W ' \ H ‘l w W 1“ H J! N MM Ll L HM ‘ 0N. H-PROJECTIV‘E FG-MODULES' AND THE COMPLETE REDUCJBIUTY 0F INDUCED FHvMODULES ION (DLON Dissertatic-n for the Degree of Ph. D. MICHEGAN STATE UNIVERSITY POTTER 19:75- This is to certify that the thesis entitled ON H-PHOJECTIVE FG-MODULES AND THE COMPLETE REDUCIBILITY OF lNDUCED PEI-MODULES presented by Gloria Potter has been accepted towards fulfillment of the requirements for Ph. D degree in Mathematics ,4' /’/ ~ We - Major professor Date «fa/n2 2-3! /77J'. 0-7639 ABSTRACT ON H-PROJECTIVE FG-MDDULES AND THE COMPLETE REDUCIBILITY OF INDUCED FH-MDDULES BY Gloria Potter Let H be a subgroup of an arbitrary group G and let F be a field of characteristic p. Let c. denote the class of pr0per subgroups H of G for which the induced FG-module NG = N GTH FG is completely reducible for each irreducible FH-module N. Our first result shows that if H 6 C' then H must have finite index in G. Moreover, we show that the index is a unit in the base field F, thus proving that if H 6 Cu then every FG-module is H-projective, i.e. that (G,H) is a projective pairing in the sense of Khatri. For finite groups and normal subgroups it is known that these two pr0perties are equivalent to a third property of the Jacobson radical namely that Rad FGIE.Rad FH-FG.. Rad denoting the Jacdbson radical of the ring concerned. we will say that H E R if this inclusion holds. Now the necessity of finite index for projective pairing prevents such a theorem from holding for algebras over infinite groups. However, we do give suffi— cient conditions for the three classes of sUbgroups to coincide for normal subgroups. Gloria Potter ii Now suppose that we have a group G for which H 6 Cr: H E E a (G,H) is a projective pairing. Then we say that G is a GWCPgroup. we begin chapter two by showing that infinite GWCPgroups do exist in the form of locally finite p- groups. Then we show that we can construct more GWCPgroups by taking either extensions of locally finite p-groups by locally finite éWUPgroups or extensions of locally finite GWC—groups by finite p'—groups. Next we show that quotients of infinite GWCP groups by p-groups are éfibrgroups. Finally we extend the list of known OECPgroups to include the dihedral and dicyclic groups. Chapter three builds up machinery on induced modules for finite dimensional group algebras and ends with a counterexample to Khatri's conjecture that extensions of éwcpgroups by 9EC~ groups are OECPgroups. In chapter four we investigate replacing the base field F by a commutative ring R. In the main theorem of the chapter, we give conditions under which Rad FG'E Rad FH-FG implies Rad RG 5 Rad RH°RG for an F-algebra R. ON H-PROJECTIVE FG-MODULES AND THE COMPLETE REDUCIBILITY OF INDUCED FH-MODULES By w Gloria\Potter A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1975 ACKNOWLEDGEMENTS First I would like to thank Dr. Fred Connell for his constant encouragement during the early portion of my graduate career. I am also indebted to Dr. Steven Gagola for supplying me with pertinent information on irreducible representations of finite group algebras. Most of all, I would like to thank my thesis advisor, Dr. I. Sinha, whose patience and many helpful suggestions made my research an enjoyable project. iii TABLE OF CONTENTS Chapter Page I. PRELIMINARY RESULTS ON PROJECTIVE PAIRING AND THE COMPLETE REDUCIBILITY OF INDUCED FH-MODULES 1 Section 1. Introduction . . . . . . . . . . . 1 Section 2. Relations between projective pairing pr0perty p, and the complete reduci- bility of induced modules. . . . . 8 Section 3. Normal subgroups . . . . . . . . . 12 II. QECPGROUPS 17 Section 1. Extensions of éfibrgroups . . . . . 17 Section 2. Quotients of éflbrgroups. . . . . . 22 Section 3. Dihedral and Dicyclic groups . . . 25 III. ANSWERS TO TWO QUESTIONS OF KHATRI 28 IV. FURTHER RESULTS ON PROPERTY 9 37 Section 1. Introduction . . . . . . . . . . . 37 Section 2. PrOperty p and locally finite abelian groups . . . . . . . . . . 39 Section 3. Property p and the subgroup fp(cs)............... 43 BIBLIOGRAPHY 47 CHAPTER I PRELIMINARY RESULTS ON PROJECTIVE PAIRING AND THE COMPLETE REDUCIBILITY OF INDUCED MODULES §l. Introduction Let F be a field and G a multiplicative group. Then the group ring F[G] of G over F is an F—algebra with the elements of G as a basis. To be more precise, F[G] consists of all formal finite sums a = Z)agg with g E G, ag 6 F. Then F[G] is a ring where the addition is given componentwise and multiplication is defined distribu- tively using the multiplication of the group G. If G is a finite group then F[G] is a finite dimensional F—algebra and there are fairly strong structure theorems for studying such algebras. On the other hand, if G is infinite, then these methods are no longer available and the prdblems of studying the group ring are correspondingly more difficult. For this reason the first papers (with a few exceptions) on infinite group rings did not appear until the early 1950's. we will be studying infinite group rings as well as finite dimensional algebras. A few simple but useful observations can be made immediately. Suppose H is a subgroup of G. Then the inclusion H's G gives rise to the obvious inclusion F[H] g F[G]. In fact, if {Xi} is a set of right coset representatives for H in G, then F[G] = Z)F[H]xi exhibits F[G] as a free left F[H]-module. Since group rings are rings after all, the questions we ask about them must necessarily be ring theoretic although the techniques involved often are strongly group theoretic in nature. One of the questions that frequently arises in the study of rings is "What does the radical of the ring look like?", where by radical we shall mean Jacdbson radical, abbreviated Rad FG. According to Passman, the semi—simplicity prdblem seems to be the most difficult of all the group ring problems. It is the problem of classifying all semi-simple group rings and it is the prdblem of determining the structure of the radical of a non-semi-simple group ring. The semi-simplicity problem for infinite group rings has been studied for nearly 21 years. For finite groups we do know: Theorem 1.1. (Maschke) If G is a finite group, then F[G] is semi-simple if and only if char F = O or char F = PXIGI- But very little is known about Rad FG itself if [G] is divisible by p. Therefore, if H.S G it could be useful to know when Rad FG.§ Rad FH-FG. We will be studying this inclusion giving conditions under which it holds. Since the radical is that ideal which annihilates every irreducible FG—module we are led to consider the irreducible FH-and irreducible FG-modules. This brings us to the notion of an induced module. Throughout this paper, by the way, all modules under consideration will be right modules unless specified otherwise. Let W 'be an irreducible right FH-module. Then W can be made into a module for PG by tensoring with FG. Namely, W GFH FG is a right FG-module called the induced module. Although W is irreducible, the induced module W ®FH FG is rarely irreducible. But it is often completely reducible as we will see later. For now it will suffice to give the connection between these induced nodules and the inclusion Rad FG _c_ Rad FH-FG. Theorem 1.2. (Sinha and Srivastava [9]) If for every irreducible FH-module W, the induced module W 8TH FG is completely reducible, then Rad FG _c_: Rad FH-FG. In keeping with Sinha's and Srivastava's terminology we will say that the pair (FG,FH), or simply (G,H) if the field is clear from the context, has property p whenever Rad FG‘EfRad FH-FG. It is easy to see that property p is equivalent to the following criterhni: If {xi|i 6 I} is a right transversal for H in G and if Z3pixi e Rad FG. with pi 6 PH, then pi E Rad PH. 1 As a result of Theorem 1.2 it is clear that we need to study induced modules. The following elementary results con- cerning induced modules will be crucial: (See for example [2]). Lemma 1.3. If W is an FH-module and W = Wl e>W as FH-modules, then WG = W? 9 W? as FG-modules. 