PROC. 18th SRI, 20C15 CANBERRA 1978, 1-7. GNIDNETXE SRETCARAHC MORF LAMRON SPUORGBUS Robert .B Howlett The purpose of this note is to give a short proof of the main theorem of i. Essentially the same simplification has also been discovered (independently) by Isaacso The theorem is THEOREM . Suppose that A is a ~nite group which acts on the finite group H , and (l) for some prime r , IH'I = r and H/Z(H) is an elementary abelian r-group, (2) A centralizes Z(H) , (3) A has a soluble normal subgroup B with order prime to r and satisfying H, BZ(H) = H . Then any non-linear irreducible complex character of H can be extended to a character of AH . We treat even and odd r simultaneously, although a short proof of a different kind is available for odd r (see i). One application of the theorem is the proof for soluble groups G of the following (McKay's conjecture): if N is the normalizer of a Sylow p-subgroup of G , the number of irreducible complex characters with degree prime to p is the same for G as for N . This is proved in a paper by Wolf 6. Before starting the main part of the proof we collect into a lemma three well known sufficient conditions for the extendibility of a character. LEMMA 2. Suppose H 4 G , G : ~H , T o H : i . Let X be an irreducible complex character of H such that for all t 6 T , h ( H , xIt-lhtl = x(h) . Then X can be extended to G if any of the following hold: )a( T is cyclic; )b( X(1) and ITI are coprime; )c( for each prime q dividing IT I there exists a Sylow q-subgroup Q of T such that X extends to QH . Proof. Part ~) is easy. For part ~) see 3. (Note that )b( includes the special case X(1) = i ). For part )c( see 5. LEM~ 3. Let H ~ G , X an irreducible character of H , and K = {g 6 G x~g-lhgl = x(h) for all h 6 H} . Then if ~ is any irreducible component of the induced character X K then t G is irreducible. Proof. See 2, Theorem i. Now let A and H satisfy the hypotheses of Theorem i. The proof proceeds by induction on IAHI We may regard H* = H/Z(H) as a vector space over F , the r field with r elements. We use additive notation in H* and use stars to denote images in H* of subgroups of H . Conjugation of elements of H by elements of A induces an action of A on H* , making H* into an F A-module. As an F B-module r r H* is completely reducible (by Maschke's Theorem). PROPOSITION 4. Suppose that Z(H) ~ M ~ H and that M is B-invariant. Then M, BZ(H) = M . If M is abelian M = M, B x Z(H) . Proof. Let N be a subgroup of H with H* : M* ~ N* and N B-invariant. Since H, B* = H* , clearly M, B* = M* and N, B* = N* , proving the first assertion. The second assertion follows from 4, Theorem 5.2.3. PROPOSIIION 5. Let K be any subgroup of H such that Z(K) : Z(H) . Then H i8 the central product of K and CH(K) with Z(K) = ZCH(K) 1 amalgan~ted. Proof. See 4, Lemma 5.4.6. It is easily shown that for any linear character l of Z(H) which is non- trivial on H' , h H has a unique irreducible constituent X which satisfies X(x) : 0 if x } Z(H) , X(x) : mh(x) if x 6 Z(H) , Where m = X(1) is the multiplicity of X in l G , and m 2 = H : Z(H) . Each non-linear irreducible character of H is obtainable in this way. For the rest of the proof X and 1 will be fixed, and we assume that X does not extend to AH . (Thus A, H and X constitute a minimal counterexample.) Minimality of A implies that A acts faithfully on H . NOITISOPORP 6. H* is an irreducible F A-module. r Proof. Suppose not, and let M* be an irreducible F A-submodule of H* . r Case (i). Suppose that M is non-abelian. Then Z(M) = Z(H) and so H = KIK 2 with K I : M , K 2 : CH(M) . The K. are A-invariant, and using Proposition 4 we see that the hypotheses of the theorem are satisfied when K. replaces H . So we will be able to apply induction. By the representation theory of central products there exist irreducible CK.