The Number Irreducible Constituents Permutation Representation of G

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1 Journal of Novel Aled Scences Avalable onlne at JNAS Journal / ISSN JNAS The Number Irreducble Consttuents Permutaton Reresentaton of G B. Razzahmanesh eartment of Mathematcs and Comuter scence Islamc Azad Unversty Talesh Branch, Talesh, Iran 1,..., r Corresondn author: B Razzahmanesh ABSTRACT: et the dfferent rreducble consttuents of the ermutaton reresentaton G of f G. In addton let be the deree of (1,..., r) let Tr( ) and be the character of. Let the numbern be chosen so that 1 s the dentty reresentaton. Then number the rreducble consttuents of G f such that 2 and the reresentatons 3,..., r are conjuate. In artcular f3... fr f, and f dvdes. Keywords: ermutaton reresentaton, orbts, rreducble consttuents. INTROUCTION In R. Brauer studed about ermutaton rous and nd the ermutaton rous of rme deree and related classes of Grous (See (4)), From years 1906 to W.A.Mannn studed about rmtve rous and ndn the rmtve rous of classes sx, ten, twelve and fteen. And n 1906, W.Burnsde ntroduced d researched the a bout transtve rous, of rme deree (See (2)), and n 1921, he wored about the certan smly- transtve ermutaton rou and obtaned a beautful consequences (See (3)). (See(11),(12),(13),(14),(15),(16),(17) and (18)). In 1937 J.S. Frame determned the derees of the rreducble comonents of smly transtve ermutaton rous, and n 1941, he obtaned the double cosets of a nte rous (See(5) and (6)). And also n 1952, he ndn the rreducble reresentaton extracted from two ermutaton rous (See (7)). G.A.Mller(1897&1915),(See (19) &(20)), E.T.Parer (1954),(See (21)), M. Suzu(1962), (See(23)), J.G.Thomson (1959), (See (24)), M.J.Wess(1928), (See (25)&(26)), H.Welandt (1935 & 1956) (See (27)&(28)) and H.Zassenhaus(1935), (See (30)) are studed about transtve and rmtve rous and ther obtand the beautful and more consequence. Now n ths aer we wll rove number the rreducble consttuents of G such that artcular f3... fr f, and f dvdes. f 2 and the reresentatons,..., 3 r are conjuate. In 2. Perlmnares In ths chater we study the notatons, elementary roertes,lemmas and theorems, whose we wll used n chater Elementary notons and dentons Let be a nte set of arbtrary elements whch for natural numbers 1,2,,n as the onts and subset of. Then a ermutaton on s a one-to-one man of onto tself. We denote the mae of the ont under n P. the ermutaton by n.we wrte. We dene the roduct q of two ermutatons and q on q ( ) q by the formula. trvaly q s aan a ermutaton on.

2 J Nov. Al Sc., 3 (8): , 2014 Wth resect to the oeraton above, all the ermutatons on form a rou, the symmetrc rou S G. Let G be a ermutaton rou on, n short G S. We say that a set G s a xed bloc of G or s xed by G f. We call the totalty of G. Then each nduces a ermutaton on whch we denote by 's formed for all G the consttuent G of G on ( for examle G G ). G s a ermutaton rou on.obvously the man s a homomorhsm : G ~ G G G. If ths man s an somorhsm, that s,, then the consttuents G s called fathful. Every rou G on has the trval xed blocs on. If t has no others t s called transtve. Otherwse t s called ntranstve. Accordnly, a consttuent G s transtve recsely when s a mnmal xed bloc ( ). In ths case s called an orbt or set of transtvty of G. Every ermutaton on can berearded n the follown way as a lner substtuton n =n varables. the varables X1,..., Xn are taen 1 x x 1 x, x * x xn * x as onts. We form colum vectors n,, 1,...,n where s the n by n matrx corresondn to the lnear transformaton x x. ( 1,, s the well nown Kronecer symbol). We call * the ermutaton matrx corresondn to. such a matrx contans exactly one 1 n each row and column and zeros every where else. In addton, every ermutaton matrx * s orthoonal,.e., the transose * of * s dentcal * * wth ts nverse:, we obtan a fathful reresentatonof S G * G by. Now let G S. By G* we denote G the rou of all matrces * wth. Obvously G* s somorhc to G. We call G* the ermutaton reresentaton of G. 2.2.Theorem (See (22)). If a transtve ermutaton rou G s rearded as a matrx rou G*, then the matrces whch commute wth all the matrces of G* form a rn V=V(G). We call V "the centralzer rn corresondn to G". V s a vector B( ) sace over the comlex number eld whch has the matrces corresondn to the orbts of G1 as a lnear bass. In artcular, the dmenson of V concdes wth the number of orbts of G1. See(29), Theorem 28.4) Let 1,,r be the dfferent rreducble reresentatons aearn n G* where 1 s the dentty reresentaton. In the follown we always denote by the deree of (=1,,r), and by e the multlcty of n G*. In artcular, e n we have e1=f1=1 and, the reducton of G* ves for an arorately chosen untary n by n matrx U : U G*U [ 1,2,..., 2,..., n,..., n ]. e2 e r 2.3.Prooston G) Let M be the n by n matrx whose elements are all 1.then B( ) M. (Here the summaton s over all orbts of See ((29), rooston 28.2). 916