2 Lemma 1.4. Let {xili E I} be a set of right coset representatives for H in G and let W be an FH-module. Then W’G = O '2) W'® x. as vector spaces, where, for any iEI xeG, W®x=[w®xlw€W]. Lemma 1.5. If H g_K g_G and W is an FH—module then (WK)G = wG. 1 It is clear that W ®:x is an F[x— Hx]-module and that dimF(W ®:x) = dimFW. It is also clear from 1.4 that if H has finite index in G and if dimFW < a then dimFWG = [GzH]dimFW. Hence for every FH-module W there is a corresponding FG-module W’G with the above prOperties. On occasion, however, we will be starting with an FG—module, M say. In this case if H is a subgroup of G, then M can be regarded naturally as an FH-module by simply restricting the Operators to FH. The resulting FH—module will be denoted by Mk. It is clear that (MIQH = MB for H 3 K g G and that dimFM = dimF(MH) and we shall use these properties of restricted modules freely throughout the remainder of this paper. Also we will often be called upon to refer to the well— known theorem of Clifford (see [2]) concerning irreducible modules: Theorem 1.6. (Clifford) Let M be an irreducible KG—module where K is an arbitrary field, and let H g_o be a normal subgroup of G of finite index. Then MH is a completely reducible KH-module and the irreducible KH-submodules of MH are all conjugates of each other. Remark: In [2], Curtis and Reiner prove Clifford's theorem under the assumption that G is finite and the crucial step is the existence of an irreducible submodule of MH. How— ever, the added hypothesis of finite index on H makes MH a finitely generated KH-module so we can prove the existence of an irreducible submodule of MH. (I am grateful to Steven Gagola for his observation of this fact.) His proof goes as follows: Let H'A G and [G:H] = n < a. Let M. be an irredu- cible KG-module. Since Mn is a finitely generated KH—module there exists a maximal submodule, N, of MH' Let V be the irreducible KH-module MH/N and let szH 4 V be the natural map. Since HomKH(MH/V) 2K HomKG(M,VG) there exists a non-zero map f:M.4 VG. From the irreducibility of M.'we must have ker f = O, i.e. f is one-to-one and M can be considered contained in the KG-module VG = Z) V 919 since for any 966 g E G. V Gig is a KH-module. In particular Mfi contains some irreducible KH—submodule of the form V 8 9. Another concept which is closely related to property p is that of H-projective FG-modules. Let H be a subgroup of G. Then an FG-module M is said to be H—projective if every exact sequence 0 4 N 4 L 4 M 4 O of FG—modules for which the associated sequence 0 4 NH 4 LH 4 MH 4 O splits over FH, is itself split over FG. We Observe that in a sense H- projectivity is a generalization of the usual concept of pro- jectivity. For if we take H = {l}, the trivial subgroup, then M is H-projective if and only if M is projective in the usual sense. Many people including Gustafson [3], Khatri [5], Sinha [9], and Higman [4] have studied H-projective FG— modules, but Sinha and Srivastava were the ones responsible for noticing its connection with the radical. Higman [4] was responsible for characterizing H-projective FG-modules in terms of induced modules in the following theorem that bears his name: Theorem 1.7. (Higman's Criteria) Let H be a subgroup of G of finite index. Then the following statements for an FG-module M are equivalent: (i) M is H-projective: (ii) M is a component (i.e. isomorphic to a direct )G ‘0 summand) of (MH (iii) there exists an FH-endomorphism n of M such n _-1' _ ._ that ii: xi n xi — lMt'where [Kill — l,...,n} are coset representatives for H in G and 1M is the identity map on M. Remark: When Higman originally proved the above theorem, he did so in the more general setting of a group ring RG where R is a commutative ring. We shall use this fact later on. If H is a subgroup of G and R is a commutative ring we say that the pair (RG,RH), or simply (G,H), is a projective pairing if and only if every RG-module is H- projective. Therefore, (G,H) is a projective pairing over R if every exact sequence 0 4 N 4 L 4 M 4 O of RG- modules for which the exact sequence 0 4 NH 4 LH 4 MH 4 O splits over FH, is itself split over RG. With this definition Higman [4] and Gustafson [3] showed Theorem 1.8. Let R be a commutative ring with identity, G a group, and H a subgroup of finite index. If [G:H] = n is a unit in R, then (RG,RH) is a projec— tive pairing. In the course of our investigations the notion of a Frdbenius group will come into play. Therefore we offer Definition 1.9. A finite group G is called Frdbenius with kernel M and complement K if G = MK, M A K, K n M = l, and K n Kx = l for all x 6 G - K. Finally we will make frequent use of the following easy lemma concerning radicals, whose proof is essentially that of Lemma 16.5 in [6]. Lemma 1.10. Let H be a subgroup of G and let R be a ring with identity. Then Rad RG n RH 55. Rad RH. §2. Relations between projective pairing, property pp, and the complete reducibility_of induced modules Let R denote a commutative ring with identity. Let 9 denote the class of pr0per subgroups H of G such that (G,H) is a projective pairing over R. Let R denote the class of proper subgroups H of G for which the pair (G,H) has pr0perty p, and finally let c. be the class of proper subgroups H of G for which the induced modules NG are completely reducible for every irreducible RH—module N. With this notation, Theorem 1.2 says that CgE R. In this section, we will show that if either H 6 c. or H e 9 then H has finite index in G and the index is a unit in the base ring. This will yield as a corollary the inclusion G E 9. Theorem 1.11. Let R be a semi-simple artinian ring and let G be a group with subgroup H. If H E c. (H 6 9) then H has finite index in G. Furthermore, consider the following three conditions: a) H60 (H69) b) R eRH RG = RG is a completely reducible RG-module (the trivial module, R, is H-projective) c) [G:H] is a unit in R. Then a) =‘b) = c). Further, c) = a) if H A_G. Proof: a) ='b) If H E 9 then all modules, including the trivial module R, are H-projective and there is nothing to prove. So suppose H E G. Since R is semi-simple artinian it can be written as the direct sum of fields; R = Fl @...