RELATIVE mmcnvmmn RADICAlS ‘ a m MODULAR GROUP‘ALGEBRAS _ --_ ~ Thesis for the Degree of Ph.yD. MECHIGAN STATE UNIVERSITY. 'DHARAM CHAND KHATRI 1 9 7 2 LIBRARY Michigan 8 tate University This is to certify that the thesis entitled QM Pyrificb VJ; M WCAflo M Wag presented by has been accepted towards fulfillment of the requirements for i‘lD—_ degree in W “’5 Wm. Major professor Datew ' 0-7639 : “‘filljl‘fijmflil" ‘ ABSTRACT RELATIVE PROJECTIVITY AND RADICALS IN MODULAR GROUP ALGEBRAS By Dharam Chand Khatri Let H be a subgroup of a finite group G and F a field of characteristic p. Using J.A. Green's concept of vertices, our first main result characterizes completely the class .9 of those subgroups H of G for which every FG-module is H-projective. The pair (G,H) is said to have a projective- pairing if H 6.9. We show that the class .9 consists of precisely those subgroups of G which contain a p-Sylow sub- group of G (p = characteristic F). As it turns out, two other classes - the class fi’ of subgroups H of G for which Rad FG C (Rad FH)°FG, Rad denoting the Jacobson Radical of the ring concerned, and the class C, of subgroups H of G for which the induced FG-module MG = m 8>FG is completely reducible for each irreducible FH-moduleF; - are almost equi- valent to the class .9. We show that for normal subgroups these three concepts coincide. Otherwise examples exist showing that these three classes are distinct in general. A (finite) group G is called a PRC-group if .9 = E’= Ca Our main results may be summarized as follows: C is a PRC-group over F of characteristic p if it satisfies any of the Dharam Chand Khatri following conditions: (1) p = 0 or p Y ‘G‘, (2) G is a p-group, (3) G is a Frobenius group with kernel 6' and complement a p-Sylow subgroup of G, (4) G is an extension of a PRC-group by a p'-group, and (5) G is an extension of a p-group by a PRC-group. Finally, we attempt to spot out "Projective - Sensitive" subgroups of G, which are the subgroups H of G satisfying (i) Whenever (G,M) is a projective-pairing then M 2 H or (H, H fl'M) is a projective-pairing, and (ii) If K s H and (H,K) is a projective-pairing then there exists an M S G such that (G,M) is a projective-pairing and K = M F1H. RELATIVE PROJECTIVITY AND RADICALS IN MODULAR GROUP ALGEBRAS By Dharam Chand Khatri A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1972 TO MOTHER ii ACKNOWLEDGMENTS I wish to express my deep gratitude to Professor I.N. Sinha for his continued interest, for the discussions I have had with him and for the stimulating advice and suggestions he has given me . iii Chapter I II III IV TABLE OF CONTENTS Notation and Terminology INTRODUCTION TO THE PROBLEM AND PRELIMINARY RESIH—ITS OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO ......... Section 1. Introduction .. ............. . ......... Section 2. Preliminary Definitions and Results .. GENERALITIES ABOUT THE CLASSES 9, I? AND a ..... Section 3. Projective-Pairing ..... ..... ......... Section 4. Relation between Projective-Pairing, Property 9 and Complete Reducibility of Induced Modules for Normal Subgroups Section 5. Invariance of Projective-Pairing and Property 9 .......................... EQUIVALENCE OF THE CLASSES 9, ,9 AND O .... ..... Section 6. p-groups .... ..... ............. ....... ‘ Section 7. Special Type of Erobenius Groups ..... Section 8. Extensions by Groups of Order Prime gem 9. Eitefis;;gg°;ag:g;gggg'z:::::::::::::: PROJECTIVE-SENSITIVITY ..... . . ............... .. Section 10. Projective-Sensitivity ...... ......... BIBLIOGRAPHY ..................................... iv Page 10 15 l9 19 20 24 28 33 33 42 H SIG FG [G:H] Rad FG NOTATION AND TERMHNOLOGY a finite group H is a subgroup of G H is a normal subgroup of G Base field of characteristic p Group algebra of G over F index of subgroup H in G order of G the derived group of G Center of the group algebra FG a p-Sylow subgroup of G (p = characteristic of F) H,K are subgroups of G and there is an x in G such that x-le S K x’le,x ec, HsG the Jacobson Radical of the group algebra FG dimension of the vector space V over F direct sum tensor product 92 ® FG, 91 being an FH-module, H S G anFHG-module Mt considered as an FH-module, H S G ideal in FG generated by {h-l‘h e H s G} {g E G‘g - 1 E I, I an ideal in FG} CHAPTER I INTRODUCTION TO THE PROBLEM.AND PRELIMINARY RESULTS §1. Introduction Let H be a subgroup of a finite group G and F a field of characteristic p. The H-projective FG-modules were studied by G. Hochschild and D.G. Higman [7], [6]. Green used this concept to classify indecomposable Fdeodules through the theory of Vertices and Sources [4]. Using his concept of Vertices, our first main result (Theorem 3) characterizes the class ‘9 = {H < G‘every Fdeodule is H-projective}. For a sub- group H in .9 we call (G,H) a projective-pairing. During the course of our investigations we are led to study two more classes: the class &> of subgroups H of G for which Rad FG ; (Rad FH).FG and the class C, of subgroups H of G for which mG = m.G>FG is completely reducible for each irreducible FH-modifie m. We show that these three concepts coincide for normal subgroups (Theorem 7). Otherwise examples exist showing the distinctness of these classes (Theorem 8). A (finite) group G is called a PRC-group over F if .9 = E’= Ca Our results in Chapter III may be summarized as follows: G is a PRC-group over F of characteristic p if it satisfies any of the following conditions: (1) p = O or p Y ‘G‘ — in this case each class being the class of all subgroups of G, (2) G is a p-group - in this case each class consists of singleton {G} only, (3) G is a Frobenius group with kernel G' and complement a p—Sylow subgroup of G, (4) G is an extension of a PRC-group by a p'-group, and (5) G is an extension of a p-group by a PRC-group. As a special case it follows that if G has a normal p-Sylow subgroup then G is PRC and that if Rad FG is central in PC then G is PRC. We