3 J Nov. Al Sc., 3 (8): , Theorem V s commutatve f and only f all the e=1. See ((29), Theorem 29.3) 2.5.Theorem (29.8) E * C V s commutatve f and only f the class matrces ( = 1,..,n) whose C be the th class of conjuate r B z E elements of G, enerate V,.e., when each B V has a (not necessarly unque ) reresentaton 1. See ((29), Theorem 29.8) 2.6.Theorem Let and be two orbts of G1. then Tr(B( ) B( ) n,, See ((29), Therem 28.10) 2.7.nton By a Burnsde-rou (n short : B-rou) we mean an abstract nte rou H wth the roerty that every rmtve rou contann the reular reresentaton of H as a transtve subrou s doubly transtve. (See (28),.343). 2.8.Theorem (See (27)).Every cyclc rou of comoste order s a B-rou. See ((29), Theorem 25.3) 2.9. Theorem (See (10)): G s doubly transtve. If n addton theorem). G s rmtve on, then G s even doubly rmtve (Jordan See ((29), Theorem 13.1) nton A ermutaton rou G on s called semreular f, for each, G 1; and G s called reular f t s semreular and transtve, Accordnly, every reular rou s also semreular and subrous as well as consttuents of semreular rous are semreular 1 s semreular. In the case of semreular rous, the deree and mnmal deree concde Prooston The order of a semreular rou s a dvsor of ts deree. A transtve rou s reular f and only f ts order and deree are equal. See ((29), Prooston 4.2) 917

4 J Nov. Al Sc., 3 (8): , Theorem Every normal subrou 1 of a rmtve rou s transtve. See ((29), Theorem8.8) 2.13.Theorem The reresentaton module assocated wth G* contans a one-dmensonal nvarant subsace corresondn to 1 the dentty reresentaton, namely, the one enerated by 1, and because of the transtvty of G t contans no others. The dentcal reresentaton therefore aears n G* wth multlcty exactly 1. See ((29)), theorem 29.1) Theorem G s doubly transtve f and only f dm V(G)=2. In ths case G* has exactly two rreducble consttuents. In artcular, we have r= and V(G) commutatve. See((29), Theorem 29.9) Theorem (See (1),.341). Every nonsolvable transtve rou of rme deree s doubly transtve. See ((29), Theorem 11.7 and (1),. 341) Prooston Every abelan rou G transtve on s reular. G s ts own centralzer n See ((29), Prooston 4.4) S Prooston If G s rmtve on and and are dfferent onts of, Then ether rme deree. G G or G s a reular rou of See ( (29), Prooston 8.6) Theorem Let G betranstve on,,, G not a rme number,. Let G have a subrou H ntranstve on G H wth the roertes H H. H H. and Then G s mrmtve and See ((29), Theorem 27.5) Theorem Pared orbts have the same lenth. 918