@ Fn. Now each Pi is an irreducible RH-module and hence Fi 8RH RG is a completely reducible RG-module. Thus, R ®RH RG = igd Fi GRH R6 is completely reducible. b) = c) Since R GRH RG is completely reducible the sequence 0 4 Ker fl 4 R ®RH RG I'h'Jvl'h 4 O 2 splits where fl is the RG-module epimorphism defined by fl(x Sly) = x-y with the trivial action of G. Hence there exists an RG-module homomorphism f :R 4 R 8RH RG with 2 1 0 f2 = 1R. (If H 6 9 then the trivial module is H- projective and the sequence splits since it splits as RG-modules f using the map x 4 x a 1.) Let [gili e I] be a right trans- versal for H in G and assume that 91 = 1. Then we can write f2(l) = gpiu Ogi) where pi 6 RH. Writing pi = Z )‘jihji With xji 6 R we have f2(1)= Z; xjih ji a 91 = 2233x31(1h3®g) =§Z§1ji(1®gi) =§ri® 91 where we let = Zixji 6 R. Since fl 0 f2(1) = 1 there exists some i 3 . for which ri # 0. Therefore, assume 9k is a coset representa- . . -1 -1 _ tive for which rk # 0. Then f2(1) — f2(l g.k ) _ f2(1)gk _ rku O 91) + Z (ri 8 919121) . Hence the coefficient of l a 91 i is non-zero. Now if g. is any coset representative for H in G, we have f2(l) = f2(l-gj) = f2(1)gj = r1(l Olgj) + Z) ri(1 Q‘gigj). By uniqueness of representation of f2(l) i¥1 we know r1 = rk and hence the coefficient of 1 ® gj is lO non-zero. In this way we see that every coset representative appears in the finite sum f2(l) = Z)ri(1 8’91) thus proving H has finite index in G. Let this index be n. Then n f2(1) = iii ri(l ® gi) where ri = rk V1. Applying :1 to both sides of this equation we get 1 = fl 0 f2(1) = Z, ri-gi = n i=1 r. = n-r . Hence n is a unit in R. . 1 k 1:1 It only remains now to prove c) = a) for H.A G. By virtue of Theorem 1.8 it suffices to prove 0 §<3 for normal subgroups. So let H E 9 and let xl,...,xn be a set of coset representatives for H in G. If M is an irreducible n RH—module then MG = Z} M Q xi where M @zxi is an irreduc— i=1 ible R[x;lei] = RH—module. Hence M9 is completely reducible as an RH-module. Now let N be any submodule of MG and form the exact sequence 0 4 N 4 MG 4 MG/N 4 0. Since MG is completely reducible as an RH-module the asso— ciated exact sequence G G o 4 NH 4 (M )H 4 (M /N)H 4 o splits as RH-modules. By projective pairing we have the original sequence splitting as RG-modules. Hence N is a direct summand of MG. Since N was arbitrary, M9 is completely reducible. This completes the proof of the theorem. Remark: We remark that in proving a) =‘b) = c) for H E 0 ‘we did not use the hypothesis that R ‘was semi—simple artinian. ll Corollapy 1.12. Let R be a commutative ring with identity. Then (G,H) is a projective pairing if and only if H has finite index equal to a unit in R. Corollary 1.13. 635 9. 8‘; 9. P‘é Cu EEQQE} That ng 9 is immediate from the theorem and 1.8. To see that p.g 9 and E g,c. let G be any infinite group for which FG is semi-simple. (For example G could be the additive group of integers.) Then the trivial subgroup H = [1} E E 'but since [G:H] = a, H £ 9 and H g C“ Remarks: 1) In the theorem we cannot have c) = a) in general for then c.= 9. But Khatri showed in [5] that for the group AS and a field F of characteristic 3, PAS has a subgroup H 2.A4 such that (A5,A4) is a projective pairing but H A C. 2) In chapter 3 we will give an example to show that c) )6 b) in Theorem 1.11. As was shown in the above theorem if H.A G and H 6 9, then H 6 C“ For purposes of future reference we shall give this result a number: Theorem 1.14. If H A_G and H 6 9, then H 6 c where G is any group and R is a commutative ring with l. 12 §3. Normal subgroups In [5], Khatri shows that the three classes of subgroups G»R.9 are equivalent for normal subgroups if all groups considered are finite. From Corollary 1.13 it is clear that such a statement is false in general for infinite groups. Nevertheless, we can give sufficient conditions for the three classes to coincide for normal subgroups and our result yields Khatri's theorem as a corollary. For the main result of this section we will need the following definition and a lemma of Passman. Definition 1.15. If G is a group, let AG = [x 6 G; [G:CG(x)] < a} where CG(x) is the centralizer of x in G. Lemma 1.16. (Passman [6]) Let H be a torsion sub- group contained in AG such that H A G. Then Rad FHERad FG. Let H be a subgroup of G. We denote by MG(H) the ideal in FG which is generated by the set [l-hlh E H}. It is well-known that if H.A G and m:FG 4 F(G/H) is the map obtained by extending the canonical map G 4 G/H to FG by . . __ _ , PG 3.: linearity, then Ker T - NG(H) - NH(H) PG and so [NETHTTFE _ F(G/H). Therefore 'fl%gy»e F and m(G) EgRad FG. Lemma 1.17. Let H and H1 be subgroups of G. Then 91(H1) 'FG n FH 5 ”(H n H1) 'FH. 13 Proof: Let [gili 6 I} be a set of right coset representatives for H1 in G, and let x e fl(H1)-FG n FH. Then we can write x = leigi where pi 6 ”(H1). Let i pi = ngijhij where fij 6 F, hij 6 H1’ and ZZfij O. 25’ 1 Since x 6 EH. hijgi E H V 1,]. Now write pi = jg: fijhij = f h + f. h. +...+ f h. = (fi1+f. h. th + f. h. th + il 11 12 12 imi 1mi 12 12 11 13 13 11 "' -1 . -l - -1 _ + fim.him.hil)hil ' Slnce hijhil ‘ hijgigilhil ‘ -1 -l . lgi) . the hijhil E H 0 H1 V j and therefore (h..g.)(h. ij 1 1 Pi 6 ”(H n H1)°FH V i. This shows x e ”(H n H1)-FH as desired. We can now state the main theorem of this section: Theorem 1.18. Suppose H'A G ‘with [G:H] < u and AG a periodic. Also suppose that whenever x 6 G - H and xp 61L a #’O, then x 6 AG where char F = p. Then the following are equivalent. 3) H69 b) H e c c) H 6 E Furthermore, none of the hypotheses can be dr0pped. Proof. a) 41b) is just Theorem 1.14 and b) 4 c) is always true by Theorem 1.2. So it suffices to prove c) = a). By Gustafson's result it suffices to show that G/H is a p’- a group. So let x E G - H with xp e H. Then by assumption a X 6 AG and (xp )n = l for some integer n. Since x has 14 only a finite number of conjugates in G there exists a finite group H’ A_G which contains x. (Take H’ to be the subgroup generated by x and its finitely many conjugates.) Then Rad FH' g Rad FG by 1.16 53 Rad FHoFG by hypothesis g;m(FH)'FG. Hence Rad FH’ _c_ 91(FH)°FG n FH' _c_: 91(F[H n H’])-FH’ by 1.17. Therefore, FH' H) F(—— 91(F[H nH'])-FH’ H n H’ is semi-simple since it is a homomorphic image of the finite dimensional semi-simple algebra §3§§5fi7" By Maschke's Theorem (1.1) firgffij’ cannot contain any elements of order pa. But xq is a p-element of H’ where npa = qu and (q.p) = 1. B B > a. Hence xq E H. Now 1 = aq + bp and therefore f3 B x = xaq-xbp = (xq)a(xp )b E H. Hence [G:H] is a unit in F which was to be proved. Now suppose the hypothesis of normality is dr0pped. Then Khatri's example of G==A ,H==A4. F a field of character- 5 istic 3 gives a counterexample to 9_§.C~ Nor can we drOp the hypothesis that AG is periodic for in that case we could take G = Z, the additive group of integers, and H the subgroup of Z consisting of all integers divisible by p where char F = p. Since F2 is semi-simple, O = Rad FZ g:Rad FH-FG and H 6 EL But [G:H] = p so that H g 9. 