Code of legth p, The Korea J Comput & Appl Math 8(3)(00), [7] Batra, ad Arora K ; ome Cyclc Codes of legth p, Desg Codes ad Cryptography (Accepted for Publcato) [8] Berma, D ; emsmple cyclc ad abela codes, II Cyberetcs 3(3), (967), 7-3 [9] Burto, Davd M ; Elemetary Number Theory, Mcraw Hll Book Compay, New York, (Ffth Edto), 00 [0] Pruth, M ; Cyclc codes of legth m, Proc Ida Acad c Math c (00), [] Pruth, M ad Arora, K ; Mmal codes of prme power legth, Fte Felds & Ther Applcatos, 3(997), 99-3 Receved: Jue, 00