Automorphisms of principal blocks stabilizing Sylow subgroups

Transcription

1 Arch. Math. 00 (2003) X/03/ DOI 0.007/s Birkhäuser Verlag, Basel, 2003 Archiv der Mathematik Automorphisms of principal blocks stabilizing Sylow subgroups By Martin Hertweck Abstract. Let p be a prime, G a finite group which has a normal p-subgroup containing its own centralizer in G, and R a commutative local ring with residue class field of characteristic p. In this paper, it is shown that if α is an augmented automorphism of RG which fixes a Sylow p-subgroup P of G, there is ρ Aut(G) such that xαρ = x for all x P, and αρ is an inner automorphism of RG.. Introduction. Let p be a prime, and let G be a finite group which has a normal p-subgroup N containing its own centralizer, C G (N) N. Further on, let R be a commutative local ring with residue class field of characteristic p. The aim of this paper is to prove under these assumptions the following theorem. Recall that an automorphism α of the group ring RG is called augmented if for any g G, the image gα is in the group V(RG) of normalized units in RG (those with augmentation ). Theorem. With G and R as above, let α be an augmented automorphism of RG which fixes a Sylow p-subgroup P of G. Then there is ρ Aut(G) such that xαρ = x for all x P, and αρ is an inner automorphism of RG. Certainly someone will point out that this is a special case of a theorem of K. W. Roggenkamp and L. L. Scott [8] (see [0, p. 266], [6, (4.6) Theorem], [5, Theorem 22]). However, there is no published account. According to [6], the proof of the above theorem is based on the Coleman Lemma and on Green correspondence of automorphisms. The latter can only be applied when the automorphism α fixes a Sylow p-subgroup elementwise; for that part, an extremely short argument of Maschke type is given (see (0) next section). So the difficulty is really to show the existence of the group automorphism ρ. Therefore, a constructive proof is given which contains new ideas. Note that the hypothesis on G is that G is p-constrained with O p (G) = (cf. [2, VII 3.3]), or, equivalently, that the generalized Fitting subgroup F*(G) equals O p (G) Mathematics Subject Classification (2000): Primary 20C05; Secondary 6U60. The research was supported by the Deutsche Forschungsgemeinschaft. E737

2 2 Martin Hertweck arch. math. (cf. [3, X 3.2, X 3.3]). Also note that RG consists of a single (principal) block only (cf. [2, VII 3.5]). The following lemma explains to some extent why the assumption on the stabilization of a Sylow p-subgroup is so powerful. It is an extension of a result of D. B. Coleman (in its present form, it is due to A. I. Saksonov [9,.4]; see also [4, 2.6 Theorem]). Coleman Lemma. Let P be a p-subgroup of the finite group G, and R a commutative ring with pr = R. Then N V (P ) = N G (P ) C V (P ), where V = V(RG). The next result shows how the assumption on the group G is used. For a subgroup H of G, denote by I R (H ) the augmentation ideal of RH, i.e. the R-span of the elements h, for h H. If H is a normal subgroup, I R (H )G denotes the kernel of the natural map RG RG/H. Proposition. Let G be a finite group which has a normal p-subgroup N with C G (N) N. Then C RG (N) R + I R (N)G + prg R + rad(rg). In particular, if x C RG (N) has augmentation, then x maps to under the natural map RG (R/pR)G/N, and x + rad(rg), sox is a unit in RG. Proof. Note that an R-basis of C RG (N) is given by the N-conjugacy classes x, x g N where g ranges over the elements of G. Hence it follows from C G (N) N that C RG (N) is contained in R + I R (N)G + prg, which itself is contained in R + rad(rg) (see [, 5.7, 5.26]). If x C RG (N) has augmentation, then x maps to under the natural map RG (R/pR)G/N, and x + rad(rg), sox is a unit in RG by [, 5.0]. 2. Proof of the Theorem. Assume that the finite group G satisfies the assumptions of the theorem, and let N = O p (G); then C G (N) N. Let P be a Sylow p-subgroup of G, and α an augmented automorphism of RG fixing P. The proof of the theorem proceeds in a number of steps. () Nα = N. Proof. Note that a subgroup H of G is a normal subgroup if and only if h lies in h H the center of RG. Since g = ( n)α lies in the center of RG, and Nα is a subgroup g Nα n N of G of the same order as N, it follows from the definition of N that Nα = N. Let A = C G (N) = Z(N), an abelian normal p-subgroup of G. Note that A = C P (N), so by (), Aα = C Pα (Nα) = C P (N) = A. The so called small group ring of RG with respect to A is the quotient s R (G, A) = RG/I R (A)I R (G). The automorphism α induces an