15 The hypothesis of finite index cannot be drOpped. Let char F = p and let G be an infinite locally finite abelian q-group where q # p. Then FG is semi-simple (see [6]). So 1 = H 6 R trivially but H does not have finite index in G so that H £ 0. (I. P Finally, we cannot drOp the hypothesis that x e H = x 6 AG. To see this let G be the wreath product zq ?,fl Zp. D Again from [6] (Theorem 21.2ii) we know that if F is a field of characteristic p then FG is semi-simple so that (G,H) has preperty p for any H g,G. Let W = H(Z ) . Then yea zp ‘1 Y W.A G and G/W a n z . Therefore there exists a one-to-one a P correspondence between the normal subgroups of G containing W and the subgroups of E Zp. In particular, there exists a normal subgroup H of G which has index p in G. Hence H £10 and therefore e‘g49. On the other hand, G -and H do not satisfy the hypothesis "xfia 6 H =»x 6 AG." For let x e 11 z - H. Then xp 6 H but x cannot be contained in AG. up Since G/W a n Zp acts on W, clearly Cw(x) consists of C all those elements of W’ whose projections in the factors (Zhjy are constant on the orbits of (x). Since nzp is infinite, [W:Cw(x)] = -. Hence x has infinitely many dis- tinct conjugates, i.e. x f AG. This completes the proof of the theorem. 16 We have Khatri's result as a corollary: Corollapy 1.19. (Khatri) If G is a finite group and H_A_G, then HeeeHeoeHe/E’. CHAPTER II OPO- GROU PS §2.l. Extensions of éflcrgropps In the previous chapter we studied property p and investigated its relationship to projective pairing and the complete reducibility of induced modules. we saw how the complete reducibility of induced modules implied both pro- jective pairing and property p but that the reverse impli- cations were false in general. However, there are many cases when the reverse implications do hold and in this chapter conditions will be given for which these three classes of subgroups do coincide with each other. Following Khatri [5] we make the definition: Definition 2.1. A group G is called a EWCrgroup over the field F if 0 = G= E- As Khatri pointed out, examples of ORCrgroups do exist for if F is a field of characteristic p and if p X [G] where G is a finite group, then trivially each class consists of all the subgroups of G. we also have Theorem 25;. Let G ‘be a locally finite p-group and F a field of characteristic ‘p. Then G is a afichroup. l7 18 Proof: we will show that each of the three classes of subgroups 9.K.Cv is empty. First we claim that any sub- group H of G of finite index must have index a power of p. For suppose H g_G has finite index. Then there exists a normal subgroup H of finite index such that H g.H. Since [G:H] = [G:H][H:H] it suffices to prove that the index of H is a power of p. But this is easy for if xH E G/H is an element of order n = mpY, say, with (m,p) = 1 then an = H a or xn E H. Since H is a locally finite p-group (xn)p = a a+Y xnp = 1 for some integer o. If m ¥ 1, then (xP )m = l O.+Y and xp is an element of G ‘which is not a p-element. This contradiction proves the claim. Together with Theorem 1.11 this fact shows that the classes 9 and c. are empty. To see that the class B is also empty, suppose there exists an H g G with Rad PG 5; Rad FH-FG. By Lemma 21.5 of [6] we know that Rad FG = {Z agngag = o}. If x e G - H, then 1-1 - l-x E Rad FG. But 1,x can be made part of a transversal for H in G thereby putting 1 into the radical of PH by property p. This contradiction shows that E is also empty. We can Obtain further éwcpgroups as follows Theorem 2.3. An extension of a locally finite p-group by a locally finite Oflchroup is a ORCPgroup. We need two lemmas 19 Lemma 2.4. Let I be a nil ideal in a ring R and let «:1! 4 R/I be the natural map. If Rad R and Rad R/I are both nil ideals, then -r(Rad R) = Rad R/I. Moreover: if 1-(r) = r + I ERad R/I, then r eRad R. Proof: We always have 1(Rad R) 5 Rad R/I. For the reverse inclusion, suppose rl + I e Rad R/I. Then for each r + I 6 R/I, (r1+I)(r2+I) = r r + I is nilpotent, i.e. 2 1 2 there exists an integer m such that (r +1)m = (r1r2)m+1 =I. 1‘2 Therefore, (r1r2)m e I. But I is nil, hence rlr2 is nil- potent. Since r2 was arbitrary, r1 6 Bad R and r1 + 1 .-. 1-(r1) 6 'r(Rad R). Lemma 2.5. If F is a field and G is a locally finite group, then Rad F6 is nil. Proof: Let x = alg1 +...+ angn 6 Rad PG and let H be the finite subgroup of G generated by the 91, i = l, ...,n. Then x 6 Rad FG n PH 5 Rad PH which is nilpotent. Proof of Theorem: Let G be an extension of a locally finite p-group P by a locally finite ORG-group. Since we always have c‘, _c_: R by 1.2, we need only show that O E. O and I? c: o. 0 £- 0: Let (G,H) be a projective pairing. Then since [G:H] = n a unit in F, H must contain P as a normal subgroup. Now we use a result due to Woods [12] which says if H A G and 6/3 is locally finite, then Rad PH 5 Rad PG. Therefore. 20 Rad PP = {Z aPPlZ ap = O} 55 Rad FH. So if N is an irreducible anmodule, then for every n e.N and p e P, n-l = nop since 1 - p 6 Rad PH _c_: Annih N. This shows that we can regard N as an irreducible F(H/P)-module. New (G/P,H/P) is a projective pairing and G/P is a OTC-group, so the induced F(G/P)—module NG/P is completely reducible, say NG/P = e N. . 161 1 G/P Since P acts trivially on N we can make N and the Ni into FG-modules by defining (n @ Pgl)g2 = (n O Pgl)Pg2- G/P becomes isomorphic to NG as an With this definition N FG-module via the correspondence n ® Pg 4 n a 9. Now the Ni remain irreducible as FG—modules since if M 5 Ni then M % Ni also as an F(G/P)-module. This proves the complete reducibility of NG over PG. 85 0. Suppose that (G,H) has preperty p and suppose [xili 6 I} is a set of coset representatives for H in G. Since 9 .A c, Rad FP = [Z aPPIZ aP = o} _c_ Rad PG 55 Rad FH'FG using WOods' result again. Then p - 1 = hxi - 1.1 6 Rad FH-FG implying l E Rad FH unless p 6 H. Thus P A H. Now let 2 Sipxi e Rad F(G/P) where the Pxi are right coset representatives for H/P in G/P, and Bi e F(H/P). If m:FG 4 F(G/P) is the natural map, then there exists 9i 6 PH such that cp(Pi) = Bi. and em pixi) = )3 Eipxi e Rad F(G/P). But F(G/P) a FG/HG(P) and all groups in sight are locally finite so that flG(P) = Rad FP-FG, Rad F(G/P), and Rad FG are nil by Lemma 2.5. Thus, we can apply Lemma 2.4 to conclude 21 Epixi 6 Rad PG 55 Rad FH-FG, i.e. p. 1 6 Rad FH and pi = CMPi) 6 cp(Rad PH) 5;. Rad F(H/P) . Since G/P is a OEO-group and (G/P,H/P) has been shown to have prOperty p, [G/P:H/P] = [G:H] is a unit in F. This completes the proof of the theorem. We also have Theorem 2.6. Let F be a field of characteristic p. Then an extension G of a Gwcrgroup L by a finite p’-group is a éfibrgroup. Proof: Again we need only show that 0 g G and I? E 9. 