also prove that if G is p-nilpotent or supersolvable then E’ is a subclass of .9. In the last chapter we attempt to spot out those sub— groups H of G which are "Projective-Sensitive" in the sense 4 that they satisfy the following two conditions: (i) Whenever (G,M) is a projective—pairing then M 2 H or (H, H FIM) is a projective-pairing, (ii) If K S H and (H,K) is a projective- pairing then there exists an M s G such that (G,M) is a projective-pairing and K = M n H. We give a necessary and sufficient condition for a normal subgroup to be projective- sensitive. Also a necessary and sufficient condition for G to have all its subgroups projective-sensitive is given. Finally, we give an application which determines the defect groups of blocks of G when G has all its subgroups projective-sensitive. §2. Preliminary Definitions and Results Let F be a field of characteristic p > O and G be a finite group such that p divides the order \G‘ of G. All modules under consideration will be right modules finitely generated over F. Let Tl be a G-module over F or an FG~module, where FG denotes the group algebra of the group G over the field F. If H is a subgroup of G then. M! can always be regarded as an FH-module in a natural way by restricting the domain of operators to FH. The resulting FH-module is denoted by' Th. Similarly if m is an FHdmodule, the induced FG-module m 8>FG is denoted by mG, where 8> denotes the tensor- product. FH Then the following elementary facts about the induced modules can easily be verified (See, for example, [4]): (2.1) If m is FH-direct sum of two FH-modules m1 and m2. then ERG = 91C: 3) mg : as FG-direct sum. If R is another subgroup of G such that H C R C G, then we have the transitive laws: (2.2) an ) E m , and (2.3) GHQH = {Uh . Now let G = Hx1 U Hx U...U Hxn, n = [G:H] be a coset- 2 decomposition of G over H. We shall always take x1 = 1. Then G n (2.4) 91 = G) 2, in (8 xi as vector Spaces, i=1 where, for any x E G, m G>x = {m48 x|m E M}. It is clear that ‘R 69 x is F(x-]‘Hx)-module and that dimFQn <8) x) = ding“. We G (n ) . x'lux can regard m 8>x as a submodule of From (2.4) it follows that (2.5) dimFmG = [G:H]-dimeR . The mapping m -+ m 8 x (m G St) although it is not in general a module-isomorphism, it does induce an isomorphism between the submodule lattices of m and m 8>x. Thus (2.6) SR (3 x is indecomposible if and only if in is so . Also it can easily be verified that II! (2.7) (fit (8 x)G ERG for any x E G . We make Definition 1. If T! and m are two FG-modules, we say that MI is a component of m if TI is FG-isomorphic to a direct summand of m. Definition 2. Let H be a subgroup of G. An FG-module 2m is said to be H-projective if every exact sequence 0 —+ 9t —* a8. -+ 931 -> 0 of FG-modules for which the associated sequence of restrictions 0 I’mH a ih afmh a O splits over FH, is itself split over FG. Remarks. (1) H-projectivity is a Special case of "relative- projectivity" defined by Hochschild [7]. An H-projective module m1 is (R,S)-projective in the terminology of [7] by taking R = FG and S = FH. iIn the general theory there is also the dual notion of "relative-injective" modules. (2) We observe that MI is [l]-projective if and only if it is projective in the usual sense. In this sense, therefore, H- projectivity is a generalization of (usual) projectivity. Definition 3. Let H s G. We say that (G,H) is a projective- pairing over F, if every exact sequence 0 -+ 91 .. {,4 {Uta 0 of FG-modules for which the exact sequence 0 —o 91H -o £11 -+ 53]] -+ O splits over FH, is itself Split over FG. It follows immediately from Definition (2) above that (G,H) is a projective-pairing if and only if every FG-module is H-projective. n Definition 4. Let G = U Hx i=1 of G over H. Then we can treat PC as a free FH-module with X i’ 1 = 1, be a coset decomposition basis x1 = l,x2,...,x and every element of FG can be n written in the form .glpixi where pi's are in FH. The pair (G,H) is said to havé—property p over F if ; pixi E Rad FG implies each pi E Rad FH, where Rad denotes th; Jacobson- Radical of the ring concerned. In [11] it is shown that property p is independent of the choice of coset representatives and that it is a transitive- property in the sense that if K s H s G and the pairs (G,H) and (H,K) have property 9 then the same is true for the pair (G,K). It is evident that the property p is equivalent to the requirement Rad FG ; (Rad FH)-FG. Comparing the dimensions, we see that if (G,H) has property 9, then (2.8) dimF(Rad FG) s [dimF(Rad FH)]°[G:H] . D.G. Higman's Criteria for Heprojectivity. The following char- acterization of H-projective FG-modules is due to D.G. Higman [6]: (2.9) The following statements for an FGemodule fl! are equivalent: (i) M! is H-projective; (ii) EU! is a component of mh)G; (iii) There exists an FH-endomorphism n of Mt such that where [xi] are the coset representatives of G over H and Im’ is the identity map on TL Note: Here we make the convention that if T]: gm... Tl is a map then x“ and fix, x E G, denote the mappings which take HIEEMI to mxfl and mnx respectively. In [4] Green gave the following definition of H-projectivity which can easily be seen to be equivalent to those in (2.9). (2.10) An FG-module TI is H-projective if and only if there exists an FH-module m such that fit is a component of 926. We now introduce a partial ordering S on the class of subgroups of G by saying that H S R if fog some x E G, x-le S R. It follows that if H SGR and R S H then H and R are conjugate subgroups of G find we expregs this by writing H = R. G From (2.2) and (2.7) we infer: (2.11) If an FG-module TI is H-projective, then it is R-pro- jective for every subgroup R such that H S R. G Now let P be a p-Sylow subgroup of G, then the index n = [G:P] is prime to p and taking n = n-ILm in (2.9)(iii) we obtain another theorem of D.G. Higman: (2.12) Let P be a p-Sylow subgroup of G (p being the char- acteristic of F) then