Arch. Math., Vol. 60, 1 -6 (1993) 0003-889X/93/600 1-0001 $ 2.70/0 © 1993 Birkhauser Verlag, Basel A Sylowlike theorem for integral group rings of finite solvable groups By W. KIMMERLE and K. W. ROGGENKAMP *) 1. Introduction. For a finite group G and a commutative ring R we denote by the group ring of G over R. This group ring is an augmented algebra with augmentations: RG -> R, £ rg'9~^ Z V geG geG By V(RG) we denote the units in RG, which have augmentation 1. The group of units in RG is then the product of the units in R and V(RG). A subgroup H of V(RG) with \H\ = |G| is called a group basis, provided the elements of H are linearly independent. This latter condition is automatic, provided no rational prime divisor of \H\ is a unit in R [1]. If H is a group basis, then RG = RH as augmented algebras and conversely. The object of this note is to prove the following Theorem 1. Let Gbea finite solvable group, and let H be a group basis ofZG with Sylow p-subgroup P. Then there exists a unit a e QG such that a Pa'1 is a Sylow p-subgroup of G. Remark 1 . For solvable groups it was conjectured by Hans Zassenhaus [12, 1 1] that for any finite subgroup U of V(ZG) there exists a e QG with aUa'1 c G. It is known that for a solvable group G, the Sylow p-subgroups of different group bases in TLG are isomorphic; however, the above result gives information about the embedding of these Sylow p-subgroups into ZG. The isomorphism of the Sylow p-subgroups is an immediate consequence of the following more general result: tfLp stands for the complete ring of p-adic integers.) Theorem 2 ([9J). Let Gbea finite group such that the generalized Fitting subgroup F*(G) is a p-group *). Then a group basis H of ZG is conjugate by a unit in TL G to a subgroup of G. *) The second author was partially supported by the DFG. *) This is to say that G has a normal p-subgroup N with the centralizier CG(N) c JV or that the generalized p'-core 0^(G) is trivial [2, 3]. Archiv der Mathematik 60 1  W. KIMMERLE and K. W. ROGGENKAMP ARCH. MATH. We shall state next a more general result, which does not only apply to solvable groups, and of which Theorem 1 is a special case - as will become transparent later on. For this wfr have to introduce some more notation. Definition 1 . Let G be a finite group. 1. n(G) is the set of rational prime divisors of |G|. 2. For the rational prime p, the group Op>(G) is the largest normal subgroup of G with order relatively prime to p. 3. Op*(G) is the generalized p'-core of G [4, Ch. X, Paragraph 14]. 4. Let n be a finite set of rational primes. We call a finite group G n-constrained, if for each q E n there exists a rational prime p such that Op*(G/Op,(G)) = 1 and q does not divide Remark 2 . Note that in the above definition, the prime q need not be different from p. Therefore a p-constrained group G is also 7r-constrained for n = n(G)\n(Op,(G)). Clear ly a finite solvable group is 7i-constrained for every set of primes n. However, there are many insolvable groups which are 7i-constrained for some set n (e.g. every Frobenius group is 7r-constrained for a suitable set n). It is not true though, that a Ti-constrained group G is p-constrained for every p e n [2, 3]. We can now state the result, which we shall prove here: Theorem 3. Let G be a finite n-constrained group, and let H be a group basis in ZG. For each pen and P a Sylow p-subgroup of H9 there exists a unit a e QG with a Pa'1 a Sylow p-subgroup of G. 2. Connection with the Zassenhaus conjecture. Let us return to a weak form of the Zassenhaus conjecture (cf. Remark 1): Conjecture 1 (Zassenhaus [12, 11]). Let G be a finite group. If H is a group basis in ZG, then H is conjugate in QG to G; i.e. there exists a unit a e QG such that aHa~l = G. Remark 3. It was shown in [7] that the above conjecture is true for finite nilpotent groups. However, in [8] a metabelian group was constructed, which is a counterexample to the above Zassenhaus conjecture. It is convenient, to rephrase the Zassenhaus conjecture in terms of isomorphisms over class sums. Definition 2. Let G be a finite group. 