9 g G. Let (G,H) be a projective pairing and let N be an irreducible FH—module. Since L.A G, H n L A H and by Clifford's theorem NHnL = @>Z)Ni where the Ni are irreduc- ible F(H rtLl-modules. Now observe that [G:H] = [G:LH][LH:H] = [G:LH][L:H n L] and therefore [L:H n L] is a unit in F. By the hypothesis on L, each (Ni)L is completely reducible over FL. Since (G,L) is a projective pairing, (Ni)G = (N?)G are completely reducible over PG for all i. Since (H,H n L) is also a projective pairing, a result of Higman [4] gives that N is a )H. This in turn implies that NG is a )6: component of (NLnH component of the completely reducible FG-module (NHnL ((NfitflL)H)G' Hence NG is completely reducible over FG. 22 P g 9. Let H be a subgroup of G such that (G,H) has property p. Then Rad PG 5; Rad FH-FG. By Woods' result Rad FL 55 Rad F6 and therefore Rad FL 5 Rad FH-FG n FL. Let K = H n L and let {yili 6 I} be a set of coset representa— tives for K in L, i.e. L = L) Kyi. Then the yi also iel belong to a set of coset representatives for H in G. For suppose y1 = yjh for some h e H. Thus y; or y1 = yj k for some k e K, a contradiction. Thus, if x 6 Rad FL 5 Rad FH-FG n FL, then x = Z pnyn 6 FL with yi=h eHnL=K. pn 6 FK E FH. Since the yi are coset representatives for H in G, pn 6 Rad FH. Therefore, pn e Rad FH n FK‘E Rad FK. We have shown then that Rad FL 55 Rad FH-FG 0 FL 5 Rad F(H n L) -FL, i.e. (L,H n L) has prOperty p. Since L is a GWCPgroup. H n L has index a unit in L. Now [G:H OH = [G:H][H:H nL] = [G:L][L:H n L] which is a product of units in F. Thus [G:H] isaunit in F and H69. We will show in the next chapter that it is not true in general that extensions of Okc‘rgroups by ORG-groups are 98& groups, so perhaps Theorems 2.3 and 2.6 are the best possible. §2.2. Quotients of Mel-groups It is not known whether subgroups of ORG-groups are PEG-groups. Nor is it known in general if quotients of 9/80- groups are 9R6. However we do have some partial results in this direction. 23 Definition 2.7. If G is a group and p is a prime then GP is the subgroup of G generated by the elements whose order is a power of p. Theorem 2.8. Let F be a field of characteristic pI>O and let G be a locally finite 9k’,—group. Let AAG. If either A is a p-group or A.2 GP, then G/A is a QECPgroup. Proof: The case A EEGP is clear; for if G is a QECP group then GP must have finite index a unit in F since Rad FG‘ggRad FGp-FG (see [6]). Thus G/A is a finite p’-group which is trivially a OECPgroup. Now SUppose A is a p-group. we first show elg a. Let H/A ‘be a subgroup of G/A such that (G/A,H/A) is a projective pairing and let N be an irreducible H/A-module. Note that N is also an irreducible FH-module for n-h = n(hA) for n E N and h e H. But (G,H) is a projective pairing and G is a éwcpgroup, hence NG is completely reducible over FG. As in the proof of Theorem 2.3 NG is isomorphic to NG/A G/A as FG-modules. Hence N is completely reducible. To show I? E G, let H/A be a subgroup of G/A such that Rad F(G/A) E_Rad F(H/A)-F(G/A). We*will show that (G,H) also has property p. Let {x121 e I} be a right transversal for H in G and let leixi e Rad FG ‘with pi E FH. Let m:FG 4 F(G/A) be the canonical mapping. Then ¢(Z)pixi) = Z cp(pi)hxi e Rad F(G/A) _c_ Rad F(H/A) -F(G/A). Thus. 24 Moi) e Rad F(H/A) a semi—’11“) and by Lemma 2.4, Pi e Rad FH. Hence (G,H) has prOperty p and therefore is a projective pairing, i.e. [G/AzH/A] = [G:H] is a unit in F. We have two corollaries, the first of which follows immediately from the proof of the theorem. Corollary 2.9. If G is a aficrgroup and H.A G, then G/H has the property that 9 g 6. Corollary 2.10. Let G be a finite group with H'A G. Let F be a field of characteristic p and assume [G/H: is divisible by pCL with o g a g 1. Then ms/H) _c_: 9(G/H) and hence G/H is a QECrgroup. nggf: If p X :G/HI then G/H is a p'-group and there is nothing to prove. Suppose p I [G/HI but no higher power of p divides lG/H] then Rad F(G/H) is non-zero by Maschke's theorem. So if Rad F(G/H) s Rad F(A/H)-F(G/H) then Rad F(A/H) # 0. Hence (A/H) is divisible by p and A/H contains a p— Sylow subgroup of G/H. This proves E E 9 and the second assertion follows immediately from Corollary 2.9. Example 2.11. Let F be a field of characteristic 3 and let G be the linear group SL(2,5); then G is not a GWC—group. Proof: Recall that if char F = 3, then PAS is not a SL(2,52 5 2(6) Corollary 2.10, SL(2,5) is not a éfiergroup. 6WC~group. But PSL(2,S) e.A and PSL = so by 25 §2.3. Dihedral and dicyclic groups Besides Observing that finite p'-groups and finite p-groups were QECPgroups, Khatri was able to prove in [5] that p-nilpotent FrObenius groups are éwcpgroups. Using this result we will extend the list of known éfibrgroups to include the dihedral and dicyclic groups. Theorem 2.12. Dihedral groups are GNU—groups over any field F. Proof: write G = {a,b|am = l = b2, bab = am-l}. we first suppose that char F # 2. Let H = [l,a,a2,...,am-l}. Then H is an abelian group (and therefore a PHD—group by Corollary 1.19) and G is an extension of H by a group of order 2. Therefore by Theorem 2.6 G is a aficpgroup. Hence we can assume char F = 2. Case I: [G] = 2m, m odd. In this case, if we let M = [l,a,a2,...,am-l} and P = [1,b} then we easily have G = PM and P n M,= {1}. More— over, since conjugation of b by powers of a yield elements of the form ‘bak, k #’m, each conjugate of P ‘by an element out- side of P intersects P trivially. This makes G a 2- nilpotent FrObenius group with kernel M. By Khatri's result, then, G is a ORCPgroup. 26 Case II: 1G! = 2am', m’ odd, a > 1. . . m’ 2m’ In this case conSider the subgroup H = [a ,a ,..., a-l , a2 m = l]. H is a normal 2-group and G/H is a dihedral group of order 2m’. Hence by Case I, G/H is a Qwergroup. Therefore, by Theorem 2.3, we again have that G is a QECP grOUp. One may then ask if the infinite dihedral group is a GWCPgroup. The answer is no. Theorem 2.13. The infinite dihedral group is not a éwcpgroup over any field F. Proof: Write G = {x,y[y2 = l, yxy = x-l}. Let A be the torsion-free abelian subgroup generated by the element x. Then A A G and CG(A) = A. Hence, by Theorem 21.2(iii) of [6], FG is semi-simple. By 1.11 it is clear that G cannot be a GWCPgroup. (Take H = [1}, then H e E, 'but H f 9.) Theorem 2.14. Dicyclic groups are GKC—groups over any field F. 1 Proof: We can write G= [a,b[a2m=l,am=b2,b- aba=l} where [G] = 4m. If char F # 2, let H be the subgroup [1,a2,a4,...,a2m-2}. Then H is a normal abelian subgroup of G and G/H is a group of order 4. Hence by Theorem 2.6 we are done. 27 If char F = 2, let H = [1,am}. Then H is a normal 2-group and G/H is isomorphic to a dihedral group of order 2m. By Theorem 2.12 and Theorem 2.3 G is a GWCPgroup. This proves the theorem. CHAPTER III ANSWERS TO TWO QUESTIONS OF KHATRI Let F be a field of characteristic p. Then in [5] Khatri asks whether extensions of GWCPgroups by éficrgroups are éfibrgroups and in particular if extensions of p’—groups by p-groups are éWCrgroups. we will answer both of Khatri's questions with a counterexample. In the following, then, let G be a finite p—nilpotent group with p—Sylow subgroup P isomorphic to a cyclic group of order p. Let P have normal p-complement K. Notice that G is an extension of a p’-group and hence falls into the category of groups under discussion. we will also be assuming throughout this chapter that F is a finite splitting field for the group G. Next we Observe that the trivial module F is the unique (up to isomorphism) irreducible FP—module. We shall study FG under the assumption that it is completely reducible. Let T be a fixed irreducible FG-module such that TK is also irreducible and G-invariant, i.e. for any 9 e G, the FK-module TK 8 PK 9 is isomorphic to TK' Definition 3.1. Let G be a group and H a subgroup of G. If L is an FH-module, we say that the FG-module M is an extension of L if MH 21L. 28 29 Lemma 3.2. FG a FK as FK-modules. 2,k3,...,k‘}. the ki form a set of coset representatives for P in G. Proof: Let K = [1,k Then, since G==PK . G Therefore, F6 = F O 1 +...