every FG~modu1e is P-projective. Combining (2.11) and (2.12) we obtain (2.13) If H is a subgroup of G such that H 2 P then (G,H) G is a projective-pairing. In order to save Space and avoid lengthy repititions we abbreviate the following three classes of subgroups of G. Definition 5. Let G be finite and F have characteristic p. We denote by 9 = 9(G,F) = {H < G‘ (G,H) is a projective-pairing over F] E’= EKG,F) = {H < G\(G,H) has preperty 9 over F} CI: C(G,F) = {H < G‘For every irreducible FH~module m, the induced Fdeodule TC is completely reducible}. Note that here H < C means "proper" subgroup, {1} being considered as proper subgroup. For the class .9 we have Definition 6. A minimal member B of .9 under the partial ordering S is called a projective-foot of G and a maximal element TG is called a projective-tOp of G. These exist, because .9 is a finite class. In the course of our investigations the notion of Frobenius groups will be important which we define next. Definition 7. A finite group G is called Frobenius group with kernel M and complement K if G = KM, M'Q G, K H‘M = 1 and K n KX = 1 forall x E G - K. For group-theoretic prOperties of Frobenius groups we refer to Scott [10]. The importance of representation theoretic properties of Frobenius groups was realized by Wallace ([14] and [15]) in connection with the centrality of the radical of the group algebra. We state below his results which are pertinent to our investigation. (2.14) If ‘G‘ = pam, (p,m) = l, and Char F = p then dimF(Rad FG) 2 pa - 1, the equality holds if and only if G = PM, M a G, P n M = l, is Frobenius group with kernel M and complement a p-Sylow subgroup P of G. (2.15) Rad (FG) ; Z(FG) if and only if G is one of the following three types: (i) G has order prime to p (ii) G is abelian (iii) If P is a p-Sylow subgroup of G then G'P is a Frobenius group with kernel G', the derived group of G. We will also use the result about the dimension of the radical of FG due to Brauer and Nesbitt [2]: *1 2 (2.16) dimF(Rad FG) = ‘G‘ - kglfk, Where f1,f2,...,fL are the degrees of absolutely irreducible p-modular representations occurring in the regular representation of G. For most of the other so called "standard" results of re- presentation theory we refer to the basic text by Curtis and Reiner [3]. CHAPTER II GENERALITIES ABOUT THE CLASSES 9, I? and a In this chapter we characterize the class .9 completely and show that when subgroup under consideration is normal in G, the three properties are equivalent. We also prove, by exhibiting examples, that the classes .9 and &> and .9 and C, are entirely independent of each other while CI is a subclass of Rh 1| §3. Proiective-Pairing We devote this section to classify the class ~9 of all proper subgroups H of G such that (G,H) is a projective- pairing. By (2.13) if H 2 P, P a p-Sylow of G, then H 6'0. It is proved in Khatri andGSinha [8] that the converse of (2.13) is also true. For non-p-group G we state the main results of [8] below and refer to the same for the proofs. Lemma 1. Let H S S S G such that (G,S) and (S,H) are pro- jective pairings over F. Then (G,H) is a projective-pairing. Lemma 2. If (G,H) is a projective-pairing then for every sub- group S of G such that S 2 H, (G,S) is a projective-pairing. Theorem 1. If H E éKG,F) thin p divides ‘H‘. Theorem 2. A projective-foot B of G is a p-Sylow subgroup of G. Theorem 3. ‘9(G,F) consists of precisely those subgroups of G which contain a p-Sylow subgroup of G. 9 10 Theorem 4. A subgroup T of G is a projective-top if and only if it is a maximal subgrOup containing a p-Sylow subgroup of G. Clearly T is not unique up to conjugacy while B is. The results about p-groups we will include in the next chapter along with the other results. §4. Relation Between Projective-Pairing, Property 9 and Complete Reducibility of Induced Modules for Normal Subgroups Having characterized the class 9 in the above section, we turn our attention to its connection with the representation theory. As was realized by Sinha [12] and as we shall see later that the property of complete reducibility of induced modules is closely related with these concepts. We state below a theorem of Sinha [12]: Theorem 5. Let H be a subgroup of G. If for each irreducible FH-module m, the induced module m6 is completely reducible over FG then (G,H) has property p. Conversely if H S G then (G,H) property 9 implies that MG is completely reducible for every irreducible FH-module m. We now prove the equivalence of projective-pairing and property p for normal subgroups. Theorem 6. If H is a normal Subgroup of G then (G,H) is a projective-pairing if and only if (G,H) has property p. 2322;. First suppose H S G and (G,H) is a projective-pairing. Let G = 3 Hxi, x . 1 i=1 We observe that for h E H, xih = (pi(h)xi for all i, where each = 1, be a coset decomposition of G over H. mi(h) = xihx;1 induces an automorphism of the normal subgroup 11 H. We can extend mi by linearity to an automorphism of FH. Now let m be an irreducible FH-module. Then ERG = @291 ® xi, where each in ® x1 is an irreducible FH~module. Thus $61 is completely reducible over FH. Hence given any exact sequence of FG~modules the sequence splits over FH. Then by the projective-pairing (G,H) the sequence splits over FG as well. Thus m9 is completely reducible over FG. Now let ; pixi E Rad FG, pi E FH. Since NC is completely reducigle, we have mG(g pixi) = O. In particular, we have 0 = (n ® 1) (5:3 pixi) = 2 nrlai 8 xi, n E m. This implies that each npi = O fir all n :_m. That is, Mp1 = 0 for all k .E .\ 1. Since m was arbitrary irreducible FH~module, so each pi 6 Rad FH. This gives prOperty p for (G,H). Conversely suppose that (G,H) has property p. Let M} be an irreducible FG-module. By Clifford's theorem we have 93% :91]. @oooGD‘Rka where each mi is an irreducible FH-module. Consider m1. We g G . . have HomFHthflJh) HornFGCRlJR). Since ‘fil IS a component of Th, the left hand side is non-zero and so is the right hand side. Thus MI is a composition factor of mi. Since (G,H) has pro- perty p, by Theorem 5, mg is completely reducible over FG. Hence MI is a component of mi. Therefore, by Green's criteria for H-projectivity, ml is H-projective. 