- modules V. (i = i, 2) such that under the action XzX2¢ l e~2) : Xzh eX2V 2 x( i ~ %, v i ~ V )i V I ® V 2 is a well defined @H-module affording the character X • By induction V i can be made into a @AK.-module. Now if we define a(v I ® v2) = av I Q av 2 (a ( A, v i E Vi then V I ® V 2 becomes a @AH-module, and its character extends X - a contradiction. Case (ii). Suppose that M is abelian, and let N be a maximal abelian A-invariant subgroup of H containing M . Then N = B, Nx Z(H) . Define U to coincide with 1 on Z(H) and have kernel L containing B, N . Then ~ is fixed by A , and so the inertia group K = {x ( H I pX = U} is A-invariant. Let be an irreducible constituent of U K . By Lemma 3, ~H is irreducible, and since it is a constituent of h H , ~H : X • Suppose firstly that K = N . By Lemma 2 (b), ~ : ~ extends to a linear character "~ of AK , and obviously ~AH extends ~H = X - a contradiction. Suppose on the other hand that K is non-abelian. Since L ~ K we may set : K/L . Because L n K' : 1 , IK'I = 1 K'I : r . Moreover, Z(K) = Z(K)L/L = N/L = Z(H)L/L , and so A centralizes Z(K) . Now we may apply induction (with A, K, U, ~ replacing A, H, h, X ) to conclude that ~ extends to a character ~ of AK . As before, ~AH extends @H , a contradiction. PROPOSITION 7. If L is a Sylow r-subgroup of A then A = LB . Proof. If LB # A then by induction X extends to LBH . If q is any other prime dividing IAI and Q a Sylow q-subgroup of A then by Lemma 2 (b), X extends to QH . By Lemma 2 )c( it follows that X extends to AH , a contradiction. PROPOSITION .8 B is a minimal normal subgroup of A . Proof. Suppose not, and let B 0 be a minimal normal subgroup of A contained in B . Let A 0 = LB 0 , so that IA01 < IAI . Since it suffices to prove that X extends to LH it suffices to prove that X extends to AoH . If ~0' HJZ(H) = H this follows by induction since all the hypotheses of the theorem are satisfied when A 0 replaces A . If B0, HZ(H) < H then by Proposition ,6 ~0' HI ~ Z(H) . By 4, Theorem 5.3.5, CH(Bo) = H . But this is impossible since B 0 is a nontrivial group of automorphisms of H . Hence ~0' ~Z(H) = H , as required. In view of Proposition 6 we may apply Clifford's Theorem 4, Theorem 3.4.1 and write H* as an F B-module direct sum r H* = HI *® H2 *® "" . ® H* n where the H. are permuted transitively by L . The H~ are the LB-primary components of H* (that is, Hi is the sum of all the irreducible F B-submodules of r H isomorphic to a given irreducible module. NOITISOPORP 9. If B is not cyclic there exist B-invariant proper subgroups KI, K 2 ..... K m of H such that (I) KI, K2, ... K are permuted transitively by L ' m (2) H is the central product KIK 2 ... K m with z(%) = zIKg) = ... = = amalgamated. Proof. By Proposition 8 and the fact that B is soluble, B is abelian. Assuming that B is not cyclic it follows that the subgroups CH(X) (x (B) generate H 4, Theorem 5.3.16. Let B I be a maximal subgroup of B such that K I : CH(BI) # Z(H) . If M* is any ErB-submoduie of H* isomorphic to some submodule of K~ then B I acts trivialiy on M* , and by 4, Theorem 5.3.15, M ~ c K~ . Thus K~ is a direct sum of some subset of the primary components H{ • Since H* = L~( ~ BI' HI* (4, Theorem 5.2.3) it follows that B I, HI* is the sum of the remaining primary components. Now since K I (H and If I, BI = i , Vl' h' : h : 1 and so by the three subgroup lemma (4, Theorem 2.2.3), BI, HI and K I centralize each other. Let I ( L , 1 ~ NL(BI) . Then by choice of B I . Hence is a sum of primary components distinct from those in Ki . So K I and K I l centralize each other. Similarly, if T is a set of representatives of the right eosets of NL(BI in L then distinct members of do not have a primary component in common. Hence their sum is direct. Furthermore, KI, K2, ..., ~ centralize each other. Since L pe~utes KI' K2' "''' mK