5 J Nov. Al Sc., 3 (8): , 2014 See ( (29), Theorem 16.3) nton Let G s transtve and consder the orbts of G1. Wth each of these orbts {1} (ncludn the trval ) we B( ) V assocate n the follown way a matrx,, 1,..., n,, wth elements: 1, f th ereexsts G and wth 1 and. V, 0, otherwse B( ) V 1 Thus, n the rst colum of we have exactly those, for whch holds. If, B( ) the ones of B( ) (, ) and do not occur n the same lace. On the other hand, for each lace there s an orbt of G, (namely, the one n whch the 1 les) B( ) wth such that has 1 n ths oston Theorem The matrces corresondn by denton 2.18 to ared orbts of G1 are transoses: B( ) (B( )). See ((29), Theorem 28.9) 2.22.Theorem If G has an orbt wth rou. 2, then G contans a reular normal subrou R of ndex 2. G s a Frobenus See ((29), Theorem 18.7) Theorem (A) If the rreducble consttuents of G* are all dfferent,.e., f all the multlctes e=1, then the ratonal number -2 n q n f 1 s an nteer. (B) If n addton the numbers n are all dfferent, then q s a square. (C) If the rreducble consttuents of G* all have ratonal characters, then q s a square. The hyothess s always fullled f the derees are all dfferent. of 2.23.(A) It sufces to show that q s an alebrac nteer. The notaton of the recedn secton s contnued. Let U aan be the untary transformaton matrx ntroduced n 2. From the hyothess e=1 t follows that every BV has daonal form. Because B B( ) V(G) matrx M=U BU wth we have n artcular M U BU [w1,w2,..., w ], f2 f f whch s the by dentty matrx. Let w be the daonal elements of the matrx M U BU, for arbtrary B V. We ut B zb, z1 z, z w w1, w 919

6 J Nov. Al Sc., 3 (8): , 2014 N [ 1,n2,..., n ], F [1,f2,..., f ], and I=(wj);,j=1,,. From M U BU t follows that M M U B BU, snce U was assumed untary. Wth the ad of 2.4 we now obtan znnz zzn n z z jtr(b Bj ) Tr(B B) Tr(M M) w w wfw. 1, j w z Because j jwj,.e., w Iz, n n Nn F II we therefore have I I. Nn I FI. By tan the determnant we et The wj, as eenvalues of the matrx Bj whch has nteer coefcents, are alebrac nteers, and therefore I I and are also alebrac nteers. I We wsh to show that s dvsble by n. by 2.3, j B j =M where M s the n by n matrx consstn of n 2 ones. 1 M has the eenvalue n occurrn wth multlcty 1 belonn to the eenvector 1. Its remann eenvalues are 0. In the daonal matrx j M j,n therefore aears exactly once, the remann elements ben 0. Therefore j w j n for =1 and =0 for the remann. Ths mles Hence -2 q n I I n w12 0 w22 I w1 w2 0 (modn). 0 w2 w s an alebrac nteer, hence also a ratonal nteer. of 2.23(C). (a) e1 e 1, Because of the hyothess the commutatvty of V follows by 2.4. Theorem 2.5 yelds the B xjc j ( 1,..., ). C exstence of class matrces 1,..., C x and of comlex numbers j such that j1 C x xj Bj. j x Conversely by 2.2 there are also wth j1 j The are, by well-nown theorems of lnear alebra, unquely determned and ratonal, snce the matrces B and Cj are ratonal. (b) By hyothess all the rreducble characters aearn n G* are ratonal. Thus the matrcs U Cju aearn M U BU xju CjU n the roof of 2.5 are also ratonal. By (a) the matrces are then ratonal. The wj are w therefore ratonal. Snce the j were already n the roof of 2.23 (A) shown to be alebrac nteers, they are ratonal ] w j ], n ] nteers. s therefore a ratonal nteer. Snce n dvdes s also a ratonal nteer, and q s therefore a square as was asserted. 920