12 As MI is an arbitrary irreducible FG-module, it follows that every irreducible FG~module is H-projective. In particular, the trivial FG-module, F, is H-projective. Hence vertex of F - namely a p-Sylow subgroup P of G - is contained in H. The projective pairing for (G,H) now follows from Theorem 3. This completes the proof of the theorem. Corollary,l. If H S G such that (G,H) is a projective-pairing then the representation of G, induced from an irreducible repre- sentation of H, is completely reducible. Corollary 2. A necessary and sufficient condition for (G,H) to have a projective-pairing is that every irreducible FG-module is H-projective. Corollary 3. Let 'H S G and. $2 an irreducible FG~module. If m is a composition factor of Eh then. MI is a composition factor of NC. Though some of the implications in the next theorem go through without requiring the subgroup H to be normal in G, we state our main theorem for normal subgroups. Theorem 7. Let H SIG. The following statements are equivalent: (1) H contains a p-Sylow subgroup of G. (2) (G,H) is a projective-pairing. (3) (G,H) has property p. (4) For every irreducible FH-module m, the induced FG-module m6 is completely reducible over FG. Prggf. It follows from (2.13) and Theorems 3, 5, and 6. We now drop the normality condition on the subgroup H and ask ourselves the question about the relation between the three properties under consideration when H is not normal in G. 13 By Theorem 5, c. is a subclass of Rh Our next theorem shows that, in general, the classes .9 and k’ are independent. Theorem 8. In general, the property of projective-pairing and property 9 are independent of each other. nggf, 'We prove this theorem by exhibiting examples of group G and subgroup H such that the pair (G,H) enjoys one property but not the other. Our first example Shows that projective-pairing does not imply property p. Take G = A5, H = A4, p = Char F = 3, where An denotes the alternating group on n symbols. Since H contains a 3-Sylow subgroup of G, (G,H) is a projective-pairing by (2.13). We Show that (G,H) cannot have property 9. We may take F to be a splitting field for G. It is well-known that over complex field A has five 5 irreducible representations, say, T1,T2,T3,T4,T of degrees 5 1,3,3,4,5 respectively. Taking mod 3, it follows from the results of Brauer and Nesbitt (Theorem 1 of [2] and a remark on p. 936 1,T2,Ts and T; remain irreducible while T3 2 T] + T]. Since A5 has four 3-regular classes, these are all the irreducible representations of A5 together with Theorem 5 of [1]) that :T over the field F of characteristic 3. Now by Brauer and Nesbitt's result (2.16) dimF(Rad FAS) = 60 - (l2 + 32 + 32 + 42) = 25 . Also since H = A4 is a Frobenius group with complement a 3-Sylow subgroup of H, 14 dimF(Rad FA = 3 - l = 2, by Wallace's result (2.14) . 4> Now if (G,H) has property p, then we must have Rad FG C (Rad FH)-FG . Considering dimensions, we must have dimF(Rad FG) S (dimF(Rad FH))-[G:H] . That is, we must have 25 S 2.5 = 10, which is impossible. Thus (G,H) does not have property p. Though we are not able to find an example to show that property 9 does not imply projective-pairing in finite groups, we give an example exhibiting the same when G is an infinite group and 'H a subgroup of finite index. Let Char F = 2 and G a group generated by elements a and b subject to the relations b2 = l, babm1 = a-l. That is, G is an extension of infinite cyclic group <8) by the group of order 2. It is known that Rad FG = (0) (See, for example, Passman [9]). Let H = . Since Rad PC = (0), (G,H) has property p over F. Now Gustafson [5] communicated to me that if G is an infinite group and H a subgroup of finite index then (G,H) is a projective-pairing is equivalent to the index [G:H] being unit in the base field. In view of this result clearly (G,H) cannot be a projective-pairing. This completes the proof of the theorem. Corollary . In general the classes 6’ and C. are independent of each other. 15 Proof. By Theorem 8, 9 and R are independent. But a: ,9, therefore .9 ¢ Ca The second example in the above theorem shows that H = E C“ by Theorem 5, but H (‘9. This is what we wanted. §5. Invariance of Projective-Pairing_and Property 9 In the previous sections we characterized projective- pairing completely and showed that when H was normal subgroup of G then projective-pairing was equivalent to property p. Also we showed that these two properties are, in general, independent of each other. A natural question now arises that how far these properties can be carried over under different algebraic transformation; for example, under taking direct pro- ducts of groups or under homomorphisms. It is this question we now turn our attention to. (A) Field4ExtenSions We Show that the extension of ground field F has no effect on projective-pairing and when F is splitting field for G, on prOperty p. Indeed, Theorem 9. Let G be a finite group and F a field of char- acteristic p > O and L2: F be a field extension. Then if for a subgroup H of G (G,H) is a projective-pairing over F, it has projective-pairing over L as well. If F is a splitting field for G, the same is true for property 9. nggf. The proof for projective-pairing is immediate, since the characteristic of the field is preserved under field-extensions. 