Torsion units in group rings

Publicacions Ma emá iques, Vol 36 (1992), 47-50 . A bs ac TORSION UNITS IN GROUP RINGS VIKAS BIST Le U(RG) be he uni g oup o he g oup ing RG . In his pape we s udy g oup ings RG whose suppo elemen s o e e y o sion uni a e o sion, whe e R is ei he he ing o in ege s 7L o a ield K . Le R be a commu a i e ing wi h iden i y, G be a g oup and U(RG) be he g oup o uni s o he g oup ing RG . Deno e by T(G), he se o o sions elemen s o G . I is p o ed in [2], ha i T(U(7LG)) is a subg oup, hen T(U(3G)) = ±T(G) . In his no e we s udy g oup ings RG whose suppo o e e y o sion uni is in T(G) . Theo em 1 . Le R be an in eg al domain, F be i s quo ien ield and G be a non o sion g oup . I he suppo o e e y o sion uni o RG is in T(G), hen T(G) is a subg oup wi h e e y subg oup o T(G) no mal in G and e e y idempo en o FT(G) cen al in FG . P oo .. Le E T (G) be o o de n and le x E G T (G) .  Then a = + (1 - )x(1 -1- + . . . + n -1 ) E U(RG) andan= 1 . Since supp(o) C_ T(G) ; x = x k o some k = 1, 2, . . ., n - 1, hus x -1 x = x E ( ) . +3 i x E G T(G), hen x E NG(( )), whe e NG(( )) is he no malize o ( ) in G . I y E T(G) and x E G T(G), hen x E NG((y)) . Since NG((y))ICG(y) is ini e, so x' c CG(y) o some posi i e in ege m . Now (xy) m = xmYX --l YX --2 . . . yx y and as y' E (x), so (xy) TnI = xmk, whe e k is he o de o y . Hence xy is o in ini e o de . Thus xy E NG(( )) and so y E NG(( )) . Hence ( ) is no mal in G o e e y E T(G) . 48  V . BIST Le e be an idempo en in FT(G) and x E G T(G) . The e exis s E R such ha ex(1 - e) E RT(G) and so 1 + ex(1 - e) E U(RG) wi h (1 + ex(1 - e)) -1 = 1 - ex(1 - e) . Now o any E T(G), (1 - ex(1 - e)) (1 + ex(1 - e)) E T(URG) and + xb -x 2 0, whe e b = ( x e x (1 - e) - e x (1 - e) ) and O = 2e x2 (1 - e x ) ( e)x(1 - e), 6, O E RT(G) . Since ,3 E TU(RG) and T(U(RG)) C (RT(G)), so supp(0) C T(G) . Thus, b = 0 and O = 0 . Now xS = 0 implies ha ex(1-e) = ex(1-e) . Hence ex(1 - e) commu es wi h e e y elemen o RT(G) . Thus ex(1 - e) = 0 . Simila ly, we ha e (1 - e)xe = 0 . Thus i ollows ha ex = e o e e y x EG T(G) . Now i y E T (G) and x E G T (G), hen xy is also o in ini e o de . So ey = (ex)y = ex!' = e . Thus e is cen al in RG . This p o es he esul . By i ue o he abo e heo em he p oblem hus educes o de e mine RG such ha T(U(RG)) C_ U(RT(G)) . We now assume ha R is ei he he ing o in ege s 7L o a ield K . Fo he in eg al g oup ings 7G, we ha e he ollowing si ua ion . Theo em 2 . Le G be a non o sion g oup such ha T(G) is a sub- g oup and ha G/T(G) be igh o de ed . Then, he ollowing condi ions a e equi alen : (1) TU(7LG) C U(7LT(G)) (2) T(G) is ei he abelian o a Hamil onian g oup such ha i T(G) is nonabelian, a E T(G), o odd o de n, hen he mul iplica i e o de o ,2 in 7L n is an odd numbe (3) U(7ZG) = U(ZZT(G))G . P oo . (1) implies (2) . I T(G) is non abelian, hen by Theo em 1, T (G) =A x E x Kg, whe e A is abelian wi h e e y elemen o odd o de , E is elemen a y abelian 2-g oup