1. A class sum in 7LG is an element of the form CSG(g)= £ '0; xeGfCG(g) i.e. the sum of the different conjugate elements of g. 2. Let H be a group basis in ZG. Then there is a class sum correspondence [1]: For every h e H there exists an element y(h) e G, such that CSH(h) = CSG(y(h)) inZG. Note that y(h) is only determined up to conjugacy. Since the conjugacy class of h and y(h) must have  Vol. 60, 1993 Zassenhaus conjecture for Sylow subgroups 3 the same cardinality - use the augmentation - the map y can be extended to a bijection y: G-> H. We shall call such a map a class sum correspondence. Note that y is in general not unique and is in general not a homomorphism of groups; however, it sends p-power elements of G to p-power elements of H; it even preserves the order of the elements [6]. 3. This class sum correspondence induces a correspondence between the normal sub groups of G and H, essentially since a normal subgroup is a union of conjugacy classes, cf. e.g. [10]. 4. Let H be a group basis in ZG. An isomorphism Q: H -> G is called an isomorphism over the class sums provided the induced automorphism - note ZG = ZH - which we shall also denote by Q Q-.ZH-+ZG, £ vfc-* I rk-Q(h) heH heH has the property q(CSH(h)) = CSG(y(h)). We can now reformulate the Zassenhaus conjecture - using the theorem of Skolem- Noether: Proposition 1. The Zassenhaus conjecture is equivalent to the statement that for each group basis H of ZG there exists an isomorphism - this means that the isomorphism problem has a positive answer - which is an isomorphism over the class sums; with other words the above bijection can be chosen to be a group isomorphism. Remark 4. We shall collect here some observations: 1. Theorem 2 thus states, that in case F*(G) is a p-group, then for every group basis H there exists an isomorphism over the class sums. 2. In our Theorems 1, 3 we are not dealing with the group basis, but rather with a subgroup of a group basis H. Thus we are looking for an extension of Proposition 1 to a subgroup U of the group basis H (cf. Remark 1). 3. The obvious extension would be to require that the bijection y in the Definition 2,2 could be chosen in such a way that it is a group isomorphism when restricted to U. Theorem 4. Let G be a finite group and let U be a finite subgroup of F( (2): If we take H = a~ 1 Ga, then the conjugation by a is the desired map Q. (2) => (1): Let L e K be an algebraic number field, which is a splitting field for G and choose a simple Wedderburn component A of KG = KH. Via the projection onto A we obtain two representations of 17, denoted by 0^ and $Q(U) resp., where v and 0ff(l/) coincide. In fact, by assumption CSH(u) = CSG(Q(U)) and so we have for the trace of v and 0ff(l/) resp. with / = \CSH(u)\ = \CSG(Q(u))\: This holds for every u e 17, and since the characters determine a representation up to isomorphism (conjugacy), we conclude, that v and c(l7) are conjugate in A. Since this can be done for every simple Wedderburn component of KG, we conclude that there exists b e KG such that b Ub~ 1 = 0(17). It remains to show that this conjugation can already be achieved in LG. We shall be using bimodules to reach this goal: We consider M = LG as L(U x G)-bimodule, by letting U act in its natural way on M from the left and G acts on the right by its natural action. QM has the same right action as M, but the left action is twisted by Q: Since U and 0(17) are conjugate in KG, the bimodules K®LM and K LQM are isomorphic. Invoking the Noether-Deuring theorem, we conclude that the bimodules M and QM must be isomorphic. Let  Vol. 60, 1993 Zassenhaus conjecture for Sylow subgroups 5 be an isomorphism of L(U x G)-bimodules. We put a = t(l). Then a is a unit in LG and moreover, Q(U) - a = a - u for every u e U. q.e.d. The proof of Theorem 3 will now follow from Theorem 4, if we can show Proposition 2. Let G be a finite it-constrained group for n a finite set of rational primes. H is a group basis in %G. For qen there exists by Definition 1,4 a prime p such that Op.