16 We give a proof for property 9. For standard results over radicals under field extensions we refer to [3]. Suppose now that (G,H) has property p over the splitting field F and L D F. Writing G = U RX 1, let 2 pi xi 6 Rad LG with each pi E LH. Since LH= FH 8%L, we may write,for each i, p‘=§:Lij qij, LijEL’qijEFI-I' Thus we have ZjLijqijx i E Rad LG= (Rad FG) 8%L (See [3], §29) . This implies that, for each j, ,_ , R d F 'th _ FH . E qlel E a G w1 qu E The property 9 for (G,H) over F yields each qij 6 Rad FH so that = 2 L .q_ E Rad FH 8%L = Rad LH, for ij ij each i. This is what we set out to prove. (B) Direct-Products We prove the following theorem: Theorem 10. Let H1,H2 respectively. Then if (Gl,H1) and (GZ’H be subgroups of the groups G1,G2 2) have property 9 (projective-pairing) so does the pair (G1 X G H X H2). 2’ 1 Proof. As before the proof for projective-pairing is easy; for if Hi 2 Pi, i = 1,2, Pi = p-Sylow subgroup of Gi then H1 X H2 2 P1 X P2 = p-Sylow subgroup of G1 X G2 g1v1ng what 1S required. We turn to the proof for property 9. n Write G1 = iilHlxi and G2 = jglflzyj . Then we have 17 n m G XG = U U (H XH)(x.,y.). 1 2 i=1 j=1 1 2 1 J Now it is well-known that F(G1 X G2) = FG1 <8>FG2 as F-algebras and that Rad F(G1 X Gz) = Rad FG1 G>Rad FGZ' Suppose now that 2 (p.,q.)(x.,y.) E Rad F(G X G ) where for each i,j, i j 1 J 1 J l 2 ’ (pi,qj) 6 F(H1 X H2). Thus i2j(pixi,qjyj) E Rad FG1 8>Rad FGZ’ 3 which gives 2 pixi E Rad FGl’ ; qjyj E Rad FG with p, E FH 1 J 1 and qj 6 FR 2 l 2 Since (G1,H1) and (G2,H2) have property 9, so we have each pi E Rad FH Hence for each i,j, 1 and each qj E Rad FHZ' ® Rad FH = Rad Full X H2) , l (pi’qj) E Rad FR 2 which is what we wanted. (C) Homomorphic images We have Theorem 11. Let ¢z G1 ~»G2 be an epimorphism, H a subgroup l of G1 and H2 = ¢(H1). Then (G1,H1) projective-palring implies that for (G2,H2). If, in addition, ¢ maps Rad FG1 onto Rad FG2 then property p for (Gl,H1) yields that for (G2,H2). Egggf. Once again proof for projective-pairing is immediate; for a p-Sylow subgroup is mapped onto a p-Sylow subgroup under an epimorphism and inclusion is preserved under ¢. We now give a proof for property 9. Suppose that G = U H x and G2 = U szj and that E q,y, E Rad FG , where i . . l ' 2 1 J J J J l8 qj 6 FHZ’ for all j. Since ¢(Rad FGl) = Rad FG2 by hypothesis, there exists an a 6 Rad FG such that ¢(a) = Z q.y.. Write a = 2 p,x, l j J J i 1 1 with pi E FHl' Then, by virtue of property p for (G1,H1), each pi is 1n Rad FHl' Let ¢(xi) = hiyi’ hi 6 H2. Then ilqjyj = ¢(§ Pixi) = E ¢(pi)¢(xi) = E ¢] xEM X€M XEM and [( 2 x)(g-1)]hy [(g-1)( 2 x)]hy = (s-1)h( 2 x)y XEM xEM XEM = [(gh-l) - (h-1)]( 2 x) = ( z x)[(gh-1) - (h-l)]. XGM XEM Thus I is an ideal in FG. But it is well-known that Rad FP = , hence I = ( 2 x)Rad FF is a nilpotent ideal of dimension \P‘ - 1. Thisxproves the lemma. We now prove Theorem 13. If G = PM, M <1G, P FIM = l is a Frobenius group with kernel M, then the three classes .9, fi’ and C. coincide. 2392;. First we observe that G' = M (Wallace [15]) so that by (2.15) Rad FG : Z(FG). 22 Now suppose that (G,H) is a projective-pairing. We wish to show H E Ca In the factorization G = PM, pick a p-Sylow subgroup P which H contains so that H = P(H H M) is Frobenius group with kernel H n M. Let G = U Hxi and M = U (H n M)yj j be coset decompositionsof G and M respectively. By Lemma 3, a typical element r of Rad FG can be written as r = a( Z x) = a( 2 h)(2 y_), where a E Rad FP . XEM hEHrM j J Let H be an irreducible FH-module. Since Rad FG is central, we have mG-r= (@zm®xi)r=®zm®xir=@2m®rxi i i ' 1 ezm®[a( z: h)(zy.)]xi i hEHr‘M j J -@zm-[a( E h)]®(2y.)xi i hEHFM j J = 0, since a-( 2 h) E Rad FH, H being Frobenius with kernel H 0 M, hEHflM G and m irreducible over FH. Thus m is completely reducible over FG. Since m was arbitrary, H 6 Ca We have thus shown the inclusion 9 C Ca Since Ca; E always by Theorem 5, our proof will be complete if we prove gag. Let H be a subgroup of G with (G,H) property 9. Then p divides ‘H‘; for it not, then FH is semi-simple and 23 if E pixi E Rad FG, pi E FH, G =lJ Hx ° coset decomposition of i . i' G over H, then each pi E Rad FH = (O), and this implies semi- simplicity of FG, contrary to our hypothesis that p ‘ ‘G‘. In the factorization G = PM, pick a p-Sylow subgroup P such that H n P is a p-Sylow subgroup of H. Then = (H H P)~(H FIM) is a Frobenius group with kernel H FIM. Let = U (H n P)xi and M = U (H FIM)y, i j J be coset decompositions of P and M respectively. We first show that G = U Hy,xi is a coset decomposition of G over H. isj To see this, first observe that [G:H] = [P: H 0 P]-[M: H FtM]. Suppose now that 11ij1 =Hy x then Lk’ 1)€H= (H nP)(H mu). 1 (XiXk1k)(X X 1ijlxk 13"; Therefore, there exist a E H n P and b E H fl‘M such that -1 -1-1 _ (xixklki)(xx iji xky&)—ab . . ‘1 'l _ '1 -1 -1 -l J and thlS gives a (xixk ) — b(xkxi ijixk yL ) E P FIM - {1]. 1 6 H FIM. This gives Hence Xixk 1 E H n P and xkxi yj x = k and consequently j = L. Thus {Hiji} are distinct and i xk yL cardinality argument proves our assertion. Suppose that (G,H) does not have projective-pairing. Then there exists g E P such that g 4 H. Write g = hpxo where hp 6 H H P and xO # 1 is in {xi}. Then by Lemma 3 = (g-l)( 2 x) E Rad FG . XEM We now have 24 r=(hX-1)(ZX)=hX(2X)-(ZX) 9° x€M P°xeM x€M =h(2x)x -(2x), since M q, then P 4 G and Corollary 1 above is applicable. On the Other hand, if G is non-abelian and p < q, then G is Frobenius group with P as a complement (Scott [10]) and Theorem 13 takes care of the proof. Corollary 4. If \G‘ = pqz, p X q - 1, then G is PRC. 3529:. It is easy to see that G satisfying the above hypothesis has either P normal in G or is a Frobenius group with complement P and in both cases we are done. Corollary_5. If G is supersolvable then property 9 implies projective-pairing. Pooof. If p is the highest prime dividing the order of G then Pid G and we are done by Corollary 1 above. On the other hand, if p is the smallest prime divisor of ‘GI, then G is p-nilpotent and property p implies projective-pairing for p-nilpotent groups follows from the Remark following Theorem 13. Suppose now that p is an intermediate prime divisor of ‘G‘. In this case, there exists a normal p-nilpotent subgroup K of G (Sylow Tower Theorem: See Scott [10]) and G is an extension of K by a p'-group. The result now follows from the Remark to Theorem 13 and the above theorem. 