it follows that KIK 2 ... K m is A-invariant. Hence H = KIK 2 ... K m , as required. NOZTISOPORP 10. B is cyclic of prime order p . Proof. By Proposition 8, B is an elementary abelian p-group (for some prime p ) and if IBI > p then H may be expressed as a central product as in Proposition .9 Clearly all the hypotheses of the theorem are satisfied when A I = NAKI) replaces A and K I replaces H . Since K I # H we may apply induction to conclude that if XI is any irreducible constituent of the restriction of X to % then XI extends to AIK I . Let V I be an irreducible @AiKl-mOdule affording this character. Choose representatives 1 I, 12, . • . , I m for the left cosets 151 of A I in I. A , so that K. ~ = K I (i = i, 2 .... , m) . Let Yi be a vector space over isomorphic to Y I and let Oi : Yi ÷ Yl be an isomorphism• Then Y i is an irreducible ~K.-module under -i 1. = Pi x ~Ipi ~) tx ( ~, v ( Vi) . Clearly, the tensor product VI ® V2 @ "'" ® Vm is an irreducible ~H-module under XlX 2 ... Xm{V 1 ® V 2 ~ ... ® V m = XlV 1 Q x2V 2 ® ... Q XmV m x( i ~ %, vi ~ V i , and this module affords the character X (since X is the unique irreducible H constituent of XI .) But now V I ® V 2 ® ... @V m becomes a @AH-module if we define for a ( A , va i®v 2®-.. ®v m =u I ®u 2®... ®um where u. is defined by %u i = lilalj(%vj (i = i, 2 ..... m) , j being the unique index such that al~l : liA I . Thus we have contradicted the assumption that X does not extend, and the proposition is proved. (The above construction appears in i.) We now complete the proof of the theorem by deriving a final contradiction. Let L I : CL(B) . Then ~i' HZ(H) # H and is A-~nvar~ant; so by Proposition ~, ~i' 4 ~ z(~) . Now ~, ~, B : ,B LI0 4 : 1 hence i= <B,L I = H,L I by hypothesis (3) of Theorem .i But L 1 <_ aut(H) ; so L 1 = 1 . Since IBI = p and OL(B) = 1 it follows that L , being a group of automorphisms of B , is cyclic. By Lemma 2 (a), X extends to LH , and using Lemma 2 (b) and )o( as in the proof of Proposition 7, X extends to AH • This is the required contradiction. References i Everett C. Dade, "Characters of groups with normal extra special subgroups", Math. .Z 52 (1976), 1-31. 2 P.X. Gallagher, "Group characters and normal Hall subgroups", Nagoya Math. J. 2 (1962), 223-230. 3 George Glauberman, "Correspondence of characters for relatively prime operator groups", Cared. J. Math. 20 (1968), 1465-1488. 4 Daniel Gorenstein, Finite Groups (Harper and Row, New York, Evanston, London, 1968). 5 I.M. Isaacs, "Invariant and extendible group characters ,T' Illinois .J Math. 4 (1970), 70-75. 6 Thomas R. Wolf, "Characters of p'-degree in solvable groups", Pacific .J Math. 4 (1978), 267-271. Department of Pure Mathematics, University of Sydney, Sydney, New South Wales. PROC. 18th SRI, 16-02 CANBERRA 1978, 8-46 (16A38) EMOS TNECER STNEMPOLEVED IN EHT YROEHT FO SARBEGLA WITH LAIMONYLOP IDENTITIES .N Jacobson The author gave six lectures at the Summer Research Institute: two survey lectures and four lectures on algebras with polynomial identities. The survey lectures had the titles: Development of eht Concept of a Jordan Structure and History of Algebras with Polynomial Identity. The author decided not to include these lectures in the Proceedings since these would have overlapped substantially with surveys that have appeared or are about to appear, notably, a survey article on Jordan algebras by McCrimmon that will appear shortly in the Bulletin of the American Mathematical Society, a survey lecture "Pl-algebras" by the author appearing in the Proceedings of a ring theory conference at University of Oklahoma published in 1973 by M. Dekker, and a survey article "Polynomial identities" by S.A. Amitsur in Israel Journal of Mathematics, 19 (1974), 183-199. The lectures on recent developments in the Pl-theory have been written