(G/Op,(G)) = l. Let S be a Sylow q-subgroup of H. Then there exists a class sum correspondence Q'.H^G such that QS: S -> Q(S) is a group isomorphism. Proof. Let be the augmented ring homomorphism induced from reduction modulo Op,(G). Since G is 7c-constrained, q does not divide |0P'(G)|, and so /c,s injects S into ZG/Op,(G). By the choice of p, we may apply Theorem 2, to conclude that the Zassenhaus conjec ture holds for ZG/0P>(G), and so there exists a class sum correspondence in inducing an isomorphism of groups Q'.K(H)^K(G). With the correspondence of normal subgroups (Definition 2,3) we conclude that ker(K|H) = Op,(H) and that \0P.(H)\ = \0,.(G)\. Thus we can find a Sylow ^-subgroup of G, say, T such that is a group isomorphism. Summarizing, we have now constructed a group isomorphism Qs = : KI V ° Q ° *\s from StoT. Claim 1. Let now y://->G be a class sum correspondence (Definition 2,2). Then = CSG(Qs(s)).  6 W. KIMMERLE and K. W. ROGGENKAMP ARCH. MATH. Proofofthe claim. Because of the class sum correspondence y, there exists for every s e S an element t e T such that CSH(s) = CSG(t) - note that y sends g-power elements to g-power elements (Definition 2,2). On the other hand, Q induces the class sum correspondence on 7LG/Op>(G\ and so we must have CSG/0p,(G)(K(t)) = CSG/0p>(G}(Q o K(S)). Thus t is conjugate in G to a g-power element of the form w • QS(S) for some w e Op,(G). Note that we still have freedom in choosing t in its conjugacy class. Thus we can assume that t is such that K(t) = K(Q(S)). In Op,(G) • T the element w • QS(S) is - by Sylow's theorem - conjugate by an element H^ e Op,(G) to an element tl e T. But then K(I) = K^) and so we must have t = tl9 since K\T is injective. Consequently QS(S) and t are conjugate. This proves the claim and also finishes the proof of Proposition 2, and hence completes the proof of Theorem 3 and consequently of Theorem 1. References [1] S. D. BERMANN, On a necessary condition for isomorphism of integral group rings. Dopovidi Akad. Nauk Ukrain. RSR, Ser. A, 313-316 (1953), MR 15,599. [2] B. HUPPERT, Endliche Gruppen. I. Berlin- Heidelberg-New York 1967. [3] B. HUPPERT and N. BLACKBURN, Finite Groups. II. Berlin-Heidelberg-New York 1982. [4] B. HUPPERT and N. BLACKBURN, Finite Groups. III. Berlin-Heidelberg-New York 1982. [5] W. KIMMERLE, Zum Isomorphieproblem ganzzahliger Gruppenringe. Sylow- und Jordan Holder Theorie. Bayreuth. Math. Schr. 33, 91-107 (1990). [6] D. S. PASSMAN, Isomorphic groups and group rings. Pacific J. Math. 15, 561-583 (1965). [7] K. W. ROGGENKAMP and L. L. SCOTT, Isomorphisms of/?-adic group rings. Ann. of Math. (2) 126, 593-647 (1987). [8] K. W. ROGGENKAMP and L. L. SCOTT, On a conjecture on group rings by H. Zassenhaus. Manuscript 1987. [9] K. W. ROGGENKAMP and L. L. SCOTT, A strong answer to the isomorphism problem for finite /^-solvable groups with a normals-subgroup containing its centralizes Manuscript 1987. [10] R. SANDLING, The isomorphism problem for group rings: a survey. LNM 1142, 239-255. Berlin-Heidelberg-New York 1985. [11] S. K. SERHGAL, Torsion units in integral group rings. Proc. Nato Institut on methods in ring theory, 497-504. Antwerpen-Dordrecht 1983. [12] H. ZASSENHAUS, On the torsion units of finite group rings. In: Estados de matematica em homenagem ao Prof. A. Almeida Costa, 119-126. Lisbon 1974. Eingegangen am 5.4. 1990 Anschrift der Autoren: W. Kimmerle K. Roggenkamp Mathematisches Institut B Universitat Stuttgart PfafTenwaldring 57 DW-7000 Stuttgart 80