28 §9. Extensions of o-grouos The aim of this section is to further extend the class of PRC-groups. In fact, we prove that any extension of a p-group by a PRC-group is PRC. Let H be a subgroup of G. We denote by ”C(H) the ideal in PC generated by the set {h-l‘h 6 H}. Similarly if I is a two-sided ideal of FG then the set {g e G‘g-l e I} is denoted by mél(I). It is easy to check that flé1(1) is a normal subgroup of G. It is well-known that if Q: FG a F(G/H) is the map obtained by extending the canonical map G a G/H (H 4 G) to FG by linearity, then Ker Q = mG(H) and so FG/mG(H) E F(G/H). We now prove Lemma 5. Let P be a p-Sylow subgroup of G. Then m-1(Rad FG) = c (P) G oreG 2329:. Let A = M61(Rad FG) = {g E G‘g-l E Rad FG}. Then A 4 G, since Rad FG is a two-sided ideal in FG. Since Core P 4 G, by Lemma 4, Rad(Core P) ; Rad FG. But it is well-known that Rad(Core P) is ideal in F(Core P) generated by {h-l‘h 6 Core P}, as Core P is a p-group. Hence for each h 6 Core P, h-l E Rad FG and so Core P C A. For the reverse inclusion, observe that for g E A, g-l is nilpotent. Thus there exists a least integer k such that o = (g-1)k = gk - (1;)gk'1 +...+ (-1)k‘1-1 . 29 Since FG is free over F, each (E) E O(mod p), i = 1,2,...,k-l and gk = 1. In particular, (T) = k E 0(mod p) and so p‘k. Thus g E A implies p ‘ \g‘. Suppose that ‘Al = pb-m with (p,m) = 1. If m > 1 then A has elements with orders not divisible by p, contrary to what we have just proved. Hence A is a p-group. Since A is normal p-group and Core P C A we have A = Core P and this proves the lemma. We can now prove Theorem 15. Let H < G such that (G,H) has property 9. Then H 2 Core P. Egoof. If the assertion of this theorem is false then there is a g 6 Core P and g é H. Write G = G Hxi, coset decomposition: so that g = hxi for some h E H and 1x,L # 1. By Lemma 5 we have g - 1 = hxi - 1.1 6 Rad FG. Since (G,H) has property p, we must have h,1 E Rad FH which is impossible. This completes the proof. Before we come to the main theorem, we need Lemma 6. Let I be a nilpotent ideal in a ring R and m: R a R/I the natural map. If Rad R denotes the nilpotent radical of R then m(Rad R) = Rad(R/I) = (Rad R)/I. Egoof, We always have m(Rad R) C Rad(R/I). Now let a + I E Rad(R/I). Then for every b + I E R/I, (a+I)(b+I)=ab+I is nilpotent; thus there is an integer m such that m m m . (ab + I) = (ab) + I = I. Therefore (ab) E I. But I is a nilpotent ideal of R, hence ab is nilpotent. Since b in 30 R was arbitrary, a E Rad R and so ¢(a) = a +-I 6 Rad(R/I). This proves the first equality. The second is clear, since Ker ¢ = I and I c Rad R. Remark. Actually our proof shows little more than the assertion of the lemma. In fact, we have shown that a +*I is in Rad(R/I) if and only if a is in Rad R. We shall use this feet. We now come to the main theorem of this section. Theorem 16. An extension of a p-group by a PRC-group is PRC. Eroor, Let G be an extension of a p-group A be a PRC-group. Once again we prove the inclusions 9 C (L and ,Q s: 9. 9KG,F) nggG,F): Let (G,H) be a projective-pairing. Then H contains a p-Sylow subgroup of G and hence contains A. Since A.4 G, mA(A) : Rad FG by Lemma 4 and so if m, is an irreducible FH~module, each a E A acts trivially on m and m can be regarded as an irreducible F(H/A)~module. Now clearly (G/A,H/A) is a projective-pairing; hence, since G/A is PRC by hypothesis, the induced F(G/A)-module ERG/A = ‘Jt a F(G/A) F(H/A) is completely reducible over F(G/A). Since a E A acts /A trivially on. m, we can vieW' MG as FG~module in a natural way by defining (n ® Ag)h = (n ® Ag)Ah, n E m, g,h E G. m, Define cp: ERG 91G by cp(n ® Ag) = n ® 3 and extend by linearity. Since «maAam nanaamm=mmeia>=n®a=cn®ah cp(n 69 Ag) -h 31 and since (obviously) m is l-l and onto, m is an FG- isomorphism. Thus mG is completely reducible over FG implying H E C(G,F). E(G,F) CQ(GLF): Let H be a subgroup of G with (G,H) having prOperty 9. Then H 2 Core P 2 A by Theorem 15. Let W: G a G/A be the natural map. We extend m to PC by linearity. Let G = U Hxi be coset decomposition of C over i H. Then G/A = u(H/A)Axi 1 is a coset decomposition of G/A over H/A. We now show that (G/A,H/A) has property 9. Suppose that g piAxi E Rad F(G/A), pi E F(H/A). Extend- ing m to group algebfa FG, there exist pi E FH such that cp(pi) = 131. Thus no; pixi) e Rad F(G/A) = Rad(FG/QIG(A)). Since Ker m = mG(A) ; Rad FG, Lemma 6 is applicable and we conclude that ; pixi E Rad FG, with pi E FH. By hypothesis (G,H) has 1 property 9 and so each pi is in Rad FH. Hence Pi = ¢(Pi) E m(Rad FH) = Rad F(H/A). This shows that (G/A,H/A) has property 9. Since G/A is PRC by hypothesis, (G/A,H/A) is a projective- pairing. Thus H/A contains a p-Sylow subgroup P/A of G/A, P being a p-Sylow of G. Hence H 2 P and (G,H) is a pro- jective-pairing. This completes the proof of the theorem. 32 Corollary 1. If P q then np E 1(mod p) and np‘q yields np = 1. Thus P44 G and we are done. Suppose that p < q. Then nq = l or p2. If nq = p2 then np = l and so P14 G and we are done again. If nq = 1 then G = PQ, Q 4 G, Q the q-Sylow subgroup of G. Suppose now that G has an element x of order pq. Then H = - has index p, the smallest prime dividing ‘6‘; hence Hi4 G. Since H is cyclic, K = is a normal p-sub- group of G and G is an extension of K by a group of order pq. But by Corollary 3 to Theorem 14, a group of order pq is PRC, hence G is PRC by above theorem. In case G has no element of order pq, then every element of G is either a p- or a p'-e1ement and so G is a Frobenius group with Q as kernel and P as complement (Scott [10]). That G is PRC follows from Theorem 13. CHAPTER IV PROJECTIVE-SENSITIVITY §10. Proiective-Sensitivity We start with Definition 9. Let H be a subgroup of G. We call H to be projective-sensitive if the following holds: (i) Whenever (G,M) is a projective-pairing then M Q'H or (H, H FlM) is a projective-pairing. (ii) If K s H and (H,K) is a projective-pairing then there exists an M s G such that (G,M) is a projective-pairing and K=MflH. In this chapter we attempt to characterize projective- sensitive subgroups of an arbitrary finite group and give examples to show the limits of the results proved. We also give a necessary and sufficient condition for a group to have all of its subgroups projective-sensitive. Before coming to the main results, we prove two lemmas: Lemma 8. If H 9 G then condition (1) of Definition of projective- sensitivity is satisfied. 