+ F ®Zk Define x:FK 4 F by 1' x(k) = 1 92k and extend linearly. Since x(kilki2) 1 O kilkiZ = (l O kil)ki2 = x(kil)kiz, l is an FK-module homomorphism which is easily seen to be one-to-one and onto. The next lemma is a well-known fact about irreducible modules and we refer the reader to [10] for a proof. Lemma 3.3. Let H.A G and let [G:H] = pa. Let N be an irreducible FH—module such that N a N e g for all g E G. Then N can be extended uniquely to an irreducible FG-module M. We also need some results on homogeneous modules. An FG-module V is said to be homogeneous if it is a direct sum of, say, d copies of an irreducible FG-module W. Still assuming that F is a splitting field for G, let a = (a1,a2,...,ad) e F x F x...x F where not all of the ai are zero. Define W5. a submodule of V, by W5 = [(aiw,a2w,...,adwlw e w}. With this notation we have Lemma 3.4. Every irreducible submodule of V has the form W5 for some a, and W5 = W5. if and only if a = xa’ for some i # O in F. 3O Proof: Let W’ be an irreducible FG-module and W5 as defined above. we must first show that W5 is an irreduc- ible submodule of V. It is clear that Wé is a submodule so it suffices to prove W5 is irreducible. Let NEWa Tbe a submodule and let 0 # (a1w1,...,aaw1) e N. Since 0 #'wl and since W is irreducible, FG-‘w1 = W, i.e. for any w e W, there exists x 6 FG such that xwl = W. Therefore x(aiw1,a2wl,...,adw1) = (aiw,a2w,...,adw) E N. Since w was arbitrary, N = Wé and W6 is irreducible. Next we show that any irreducible submodule N of V occurs in the form Wé° Let N be an irreducible submodule of V and let 0 # (w1,...,wd) e N. we need to show thatmw1.w2,..., w' 6 W are in the same one-dimensional subspace of W; With- d out loss of generality assume w:L 7! 0. Let 1rj :W O. . .e w 4 W denote the projection onto the jth coordinate. Then since N W‘4 W' belongs to EndFG(W) = F.1w since F is a splitting ‘wl # 0, v11 :N 4 W' is an isomorphism. Therefore rj°[wllN]-1: field, i.e. ”j 0 ['n'llN]—l is scalar multiplication. But -1 _ . . ”j o [wllN] (wl) —.wj E le. i.e. there exists aj E F such that aj‘wl = wj. we'have shown, then, that there exists 1 = a1,a2.....ad 6 F such that (wi....;wd) = (aiw1,a2w1,..., adwll 6 W5. Now suppose W5 = Wé.. where a = (a1,...,ad) and o _ I o a - _ a — (a1,a2,...,ad). Since wa — Wé,, for any ‘w 6 W there exists w 6W such that (alw,a2w,...,adw) = (alw ,a2w ,...,adw ). 31 So if 0 7! aiw, then aiw = ai’w’ =w’ = (ai/ai')w = bw, say. Choose l = b. 0n the other hand, if a = xa’ for some I E F and if (alw,a w,....a w) 6 W5, then (ha w,la2w,...,xadw) = 2 d l (aiw,a2w,...,adw) 6 W5. But (al(xw),a2(xw),...,ad(xw))6 Wé*' Therefore, Wa E wa,. By the irreducibility of Wa ., W = W ,. This completes the proof of the theorem. As an easy consequence of this we get Corollapy 3.5. If F is a finite splitting field for G and V is a homogeneous FG-module containing d isomorphic constituents of an irreducible submodule W, then there are l 'd- exactly 5 _i distinct irreducible submodules of V. 3 Lemma 3.6. (Green) Let N'A G and let V be an irreducible FG-module. Assume also that G/N is a p—group and ViN is homogeneous. Then VjN is irreducible. Proof: Since V is irreducible and N.A G, VN is completely reducible into (conjugate) F[N]-modules by Clifford's theorem. Let VN = Wl O...® Wd where all the W1 are isomor- phic as F[N]-modules, and W1 = W1 3 g for some 9 6 G. Since VN = 2 W1 8 9 (not necessarily a direct sum), G acts on 96G ‘ the irreducible submodules of VN. But all the irreducible submodules of VN are F[N]-modu1es, so that G/N acts on the set of all irreducible submodules of VN. Suppose that under this action no orbit is of size 1. Then, since the size of an 32 orbit divides the order of the group and since G/N is a p— group, all orbits will have size a power of p. This implies that the number of irreducible submodules of VN is divisible by p, contradicting Corollary 3.5. Therefore some orbit has size 1 and some irreducible submodule, Wa' is fixed by G. Thus, W5 is an irreducible FG—module contained in V. Thus W6 = V by the irreducibility of V. This proves the lemma. Lemma 3.7. The multiplicity of T as a composition factor of FG equals the multiplicity of T as a composition G) K factor of (F K G . . Proof. Let 0 — Vl ; V2 g. . .; VN — F be a compos1tion . G _ _ G series for F and let 0 - Ul ; U2 g”; Um — (F )K be a composition series for (FG)K refined from the composition G Ui+1 V'+i .' Ui+1 series for F . Then either -—U-— 52-6—- 1K for 3 or 7-1—— V.+1 i j i is a direct summand of —%r—-IK 'by Clifford's theorem. Suppose j V. V. —%il»e T for some i. Then by hypothesis —%¢l-|K a TK remains I 1 irreducible so that for each T which occurs as a composition factor of PG, there exists a composition factor of (FG)K isomorphic to TK. we now show that this is the only way TK can occur as a composition factor of (FG)K. Vi+1 So suppose v i T. Then there are two cases to i V. V. consider: either —$il-' is irreducible or —$:£-l is Vi 'K Vi K completely reducible into more than one constituent. First of 33 V. all if 'HHii'2K is irreducible then it cannot be isomorphic i to TK by Lemma 3.3. Secondly, if reducible into more than one constituent then Vi+1 -§7—-}K is completely 1 Vi+l I v. 'K 1 be homogeneous by Lemma 3.6. Thus there are at least two non- leti l . 'K' 1 cannot be one of the constituents since TK is invariant under G and the constituents are not. This shows that TK precisely the same cannot isomorphic conjugate constituents of Hence, T K occurs as a composition factor of (FG)K number of times as T occurs in FG, which was to be proved. Lemma 3.8. (FrObenius, see [2]) Let H.S G, W an G FH-module, and V and FG—module. Then HomFG(W',V).1?HomFH(W,VH). Theorem 3.9. DimFT = DimF(HomFP(F,TP)). Proof: By the FrObenius relation we know HomFP(F,TP) e HomFG(FG,T). Hence it suffices to prove dimFT = . G . G _ dimF(HomFG(F ,T)). By Lemma 3.7 we have dimF(HomFG(F ,T)) — . G‘ . , . dimF(HomFK(F 'K'TK))' But K is a p -group and by assumption F is a splitting field of characteristic p for G, so that . G. _ . __ . dimF(HomFK(F 'K'TK)) — dim(HomFK(FK,TK)) — dimFTK by Lemma 3.2 and the fact that in the regular representation of a semi-simple group algebra each constituent occurs as often as its dimension. Hence dimFT = dimFTK = dimF(HomFP(F,TP)) which proves the theorem. 