The maximal quotient rings of regular group rings III

Publicacions Matem`atiques, Vol 40 (1996), 15–19. THE MAXIMAL QUOTIENT RINGS OF REGULAR GROUP RINGS III Ferran Ced´ o To the memory of Andreu Pitarch Abstract We give a new proof of the main result of [1] which does not use the classification of the finite simple groups. Introduction. In [1] the proof of the main result, [1, Theorem 2.3], is based implicitly on the classification of the finite simple groups through the use of [6, Theorem]. We give here a proof of [1, Theorem 2.3] which does not use [6, Theorem]. The proof. Let Gbe a group. Recall that ∆(G)={g∈G|[G: CG(g)] <∞} is a characteristic subgroup of G. A group Gis an FC-group (finite conjugate group) if G=∆(G). A group Gsatisfies Min (minimal condition on subgroups) if every non-empty set of subgroups of G, partially ordered by inclusion, has a minimal element. A group satisfies Min-pfor the prime pif each of its p-subgroups satisfies Min. Throughout, Kdenotes a field and Ga locally finite group with no elements of order char(K). Thus K[G] is a regular group ring (cf. [9, Theorem 3.1.5]). Lemma 1. Suppose that Gis an FC-group satisfying Min-pfor all primes pand that the type Ifpart of Qr(K[G]) is non-zero. Then Gis abelian-by-finite. Proof: Suppose that Ghas no abelian subgroup of finite index. By [9, Lemma 6.3.3] and [1, Lemma 1.3(ii)], we may assume that Gis countable. By [10, Lemma 6] and [1, Lemma 1.3(ii)], we may assume G=∞  i=1 Hi/H, 16 F. Ced´ o where ∞ i=1 Hiis a (weak) direct product of finite groups Hi.Wemay also assume that H∩Hi=1for all i. Since Hand Hiare disjoint normal subgroups of ∞ i=1 Hi, they commute and we have H≤Z∞  i=1 Hi= ∞  i=1 Z(Hi). By [1, Lemmas 1.3(ii), 1.4 and Proposition 1.1], we may assume that the Hiare not nilpotent and every proper subgroup of Hiis abelian. By [8], Hi=Piti, where Piis a normal Sylow pi-subgroup of Hi and ti∈Hiis an element of order qni i(ni≥1), where qiis a prime. Furthermore, tiis not normal in Hiand tqi i=Z(Hi). Since the type Ifpart of Qr(K[G]) is non-zero, by [3, Lemma 4.2], there exists a non-zero abelian idempotent e∈K[G]. Thus there exists an integer n, such that Supp e≤n  i=1 HiH/H ≤G. Since Gsatisfies Min-pfor all primes p, it is easy to see that there are infinitely many distinct qi. Let qjbe such that j>nand Supp ehas no element of order qj. Let ˜ tj=1+tj+···+tqnj j−1 j. Note that ˜ tj/qnj j is idempotent. Since (x, tj) = 1 for all x∈Supp e,e˜ tj/qnj j∈eK[G]e is idempotent. Since eis abelian and (x, h) = 1 for all x∈Supp eand h∈Hj, we have e˜ tjh=eh˜ tj for all h∈Hj. Using the fact that qnj j−1 i=0 K[Supp e]ti jis a direct sum, we deduce that xtj∈Supp e˜ tjfor all x∈Supp e. Thus, given x∈Supp e and h∈Hj, there exist y∈Supp eand msuch that xtjh=yhtm j. Hence tjht−m jh−1=x−1y∈Supp e∩Hj=Supp e∩Z(Hj). Since Z(Hj)is aqj-group and Supp ehas no element of order qj, we see h−1tjh=tm j. But