2322;: Let M S G be such that (G,M) is a projective-pairing. If M.é'H then, since M contains a p-Sylow subgroup P of G and H is normal in G, H n M contains a p-Sylow subgroup H H P of H and so (H, H FlM) is a projective-pairing. 33 34 Lemma 9. Let H < G. If K < H and is such that K contains a p-Sylow subgroup of H and PK is a group where P is a p-Sylow subgroup of G containing the p-Sylow subgroup of K (of H), then PK H H = K. Proor. Clearly PK 0 H 2 K. We prove the inverse containment. Suppose lP‘ = pa, \Hl = pb-s, (p,s) = 1, b s a and ‘K‘ = pb-L, (P,&) = l, L‘s. We have P - K pa-pbi a WN= PflK= b =PW' 9 Now \PK n H| divides both \PK‘ = paL and \H| = pb-S. Since {‘3 and b s a, (paL,pbs) = pbt. Since PK 0 H already has K of order pb-L we obtain PK n H = K. This proves the lemma. We first give a necessary and sufficient condition for a normal subgroup to be projective-sensitive. Theorem 17. Let H 4 G. Then H is projective-sensitive if and only if for all subgroups K of H with K 2 Hp’ a p-Sylow of H (of K), PK is a group, where P is a p-Sylow subgroup of G containing Hp; p being the characteristic of F. Proor. Suppose first that PK is a group with P and K satisfying the condition of the theorem. We show that H is projective-sensitive. In view of Lemma 8 it is sufficient to verify condition (ii) of the definition of projective-sensitivity. Suppose that (H,K) is a projective—pairing. By hypothesis PK is a group, P being appropriate p-Sylow subgroup of G. Taking M = PK it is clear that (G,M) is a projective-pairing and Lemma 9 verifies what is required. 35 Conversely suppose that H 4 G is projective-sensitive and K < H such that (H,K) is a projective-pairing. By condition (ii) there exists M s G, M 2 P and M n H = K. Since H 4 G, H fl‘M = K 4 M implying M C NG(K) = normalizer of K in G. Therefore P C NG(K) and hence PK is a group. This is what we wanted to prove. Remark. We may take PK as a choice for M in condition (ii) of projective—sensitivity. Corollary 1. Let H 4 G with Hp a p-Sylow subgroup of H. If [Hsz] is a prime then H is projective-sensitive. groor. In this case if (H,K) is projective-pairing then either K is a p-Sylow of H or K = H. In both cases PK is a group, P being appropriate p-Sylow of G, and the above theorem applies. Corollary 2. If H 4 G such that every subgroup of H is normal in G then H is projective-sensitive. In particular, if H is cyclic normal then H is projective-sensitive. Proor. Clear from above theorem. We now drop the condition of normality on H and the next theorem gives some non-normal projective-sensitive-groups. Theorem 18. (1) Let (G,H) be a projective-pairing over F of characteristic p. Then H is projective-sensitive if and only if H contains all p-Sylow subgroups of G (p fixed). (2) A p-subgroup H in G is projective-sensitive if and only if H C CoreG(P), P being a p-Sylow subgroup of G. Proor. (1) Suppose H contains all p-Sylow subgroups {pxlx 66, P fixed} of c. If M< c is such that (G,M) is x a projective-pairing then M 2 P for some x E G and so i i l J \ 36 H FIM 2 PK = a p-Sylow of H giving projective-pairing for (H, H n.M). Thus condition (i) of projective-sensitivity is satisfied. Next if (H,K) is a projective-pairing then K22 PX, some x E G. Taking M = K we have (G,M) projective- pairing and 'M n H K n H = K verifying condition (ii). Therefore H is projective-sensitive. Conversely suppose (G,H) is a projective-pairing and there exists an x E G such that H 2 PX. We show that H is not projective-sensitive. Now H 2 Py some y E G. If M = Px then (G,M) is a projective-pairing but neither M 2 H nor (H, H DIM) is a projective-pairing; the first statement M 2 H is clear while for the second H HIM f PX, and so H n M does not contain a p-Sylow of H yielding H to be non- projective-sensitive. (2) Suppose that H C Core P. Wish to show H is projective- sensitive. If (G,M) is a projective-pairing then M.2 PX 2 Core P 2 H and so condition (i) is true. Also (H,K) pro- jective-pairing implies K = H, so we may take M = P. Then (G,M) is a projective-pairing and M,fl H = H = K , thus verifying (ii). Therefore H is projective-sensitive. Conversely, suppose a p-shbgroup H is projective- sensitive and there exists an x E G such that H i PX. Then (G,Px) is a porjective-pairing but neither Px 2 H nor (H, Px n H) is a projective-pairing, since for a p-group H, (H,K) projective-pairing implies K = H. This violates the condition (i) of projective-sensitivity. The proof of the theorem is now complete. 