34 Now we arrived at Theorem 3.9 under the assumptions that FG ‘was completely reducible as an FG-module, that T was an irreducible F[g]-module, and that TK was an irreduc- ible, G-invariant FK—module. In the following example we will have all of these hypotheses holding except we will not know whether FG is completely reducible or not. We will then show that Theorem 3.9 cannot hold in our example thus proving FG is not completely reducible. Theorem 3.10. An extension of a GWCPgroup by a éfibr group is not necessarily a éfibrgroup. ggggfc Let Q ‘be the group of quaternions generated by i,j,k. Then Q is a group of order 8 and there exists a‘ cyclic group of order 3 acting on Q by permuting i,j,k. That is, there exists = [1,9,92} such that g-lig = j, g jg = k. g-lkg = i in the holomorph if Q. Let G be the extension of Q by (9) in the holomorph of Q and let F be a finite splitting field for G of characteristic 3. (E.g. F could be the field of 9 elements containing a fourth root of unity\/:I.) Then Q is equivalent to K Vin the above discussion and is equivalent to the p-Sylow P. we construct a representation p of G by defining \/-_l o 01 4/—1+1 4/-1-1 P(i)= . p(j)= ). p(g)= o 4/5I -\/:I4-1 \/:I4-1 l O 35 0 One can easily check that p(i)4 = p(j)4 = p(g)3 = (5 l) and that p(g-lig) = p(j). Moreover, we have det p(g) = 1 and tr p(g) = -1. Hence, 1 is the only eigenvalue of the matrix for 9. Therefore there is a non-singular matrix M such that M‘1 fl'i'l -\/:I-1 1a M = ( V3.1 J3+1 01 and it can be checked that a # 0. Hence there is only one ) eigenvector corresponding to the eigenvalue 1. Now let A 6 HomFP(F,TP) where T is the irreducible module corresponding to p. Since l is an FP-map, X(l) = x(l-g) = x(1)-g so that there is a correspondence between X 6 HomFP(F'TP) and the eigenvectors corresponding to the eigenvalue 1. Therefore, 1 = dimF(HomFP(F,TP)) # 2 = dim T. F This shows that Theorem 3.9 fails to hold. To see that p and le are indeed irreducible representations we observe that if p were reducible then there would be a non-singular matrix A = (2 2) such that Ap(i) = e 0 p(i)A and A-lP(j) A = (01 e ). But 2 ab t/d. o «#1 o ‘ a (Cd)o-\/:i' o-(/:'1'(° :) =Tb = c = 0. e O a... a‘lpmA -.- (I? 13.1”}: .1.) <3 2) = as}. if) 7‘ ‘3 e." 36 Finally, we must verify that TK is a G—invariant module. . 2 . That is, we must show that TK 8FK g ...TK and TK GFK g - TK. In terms of representations we must show that pK is equivalent 2 to the representations pi and p: where pg(x) = pK(g-lxg). To prove that pK and p% are equivalent we must produce a matrix T such that pg(k) = pK(g-lkg) = T-lpK(k)T fi+1 4/3-1 for all k E K. We Claim that T = makes -\/-'-T+1 \/-'-1+1 pK and p: equivalent, and we omit all the tedious matrix multiplications involved in the proof of this fact. Similarly 2 1 +./-1 1 +./—1 and p: are equivalent using T = -1 +./—l l -./-1 We have shown, then, that T and TK are PG and FK modules respectively with all of the required prOperties. Hence pK by the remark prior to this theorem, FG cannot be completely reducible and therefore P = (9) f c. But P is a p-Sylow subgroup, so clearly P E 9. Thus G is not a éfinrgroup. We have actually shown Corollary 3.11. An extension of a p’-group by a p— group is not necessarily a Ofibrgroup. Remark: The above example also shows that c) #Ib) in Theorem 1.11 by taking H = P = (9). CHAPTER IV FURTHER RESULTS ON PROPERTY p §4.l. Introduction In this chapter we would like to give some results concerning group algebras over arbitrary rings. In Chapter One we saw that projective pairing depended on the group rather than the coefficients, i.e. the coefficients could come from an arbitrary commutative ring R with 1. Also it seems likely that when talking about the complete reduci- bility of induced modules N GRH RG the most general ring we will be able to use will have to be semi-simple artinian. However, the picture is not so clear with property p. For instance, it is not even known when Rad PG 5 Rad FH-FG implies Rad KG‘E Rad KH-KG for an arbitrary field extension K of F. Therefore, we would like to give conditions under which we do have property p with the coefficients coming from commutative rings. In [11], Wallace proves that if F is a field of characteristic p and G is an abelian group with p-Sylow subgroup P, then Rad FG ggRad FP-FG. Assuming G is a locally finite group, we will generalize Wallace's result to the cases where the coefficients come from a principal ideal 37 38 domain, a semi-perfect commutative ring, a semi-local commu— tative ring, and then finally an arbitrary F-algebra. In [7] Passman shows that if F is a field of characteristic p, if G is a locally finite group, and if H is a normal subgroup of G ‘with Rad FG E Rad FH-FG then H‘g [p(G) [see page 43]. In this direction we shall prove: Theorem. Let G be a locally finite group satisfying Op(G) = l for all primes p and let [(G) S_H A G. Then if Rad PG 9. Rad FI(G) -FG for all prime fields F, Rad RG E Rad RH-RG for any commutative semi—simple ring R. We will need the following five known results. Theorem 4.1. (Connell [1]) If R is a ring with identity then Rad RG 0 R E Rad R 'with equality if either i) R is artinian, or ii) G is locally finite. Theorem 4.2. (woods [12]) Let R be a ring, G a group, and H a normal subgroup of G. If G/H is locally finite, then Rad RH _C_:_ Rad RG. Theorem 4.3. (Passman [6]) Let A be an algebra over a field K and let F be a field extension of K of finite degree, say (FzK) = n. Then (Rad(F ®.A))n E FORadAERad(F®A). 39 Theorem 4.4. (Passman [6]) Let A be an algebra over a field K and let F be a field extension of K. Then Rad(F o A) n A _c_: Rad A. Theorem 4.5. (Passman [6]) Let A be an algebra over a field K and let P be a purely transcendental field extension of K. Then Rad(F 8K A) = F Gk(Rad(F 8K A) n A). If in addition F # K, then Rad(F 8k A) fi.A is a nil ideal of A. §4.2. PrOperty_ p and locally» finite abelian groups We begin this section with the following theorem Theorem 4.6. Let G be a locally finite abelian group and R a commutative ring with JacObson radical J such that R = R/J has the descending chain condition. Let H's G and suppose Rad RG E Rad RH-RG, then Rad R6 = Rad RH-RG. Proof: Let x e Rad RG. We can write x = pl-l + p292 +...+ pngn where the g1 belong to a set of coset re- presentatives for H in G. Let W’ be the finite group gener- ated by supp p1. Let ¢:RG 4 RG ‘be the natural map. Then (p(X) e Rad RG 55 Rad RH-HG, and cp(Pl) e Rad RH-RG n EW 5.; Rad fiw. Now {W} = m < o and R has the descending chain condition so that Rad HW is nilpotent. Hence (pllm 6 ker m = JG. But by Theorem 4.1, J E Rad RG since G is locally finite. Hence JG _c_: Rad RG. 40 Now p? 6 Rad R6 or p? is in every maximal ideal of RG. But maximal ideals are prime ideals in commutative rings and therefore p1 is in every maximal ideal. Thus, pl 6 Rad RG 0 RH g Rad RH. Similarly we can show all the pi ERad RH. This proves Rad RG §.Rad RH-RG. The reverse inclusion holds by 4.2. Remark: In the above theorem the only place we used the fact that G ‘was abelian was in the claim that in commuta- tive rings, maximal ideals are prime. If RG was a ring for which this condition is satisfied, then we can dr0p the hypo- thesis that G is abelian. Using the same notation for R as above we get the immediate corollary: Corolla£y_4.7. Let R be either a principal ideal domain, a commutative semi-perfect ring, or a commutative semi- local ring, and let G be a locally finite abelian group. If for any H _<_ G, Rad RG 5 Rad RH-RG, then Rad R6 = Rad RH-RG. The next theorem not only plays an important role in tthe proof of our final result but also yields as a corollary the desired generalization of wallace's theorem. Theorem 4.8. Let R be any ring and let H be a locally finite normal subgroup of a group G. Let {Iv!v E T} be a family of ideals of R such that 41 1) 0 IV = 0, ii) R/IV = Rv has the descending chain condition V v 6 F. iii) Rad R G C Rad R H'R G. V "' V V Then Rad RG = Rad RH°RG. nggf: Let [gjlj er be a set of coset representatives for H in G. Then for any x 6 RG we can write x = Zijgj where pj 6 RH. Let W:RG 4 RH be the projection map, i.e. W(x) = p1 where we are assuming 