tjis not normal in Hj, which is a contradiction, so the lemma is proved. Let Jbe a right ideal of K[G]. As in [3], we define α(J, F ) = dim(J∩ K[F])/|F|for each finite subgroup Fof G, and α(J) = sup α(J, F ), where Franges over all finite subgroups of G. We denote by π(G) the set of all primes psuch that Ghas an element of order p.Ifπis a set of primes we say that Gis a π-group if π(G)⊆π. Maximal quotient rings 17 Lemma 2. Suppose that Gsatisfies Min-pfor all primes pand that the type Ifpart of Qr(K[G]) is non-zero. Then Gis abelian-by-finite. Proof: By [9, Lemma 6.3.3] and [1, Lemma 1.3(ii)], we may assume that Gis countable. By [1, Lemma 1.3(ii)] and Lemma 1, ∆(G)is abelian-by-finite. By [9, Lemma 12.1.2], ∆(G) has a characteristic abelian subgroup Aof finite index. Let σ=π(∆(G)/A). Since the type Ifpart of Qr(K[G]) is non-zero, by [3, Lemma 4.2], there exists a non-zero abelian idempotent e∈K[G]. Let H=Supp e and τ=π(H). Hence π=σ∪τis a finite set of primes. Now Ais the direct product A=Apof its p-primary parts. Let Aπ=p/∈πAp. Then Aπis characteristic in G. Consider ¯ G=G/Aπ. Let δ∈K[G]. We denote by ¯ δthe image of δin K[¯ G]. By [4, Lemma 7.6], ¯eis an abelian idempotent, and clearly it is non-zero. Let pbe a prime such that α(¯eK[¯ G]) >p −1and p/∈π. We shall see that p/∈π(¯ G). Suppose that p∈π(¯ G). Then there exists g∈G\∆(G) with o(g)=pn and o(¯g)=p. Let ˜g=1+g+···+gpn−1. Since K[G] is regular, there exists β∈K[G] such that ˜ge =˜geβ˜ge. By squaring it, we see that eβ˜ge is an idempotent in eK[G]e.By[2, Lemma 2.1], Supp eβ˜ge ⊆∆(G)H. Let H1=Supp eβ˜ge ∪Supp e. Thus p/∈π(¯ H1). Using the fact that we have a direct sum p−1 i=0 ¯giK[¯ H1], we deduce that pn−1¯e=pn−1eβ˜ge. Since p=0inK,¯e=eβ˜ge.Thus¯eK[¯ G]∼ =˜geβK[¯ G]. By [3, Lemma 1.2(iv)], α(¯eK[¯ G]) ≤α(˜gK[¯ G]). But an easy calculation shows that α(˜gK[¯ G]) = p−1. This contradicts the choice of p.Thusp/∈π(¯ G). Hence π(¯ G) is finite. Since ¯eis a non-zero abelian idempotent, by [3, Lemma 4.2], the type Ipart of Qr(K[¯ G]) is non-zero. By [1, Proposition 1.2], [ ¯ G:∆(¯ G)] <∞ and ∆( ¯ G)is finite. By [7, Theorem 3.13], ∆( ¯ G) satisfies Min-pfor all primes p.By[1, Lemma 1.4], [∆( ¯ G):Z(∆( ¯ G))] <∞.Thus¯ G is abelian-by-finite. By [1, Lemma 1.3(ii)], we may assume that ¯ Gis abelian. Let π1=π∪π(¯ G) and let Aπ 1=p/∈π1Ap. Then π(G/Aπ 1)⊆π1.By [9, Lemma 12.4.12], there exists a π1-subgroup Qof Gwith G=Aπ 1Q. By [1, Lemma 1.3(ii), Proposition 1.2 and Lemma 1.4], Qis abelianby-finite. By [1, Lemma 1.3(ii)], we may assume that Qis abelian. Since π(Q) is finite and satisfies Min-pfor all primes p,Qhas a minimal 18 F. Ced´ o subgroup of finite index. Thus by [1, Lemma 1.3(ii)], we may assume that Qcontains no proper subgroup of finite index. We shall see that Gis abelian. Let qbe a prime such that q/∈π1. By [1, Lemma 1.3(ii), Proposition 1.2 and Lemma 1.4], AqQhas an abelian normal subgroup Bof