7 J Nov. Al Sc., 3 (8): , 2014 (c) The hyothess that all the rreducble characters aearn n G are ratonal s fullled f the derees the rreducble consttuents of G are all dfferent. For snce G* s ratonal, wth each rreducble reresentaton all reresentatons conjuate to t aear n s therefore ratonal. G. Because all of the are dfferent, these concde wth, and of X 3. Man Result In ths chater we reare the roof number the rreducble consttuents reresentatons,..., 3 r are conjuate. In artcular of f3... fr f, and f dvdes. G such that f 2 and the For roof of man result of ths aer we rve the follown stes. (Ste1) We can assume wthout loss of eneralty that For f =2, then G s easly seen to be S or A ; hence G s doubly transtve. (Ste2) Every element dfferent from 1 of a Sylow -subrou of G s a roduct of two -cycles. Every Sylow - subrou of G s semreular and has order. Let P be a Sylow -subrou of G and 1 xp. Snce the order of P s a ower of, x conssts of -cycles and cycles of lenth 1. By 2.6 (theorem of Jordan) G has no -cycles and x s therefore a roduct of two -cycles. In artcular, x moves every ont, hence P s semreular. By 2.11, P P s a dvsor of 2, thus because 2 by (Ste1). (Ste3) If G and a s the order of, then ether a= or (a, ) =1. a h / 1 We assume that dvdes a and. Then, thus h has order. By (Ste2), h s a roduct of two -cycles. Therefore n no cycle can aear whose lenth s rme to, snce h has only cycles of lenth. If were a 2-cycle, G would contan the reular rou <>. whch s cyclc and of comoste order 2. By 2.8 (theorem of Schur) G would be doubly transtve, whch s not the case. therefore has only -cycles, hence has order, whch contradcts our assumton. 1,..., r We aan denote by the dfferent rreducble consttuents of the ermutaton reresentaton G of G. f In addton let ( 1,..., r) be the deree of let Tr( ) and be the character of. Let the numbern be chosen so that 1 s the dentty reresentaton. We now rove: Theorem We can number the rreducble consttuents of G such that are conjuate. In artcular f3... fr f, and f dvdes. f 2 and the reresentatons 3,..., r. 4(a) xg whch s the roduct of two -cycles. Wthout loss of eneralty we may ut By (Ste2) there s an x=(12 ) (+1 2). The characterstc olynomal of the ermutaton matrx x assocated wth x s (z ) 2. Hence x has the eenvalues 1, u,,u, all wth multlcty 2, where u s a rmtve th root of unty. We wsh to nvestate how these eenvalues are dstrbuted amon the (x). t(x) has the eenvalue 1 wth multlcty 1 and no others, snce 1 s the dentty reresentaton. 921