and K8 is he Qua e nion g oup o o de 8 . Le a E A be o ode n . Then by [4, 11 .2 .6] (q((a)) x Ks) = (q(a))Ks = ® YI Q(~d)Ks- dln Also C¿( ,,)Ks - Q(bn) ®  ® Q(~n) ® S, whe e S is ei he a di ision ing o M2(Q(~n)) . By Theo em 1, e e y idempo en o QT(G) TORSION UNITS IN GROUPS RINGS is cen al in q(G), so Q(~,)Ks has no noncen al idempo en s . Thus S is a di ision ing and he e o e, Q(~  ,)Ks has no nonze o nilpo en elemen s . By [4, VIT13] a 2 + b2 + c 2 = 0 has no nonze o solu ion in Q(~  ,) and by [4, VI .1 .15], his happens p o ided he mul iplica i e o de o 2 modulo n is odd . (2) implies (3) is by [1] . (3) implies (1), ollows om an easy obse a ion ha U(W)/U(ZTCG))= G/T(G) . E Finally o g oup algeb as we ha e he ollowing heo em . He e K * G deno es he c ossed p oduc o G o e K . Theo em 3 . Le K be a ield o cha ac e is ic p > 0, G be a non o sion g oup such ha T(G) is a subg oup and G/T(G) be igh o de ed . Fu he le G be such ha o e e y ini ely gene a ed subg oup H o G, T(H) is cni e . Then T(U(KG)) C_ U(KT(G)) i and only i T(G) is abelian g oup ha ing no p-elemen s and e e y idempo en o KT(G) is cen al in KG . P oo .. Suppose ha T(U(KG)) C U(KT(G)) . Then by Theo em 1, e e y subg oup o T (G) is no mal in G wi h e e y idempo en o KT (G) cen al in KG . I cha K = p > 0and E T(G) wi h o( ) = p, hen as ( ) is no mal in G, so IG : Cc ( ) i  < oo .  Siüce G is non o sion, he e exis s an elemen x o in ini e o de in CG( ) .  Then (1 + x(1 - )) = 1 and 1 + x(1 - ) ~ KT(G) . Hence T(G) has no p-elemen s . Finally i T(G) is non abelian, hen he Qua e nion g oup, Ks C T(G) and p :,A 2 as T (G) has no p-elemen s . So ZpKs = Zp ® 7L p ED Z I, ED 7Z p m M2 (7L,,), con ains anon cen al idempo en . Hence T(G) is abelian . Fo he con e se, we may assume ha G is ini ely gene a ed and so T(G) is ini e . Now KT (G) = Fi, a di ec sum o ields, since T(G) is ini e abelian and p does no di ide jT(G)j . I is gi en ha e e y idempo en o KT(G) is cen al in KG . Hence KG = KT (G) * G/T (G)  F i * G/T (G) i=I 49 5 0  V . BIST and so This p o es he heo em . Since G/T(G) is igh o de ed, ; by [4, VI . 1 .61 U (F i *G/T(G)) has only i ial uni s and so T(U(KG)) C D T(U(Fi * G/T(G))) = D T(U(F i )) C U(K(T(G)) . By Theo em 3 and [3] we ha e U(KG) =D lU(Fi * G/T(G)) . Co olla y 4 . Le K be a ield o cha ac e is ic p and G be non o - sion nilpo en o FC-g oup ha ing no p-elemen s . Then T(U(KG)) C U(KT(G)) i and onlg i T(U(KG)) is a subg oup . Re e en es 1 .  A .A . BOVDI, Cons uc ion o an in eg al g oup ing wi h i ial elemen s o ini e o de , Sibi sk . Ma . Zh . 21 (1980),-28-37- 2 .  C .P . MILICS, G oup whose o sion uni s o m a subg oup, P oc . Ame . Ma h . Soc . 81 (1981), 172-174 . 3 .  C .P . MILICS, G oup whose o sion uni s o m a subg oup II, Comm . Algeb a 9 (1981), 699-712 . 4 . , S .K . SCIIGAL, "Topics in g oups ings," Ma cel Dekke , New Yo k, 1978 . Depa men o Ma hema ics Punjab Uni e si y Chandiga h - 160014 INDIA Rebu el 2'1 de No eTn,b e de 1990