finite index. Now [Q:Q∩B]<∞, thus Q≤B.NowBis the direct product B=Bpof its p-primary parts. Let Bπ1=p∈π1Bp. Since Qis a π1-group, Q≤Bπ1.Thus AqQ=AqBπ1. Since Bπ1is a normal subgroup of AqBπ1,AqQis abelian. Hence G=Aπ 1Qis abelian. Theorem 3 ([1, Theorem 2.3]).The type Ifpart of Qr(K[G]) is non-zero iff [G:∆(G)] <∞and ∆(G)is finite. Furthermore, in this case the type Ifpart of Qr(K[G]) is isomorphic to Qr(K[G/M]), where M=∩Land the intersection is over all subgroups Lof Gof finite index. Proof: The proof of the “if” part and the second part is as in [1]. Suppose that the type Ifpart of Qr(K[G]) is non-zero. Suppose that [G:∆(G)] = ∞or |∆(G)|=∞.By[1, Lemma 1.3(i)], there exists a non-zero central idempotent u∈K[G] such that uK[G] has bounded index of nilpotence. Thus, by [9, Theorem 5.3.15], uK[G] does not satisfy any polynomial identity. By [1, Lemma 2.2], there exists an irreducible uK[G]-module Vwith representation ρ:K[G]→End Vsuch that ρ(G) has no abelian subgroup of finite index. Since uK[G] has bounded index of nilpotence, by [4, Corollary 7.10], V is finite dimensional over its commuting ring. By [5, Lemma 2.4], ρ(G) satisfies Min-pfor all primes p. Let ϕ:K[G]→K[ρ(G)] the natural projection. It is easy to see that ϕ(u) is a non-zero central idempotent of K[ρ(G)]. By [4, Proposition 7.7], ϕ(u)K[ρ(G)] has bounded index of nilpotence. By [4, Corollary 7.4 and Theorems 7.20 and 10.24], the type Ifpart of Qr(K[ρ(G)]) is non-zero. By Lemma 2, ρ(G) is abelian-byfinite, a contradiction, thus [G:∆(G)] <∞and |∆(G)|<∞. Acknowledgements. This work was partially supported by the DGICYT through grant PB92-0586. References 1. F. Ced´ o, On the maximal quotient ring of regular group rings, J. Algebra 115 (1988), 164–174. 2. F. Ced´ o, The maximal quotient ring of regular group rings. II, Proc. Amer. Math. Soc. 104 (1988), 357–362. Maximal quotient rings 19 3. F. Ced´ o and B. Hartley, Maximal quotient rings and essential right ideals in group rings of locally finite groups, Publ. Mat. 36 (1992), 473–480. 4. K. R. Goodearl,“Von Neumann Regular Rings,” Pitman, London, 1979. 5. B. Hartley, Injective modules over group rings, Quart. J. Math. Oxford Ser. (2) 28 (1977), 1–29. 6. B. Hartley and M. A. Shahabi Shojaei, Finite groups of matrices over division rings, Math. Proc. Cambridge Philos. Soc. 92 (1982), 55–64. 7. O. H. Kegel and B. A. F. Wehrfritz,“Locally Finite Groups,” North-Holland, Amsterdam, London, 1973. 8. G. A. Miller and H. Moreno, Nonabelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc. 4(1903), 398–404. 9. D. S. Passman,“The Algebraic Structure of Group Rings,” WileyInterscience, New York, 1977. 10. R. L. Snider, Group algebras whose simple modules are finite dimensional over their commuting rings, Comm. Algebra 2(1974), 15–25. Departament de Matem`atiques Universitat Aut`onoma de Barcelona 08193 Bellaterra (Barcelona) SPAIN Rebut el 16 de Juny de 1994