91 = 1. Then F(Rad RG) is an ideal of RH. We'll show that F(Rad RG) is in fact a nil ideal of RH. Let x 6 Rad RG and let W be the finite group generated by supp v(x) in H. Let mv:RG 4 RVG be the canonical map. Since mv(x) 6 Rad RvG g Rad RVH-RVG we have ch(TT(X)) e Rad RVH'RVG n va 55 Rad va. But [w] = n < a and RV has the descending chain condition so that Rad RQW is nilpotent. Hence [1r(x)]n E Ker mv V v- Therefore [1r(x)]n E n IVG = 0 and W(Rad RG) is a nil ideal in RH. Hence p1 E Rad RH. Now suppose pt occurs in the sum lejgj = x. Then xg';1 is another element of Rad RG and repeating the above argument with x replaced by xg;1 we get W(xg}1) = pt 6 Rad RH. Thus Rad RG'E.Rad RH-RG which was to be proved. The reverse inclusion follows from Theorem 4.2. Corollary 4.9. Let G be a locally finite abelian group with p-Sylow subgroup P. Also assume R is either a principal ideal domain, a commutative semi-perfect ring, or a commutative semi-local ring of characteristic p. Then RadIK3= Rad RP'RG. 42 2392;; Let R = R/Rad R and let {IV} be the collection of maximal ideals of R. Then hypotheses (i) and (ii) of the theorem are easily satisfied. Since R/IV are fields of characteristic p, hypothesis (iii) is satisfied by wallace's theorem. Hence the theorem yields Rad RG E Rad RPoRG, and therefore Rad RG‘E_Rad RP-RG 'by Theorem 4.6. The reverse inclusion holds by Theorem 4.2 again. More generally we have Theorem 4.10. Let G be a locally finite group and H a normal subgroup of G such that G/H has no elements of order p9 if char F = p. Let R ‘be an arbitrary commuta- tive K—algebra. Then Rad RG = Rad RH-RG. £1293: Let x = E) rigi e Rad RG where ri 6 R and 9i 6 G. Let K‘be the suggioup of G generated by H and the gi, i = 1,...,n. Then x E Rad RG n RK E Rad RK. But K/H is finite since G is locally finite and [K:H] is not divisible by p since G/H has no elements of order pg. Therefore, by Theorem 1.8, (RK,RH) is a projective pairing. Applying Theorem 3.3 of [9], (RK,RH) has property p; i.e. Rad RK.§ Rad RH-RK g Rad RH-RG. Hence x 6 Rad RH-RG as desired. Conversely, Rad RH-RG g Rad RG by woOds' result. Thus we have the promised generalization of Wallace's theorem: 43 Corollary 4.11. Let F be a field of characteristic p and let G be a locally finite abelian group with p-Sylow subgroup P. Then for any commutative F-algebra R, Rad RG = Rad RP‘RG. §4.3. Property_ 9 and the subgroup, [F(G) In this section we would like to consider the case where the group G is not necessarily abelian. we first need a few preliminary definitions. Definition 4.12. If G is a group then OP(G) is the maximal normal p-subgroup of G. Definition 4.13. If G is a locally finite group and A is a subgroup of G, then A is locally subnormal in [G if i) A is finite, and ii) A is subnormal in all finite subgroups of G containing it. In [7] Passman defines the characteristic subgroups I(G) = (AiA is locally subnormal in G) and IP(G) =< 0 and if G is a locally finite group with 0P(G) = 1, then Rad K[Ip(G)]-KG is a semi-prime ideal. He also showed Theorem 4.15. If G is a locally finite group and if 3.9 G such that Rad FG g_Rad FH°FG, then Hug [p(G) where char F = p > 0. Moreover, there are no known examples of locally finite groups G and normal subgroups H ‘where Rad FG g.FH-FG and H does not contain [p(G). Therefore, it makes sense to study subgroups H containing [p(G). With this background we are ready to prove the main result of this section. Theorem 4.16. Let K be a field of characteristic P p > 0. Let G be a locally finite group and I (G).g,HIA G. Also assume 0p(G) = l and Rad KG ggRad KIP(G)-KG. If R is a semi-simple K-algebra, then Rad RG = Rad RH°RG. Proof: First note that it suffices to show that Rad RG 9 Rad R[IP(G)]-RG. This is so by virtue of Theorem 4.2 and the fact that [p(G) is a characteristic subgroup of G and therefore a normal subgroup of H. Let F be a field extension of K of degree n. By Theorem 4.3 we have (Rad FG)r1 E F ®K Rad kGE F @K Rad KU‘p (G) ] -KG _c_: Rad F[fp(G)]-FG. Since F[fp(G)]°FG is a semi-prime 45 FG p has no nilpotent ideals. Therefore Rad F[] (G)]-FG ideal, P Rad PG 5 Rad F[J” (G)]-FG. Now let F be a purely transcendental extension of K. By Theorems 4.4 and 4.5, Rad FG=F®KRad FG nKG) gF®KRad KG. Now by hypothesis Rad KG E Rad K[fp(G)]-KG so again we have Rad FG‘E F sk Rad K[IP(G)]°KG.giRad F[]p(G)]-FG. Finally, let F be an arbitrary field extension of K. If x e Rad FG, then there exists a field L 'with K ggL g F such that L is finitely generated over K, and x 6 LG. Using Theorem 4.4 again we have x 6 Rad F6 0 LG.§;Rad LG. But L is a finite extension of a purely transcendental extension of K so that the first two steps yield x E Rad LG.E Rad L[]p(G)].LG _c_: F eL Rad L[]‘P(G)]-LG g Rad p[]p(G)]-FG. KG Rad K[[p(G)]-KG algebra. Since R is assumed to be semi—simple R 8k KG Rad K[]p(G)]-KG Thus, is a classically separable K- is semi-simple. Therefore, Rad RG.g R GK Rad K[IP(G)]‘KG.52Rad R[IP(G)]-RG and this completes the proof of the theorem. Corollary 4.17. Let G be a locally finite group with Op(G) = 1 for all primes p and let [(G) g_H A G. SUppose that for all prime fields K of characteristic p >0. Rad kG_c_: Rad K[jp(G)]°KG. Then for any semi-simple commutative ring R, Rad RG = Rad RH°RG. 46 2322;; First note that [(G) contains [p(G) for any prime p by definition. Therefore by the proof of the theorem we have Rad FGIE Rad FH-FG for any field F of characteristic p > 0. If F is a field of characteristic zero then FG is semi-simple by 18.7 of [6] so trivially Rad FG.§ Rad FH-FG. Now take [Ivjv E T} to be the collection of all the maximal ideals of R. Then all the hypotheses of Theorem 5.8 are satisfied and the result follows. BI BLI OGRAPHY 10. ll. 12. BIBLIOGRAPHY Connell, I.G., 0n the Group Ring, Canadian Journal of Mathematics, Vol. 15 (1963). PP-650—685. Curtis, C., and Reiner. 1., Representation Theopy of Finite Groups and Associative Algebras, Interscience Publishers, 1962. Gustafson, W.H., Remarks on Relatively Projective Medules, Mathematics Japonicae, Vol. 16, No. l, 1971, pp.21-24. Higman, D.G., Indecomposable Representations at Character- istic 2, Duke Mathematical Journal, 21 (1964), pp.377-381.‘ Khatri, D.C., Relative Projectivity, the Radical and Complete Reducibility in Modular Group Algebras, Tran- sactions American Math Society, 186, 1973. Passman, D.S., Infinite Group Ring§. Marcel Dekker, Inc., New York, 1971. Passman, D.S., Radical Ideals in Group Rings of Locally Finite Groups, to appear. Rotman, J.J., The Theory of Groups, An Introduction, Allyn and Bacon, Inc., 1971. Sinha, I., and Srivastava, J., Relative Projectivity and a Prgperpy of the JacObson Radical, Publications Mathematicae, 1971, pp.37-41. Srinivasan, B., On the Indecomposable Representations of a Certain Class of Groups, Proceedings London Math Society, 10, 1960, pp.497-513. Wallace, D.A.R., The JacObson Radicals of the Group» Algebras of a Gropp and of Certain Normal Subgroups. Mathematische Zeitschrift, vol. 100 (1967), pp.282-294. WOods, S.M., Some Results on Semi Perfect Group Rings, Canadian JOurnal of Mathematics, Vol. 26, No. l, 1974. PP.121-129. 47 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII llllllll]lH]|][|l]ll[[HJMIHMIM]Ill!I