up with the following titles: I Razmyslov's central polynomial, II The Artin-Procesi Theorem, III On Shirshov's local finiteness theorems. Central polynomials for the complete matrix algebra M(K) , K a commutative ring, have played an important role in recent developments of the Pl-theory. For example, they can be used to give an improved formulation and proof of the main structure theorem for prime Pl-algebras. The first construction of central polynomials for Pl-algebras was given in 1972 by Formanek. In Chapter I we present an alternative construction due to Razmyslov that has the advantage that the central polynomial is multilinear and is alternating in n 2 of the arguments. Razmyslov's central polynomial plays an important role in II that is concerned with a theorem connecting Pl-aigebras with Azumaya algebras. In III we give an account (with some improvements) of the best positive results that have appeared to date on a problem in ring theory, Kurosch's problem, that is analogous to Burnside's problem in group theory. RETPAHC I RAZMYSLOV'S CENTRAL POLYNOMIAL Let A be an algebra (associative with unit) over a commutative ring K and let K{Xl, x2, ...} be the free associative algebra generated by x I, x2, .... An element f(x I ..... xm) 6 K{x I, x 2 .... } is called a central polynom£al for A if f(a I ..... am ( C = C(A) the center of A for all a i ( A and there exist such that f(a I ..... am # 0 . In other words, a I a m f(xl, ..., )mX is an identity for A hut is not. 'de shall give a construction due to Razmyslov, of a multilinear central polynomial for M (K) , the algebra of n n x n matrices over K . We require first some elementary results on M (K) . n AMMEL I. a~he trace bilinear form t(A, B) = tr AB is non-degenerate an M (K) : t(A, B) = 0 for all B implies A = 0 . n Proof. If A : (aij and {ekl I 1 _< k, 1 _< n} are the usual matrix units for Mn(K) , then t(A, ekl = alk . Hence tA, ekZ = 0 for all k, 1 implies A=O . 01 M(K) is a free module with base of n 2 elements. Hence any K-endomorphism of M(K) determines an n 2 × n 2 matrix relative to a base. The trace of this matrix is independent of the choice of base and is called the trace of eht .msihpromodne In particular, we consider the endomorphism U~-+AUB defined by a given pair of matrices A and B . Then we have LEMMA .2 The trace of eht endomorphism U~-~AUB of Mn(K) si (tr A)(tr B) . This can be verified by direct calculation using the base jiel-. 1 of matrix units. We omit this simple calculation. We note next that if A is any algebra and A e = A ~K A° where A ° is the opposite algebra then we have the involution ~ a i ® b i ~+ ~~ / b i ~ a i in A e ~ ai @bi)x = ~aixb i . Moreover, A is an Ae-module relative to the action This action is by K-endomorphisms. For a certain class of algebra, the Azumaya algebras, the representation defined by this action is faithful and the image is End K A . We can verify this for A = M(K) : LEMMA .3 The map sending ~ a i @b i into eht endomorphism x~-+~aixb i of Mn(K) into itself si an isomorphism onto EndKMn(K) . Proof. Since both ~(K) ~(K) ° and End K Mn(K) are free K-modules of rank n 4 it suffices to show that the homomorphism is surjective. For this it is enough to show that for any pair (i, j), (k, )l there exists a map of the for~ U~+~A iUB i sending jie into ekl and ei,j, into 0 for every (i', j') # ,i( j) . It is clear that U~-+ ekiUejl does this. In the situation in which we have an isomorphism of A e onto End K A defined as above we can transfer the involution ~ a i ® bi~-+ ~ b i ® a i to an involution in End K A . We denote this involution as 1 +-F l* . Now let X = K$11 ..... ji~ ..... ~n~ where the ji~ are indeterminates and consider M(X) which is the same thing as X ®K M(K) . We can regard M(X) as M (K)e-module obtained by restricting the action from M (X) e to the n