37 Theorem 19. Let Hp be a p-Sylow subgroup of H. If Hp C Core P and [Hsz] is a prime then H is projective- sensitive. Proor. Let M S G with (G,M) projective-pairing. Then M 2 PX, some x E G, P a fixed p-Sylow of G. We have HflM2HflPx2HflCoreP2Hp andso (H,HnM) isapro- jective-pairing. This verifies condition (i) of projective- sensitivity. For condition (ii) note that if (H,K) is a projective-pairing then either K is a p-Sylow of H or K = H. In the former case take M = apprOpriate p-Sylow of G and in the latter case take M - G. It is trivial to verify that M has properties required by condition (ii) of pro- jective-sensitivity. Thus H is projective-sensitive. Corollary . If H is of prime order q, q # p, then H is projective-sensitive. We are now in a position to show by means of examples that normality of H is essential for both the directions in Theorem 17. In G = A4 over field F of characteristic p = 3, H = 22 is projective-sensitive (but not normal) by above corollary but (H,H) is a projective-pairing and PH is not a group (P a 3-Sylow of G), as its order is 6 and A4 has no subgroup of order 6. For the other direction take H to be a p-group not contained in Core P and let H C P. Then (H,H) is a pro- jective-pairing and PH = P is a group, but H is not 38 projective-sensitive by Theorem 18(2). We have seen in the above theorems some projective- sensitive subgroups of G. However, it may happen that G con- tains many more projective-sensitive subgroups. Indeed, Theorem 20. G has every subgroup projective-sensitive if and only if P 4 G, P a p-Sylow of G, p = characteristic of F. _P_roo_f_. Suppose, first, that P <1 G and let H s G. By Theorem 18, if H 2 P or H C P then H is projective- sensitive. So suppose H is not one of these. We first verify condition (i). If p X ‘H‘ then FH is semi-simple and so (H, H FlM) is a projective-pairing for all M C G and so (i) is trivially true. If p ‘ |H|, then, since P 4 G, the p-Sylow subgroup Hp of H is contained in P and so if (G,M) is a projective-pairing then M 2 P implying H 0 M 2 H H P = Hp. Hence (H, H 01M) is a pro- jective-pairing. Thus in all cases condition (1) is satisfied. Next we check condition (ii). Suppose (H,K) is a pro- jective-pairing, K S H. Since P 4 G, PK is a group. Taking M = PK, clearly (G,M) is a projective-pairing and, as was the case in Lemma 9, M H H = K. This completes the sufficiency part of the theorem. Suppose now that G has every subgroup projective- sensitive and P<4 G. Then there exists an x E G such that P # PX. By hypothesis P is projective-sensitive. Let M = PK. Then (G,M) is a projective-pairing but neither M.2 P nor (P, P H M) is a projective-pairing. This violates the 39 condition (i), a contradiction. The proof of the theorem is now complete. Our next theorem describes the representations of a group having all its subgroups projective-sensitive. Theorem 21. If G has all its subgroups projective-sensitive over a field F of characteristic p then every irreducible representation of G over F occurs as a component of (transitive) permutation representation induced by the trivial representation of a p-Sylow subgroup of G. Consequently, degree of each irreducible representation of G is prime to p and each block of G is of lowest kind with p-Sylow subgroup as its defect group. Proor. If G satisfies the condition of the theorem then by Theorem 20, G has unique p-Sylow subgroup P. Let T be an irreducible representation of G afforded by (irreducible) FG—module mt By Clifford's theorem we have fl$ E C>2 Fi’ each Fi = F: the trivial P~module. This gives 1 CHIP)G 5 G) 2, FE. Now since 33! is P-projective, ‘m is a i component of Gm$)G and therefore, since MI is irreducible, there exists an i such that M! is a component of F? = FG. But FG is a module which defines the representation of G as permutations of the right cosets Px. Hence T occurs as a component of the (transitive) permutation representation afforded by FG. This proves the first part of the theorem. II |-‘ 0 Since dim?FG = [G:P], we infer that (dflmgkqfl Therefore defect of each block of PC is a, where \G| a ll '0 B (p,m) = l (Brauer and Nesbitt [2]) and each block is of lowest 40 kind with P as defect group. This completes the proof of the theorem. Corollary_ . If G has all its subgroups projective-sensitive then vertex of each irreducible FG-module is the p-Sylow sub- group P. .E£22£° By the above theorem (dimfiflhp) = 1, M1 being an irreducible FG-module. But by a theorem of Green [4], if B is the vertex of' wt then.the index [P:B] divides dim5ML This proves what is desired. We now assume that P is not normal and in the next theorem we give a sufficient condition for G to have all its normal subgroups projective-sensitive. Theorem 22. Let G = PM, where P is a p-Sylow subgroup of G and M14 G is such that each of its subgroups is normal in G. Then every normal subgroup of G is projective-sensitive. §£22£3 First we observe that M and each of its subgroup is projective-sensitive by Corollary 2 to Theorem 17. Also by Theorem 18 if the normal subgroup H is a p-group or con- tains P then H is projective-sensitive. We show the same for H.4 G not satisfying any of the above conditions. Note that p divides \H‘, for otherwise H :‘M. Now if L s G then there exists g E G such that L = (L (1 Pg) (L ['1 M). For choose g E G such that a p-Sylow subgroup Lp of L is contained in Pg. If y E L we may write y = ab, a E LP, b a p'-element. Then b E L H M. If, in addition, L <1 G then L = (L 0 Pg) (L n M) = -1 -1 (L mpg)g (L mm) = (Lg nP)-(L mm = (L mm mm. 41 Let H4G,H= (HnP)(HflM) =Hp(HflM),Hp beinga p-Sylow of H. In view of Lemma 8 it is sufficient to verify condition (ii) of projective-sensitivity. Let K C H and (H,K) be projective-pairing. There exists x E H such that K=H:(KnM). By hypothesis KnM4G, so S=Px(KnM) is a group, where Px is that p-Sylow of G which contains the p-Sylow subgroup H: of K. We have s = PX(K n M) = Pxfhgm n M)] = PXK . Clearly (G,S) is a projective-pairing and by Lemma 9, SflH=K. This proves the projective-sensitivity of H and so proves the theorem. Corollary 1. If ‘G‘ = pqq, q a prime then every normal sub- group of G is projective-sensitive. Corollary 2. Dn’ the Dihedral group of order 2n, has every subgroup projective-sensitive if p is odd and every normal subgroup projective-sensitive if p = 2. Proor. If p is odd then P 4 G and Theorem 20 applies. If p = 2 then G = PM where P is a 2-Sylow subgroup of G and M 4 G satisfies the condition of the above theorem. 1_Mmeu.eo A 1 BIBLIOGRAPHY 10. 11. 12. 13. BIBLIOGRAPHY Brauer, Richard: Investigations on Group Characters. Annals of Math. Vol. 42 (1941) 2, 936-958. Brauer, R. and Nesbitt, C.: On the Modular Characters of Groups. Annals of Math. Vol 42 (1941) 2, 556-590. 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