8 J Nov. Al Sc., 3 (8): , (b) We now show that >1 holds for 2. We assume =1. Snce G s not Abelan, but (G) s Abelan, s not fathful. The normal subrou N of all n G wth (n)=1 s therefore dfferent from 1 and henc (by 2.10) s transtve. Thus (N) =1 and also 1(N)=1, whch by 2.11 cannot be the case. (Note: Ths arument s vald for all rmtve non-abelan rous.) 4(c) Let the numbern of the rreducble consttuents 2,,r be chosen so that 2(x) has 1 as an eenvalue. Because f2>1 and snce the eenvalue 1 occurs for x only twce, 2(x) has an eenvalue dfferent from 1 whch wthout loss of eneralty may be assumed to be u. 4(d) All reresentatons conjuate to a are consttuents of G* snce G* s ratonal. 2 s conjuate to tself, for otherwse a (x)wth 3 would have the eenvalue 1, whch s mossble snce x has the eenvalue 1 altoether only twce. Therefore 2 (x) has the eenvalues 1,u,,u, all wth multlcty 1, for f an eenvalue,say u, aeared n 2 (x) wth multlcty 2, then because of the ratonalty of 2 so would u2,,u. Then, however, 1 and 2 would be the only rreducble consttuents of G*, and by 2.14 G would be doubly transtve. Hence we obtan f2=. 4(e) The remann eenvalues u, u (all wth multlcty 1) of x are dvded amon the remann reresentatons 3,,r. We now rove that these reresentatons are conjuate to each other and therefore have the same deree f. For r=3 there s nothn more to show. We assume r 4. It sufces (wthout loss of eneralty) to rove that 3 s conjuate to 4. Let u be an eenvalue of 3(x), u s (1 s - 1). one of 4(x) Because (, m s() and m 1( G /). G /)=1 (by (Ste2)) there s an m whch s a soluton of the two conruences Ths yelds (m, G )=1, and therefore an rreducble consttuent of G* conjuate to 3 s dened by () ( m 3 )( G). Because of the ratonalty of 1 and 2, we have 3. In addton, u m =u s s an eenvalue of (x). It s also an eenvalue of 4 (x). Snce u s occurs altoether only once n 3,,r we have =4, hence 4 conjuate to ( f ) In artcular we obton (f). In addton, 4(d) and 4(e) show that all the occur only wth multlcty 1: e1= =er=1. REFERENCES Burnsde W "Theory of Grous of Fnte Order," 2 nd ed. Cambrde Unv. Press, London; rernted 1958, Chelsea, New Yor. Burnsde W On smly transtve rous of rme deree. Quart. J. Math.37, Burnsde W On certan smly-transtve ermutaton rous. Proc. Cambrde Phl. Soc. 20, Brauer R On ermutaton rous of rme deree and related classes of rous. Ann. of Math. 44, Frame JS The derees of the rreducble comonents of smly transtve ermutaton rous. ue Math. J.3,87. Frame JS The double cosets of a nte rou. Bull. Amer. Math. Soc.47, Frame JS An rreducble reresentaton extracted from two ermutaton rous. Ann. of Math. 55, ITÔ N On rmtve ermutaton rous. Acta Sc. Math. Szeed 16, ITÔ N. 1962b. On transtve smle ermutaton rous of deree 2.Math.Z.78, The or`e mes sur les rous rmtfs.j. Math. Pures Al.16, Jordan C Mannn On smly transtve rous wth transtve abelan subrous of the same deree. Trans. Amer. Math. Soc. 40, Mannn WA On the rmtve rous of class ten. Amer. J. Math. 28, Mannn WA On the rmtve rous of classes sx and eht. Amer. J. Math. 32, Mannn WA On the rmtve rous of class twelve. Amer. J.Math. 35, Mannn WA On the rmtve rous of class fteen. Amer. J.Math Mannn WA On the order of rmtve rous. Trans. Amer. Math. Soc. 10, Mannn WA On the rmtve rous of class fourteen. Amer.J.Math.51, Mannn WA The deree and class of multly transtve rous. Trams. Amer. Math. Soc. 35, Mller GA On the rmtve substtuton rous of deree fteen. Proc. London Math. Soc. 28, Mller GA Lmts of the deree of transtvty of substtuton rous. Bull. Amer. Math. Soc. 22, Parer ET A smle rou havn no multly transtve reresentaton. Proc. Amer. Math. Soc. 5, Schur I Zur Theore der enfach transtven Permutatonsruen. S. B. Preuss. Aad. Wss., Phys.-Math. Kl. 1933, Suzu M On a class of doubly transtve rous. Ann. of Math. 75,

9 J Nov. Al Sc., 3 (8): , 2014 Thomson JG Fnte rous wth xed-ont-free automorhsms of rme order. Proc. Nat. Acad. Sc. U.S.A. 45, Wess MJ Prmtve rous whch contan substtutons of rme order and of deree 6 or 7. Trans Amer. Math. Soc. 30, Wess MJ On smly transtve rous. Bull. Amer. Math. Soc. 40, Welandt H Zur Theore der enfach transtven Permutatonsruen. Math. Z. 40, Welandt H Prmtve Permutatonsruen vom Grad 2.Math.Z.63, Welandt H Fnte Permutaton Grous, Academc Press Inc, New Yor Zassenhaus H. 1935b. Über transtve Erweterunen ewsser Gruen aus Automorhsmen endlscher mehrdmensonaler Geometren Math. Ann