3 Vol. 00, 2002 Automorphisms of principal blocks stabilizing Sylow subgroups 3 automorphism of s R (G, A) since it stabilize both I R (G) and I R (A). Note that G embeds into s R (G, A) (cf. [7,..8]). (2) The automorphism α induces a group automorphism on (R/pR)G/N. There is a map h : G G such that h(g) (gα) C V(RG) (N) f or all g G. The map h induces a group automorphism of G/A. In addition, we may assume (and do) that h(x) = xα for all x P. Proof. This follows from (), the proposition and the Coleman Lemma. (Note that α induces an automorphism of V(RG)/C V(RG) (N), and that G C V(RG) (N) = A.) (3) Let n N with (p, n) =. Then for any a,...,a n A and x,...,x n RG with all x i of augmentation, there is a A such that n a i x i a n x i Proof. Note that from the identity xy = (x ) + (y ) + (x )(y ) it follows that ( n n ) (a i ) a i mod I R (A) 2. Let s = n x i, b = n a i and a A with a n = b. Then n a i x i s + n (a i )x i s + n (a i ) s + (b ) s + n(a ) s + (a )s a n x i For a set T of right coset representatives of P in G, define u(t, α) = h(t) (tα) RG.

4 4 Martin Hertweck arch. math. Since u(t, α) has augmentation and commutes with the elements of N by (2), it follows from the proposition that u(t, α) C V(RG) (N). (4) Let S and T be two systems of right coset representatives of P in G. Then u(s, α) a u(t, α) mod I R (A)I R (G) for some a A. Proof. Foreach t T there is x t P such that x t t S, and u(s, α) = h(x t t) (x t tα). By (2), the map h induces an automorphism of G/A. Hence for all t T, there is a t A such that a t h(t) h(x t ) = h(x t t). Since h(x) = xα for all x P, it follows that u(s, α) = a t h(t) (tα), and by (3), there is a A such that u(s, α) a h(t) (tα) a u(t, α) (5) Let T be a system of right coset representatives of P in G. Then for all g G, there is a(g) A such that h(g) u(t, α) gα a(g) u(t, α) Proof. Fix g G. It follows from (2) that for each t T, there is a t A such that h(g) h(t) = a t h(tg), and by (3), there is b A such that h(g) h(t) (tα) gα = a t h(tg) (tgα) b h(tg) (tgα) As Tg is also a system of right coset representatives of P in G, it follows from (4) that there is c A such that h(tg) (tgα) c h(t) (tα) Putting a(g) = bc, the result follows. (6) There are σ, τ Aut(G) such that for β = τ α the following hold: (i) Pβ = P ; (ii) nβ = n for all n N;

5 Vol. 00, 2002 Automorphisms of principal blocks stabilizing Sylow subgroups 5 (iii) g (gβ) C V(RG) (N) for all g G; (iv) x (xβ) A for all x P ; (v) For any system S of right coset representatives of P in G, there is a A such that σ β agrees with conjugation with a u(s, β) on s R (G, A). Proof. Keepthenotation of (5). It follows from (5) that the map g h(g) a(g) is a group automorphism τ of G, and that β = τ α agrees with conjugation with u(t, α) on s R (G, A). Note that (i) holds, and also (ii) since u(t, α) C V(RG) (N), which implies (iii) and (iv). Arguing for β in the same way as for α (assuming that the corresponding map h is the identity) it follows from (5) that for each g G there is b(g) A such that g u(t, β) gβ b(g) u(t, β) Again, it follows that an automorphism σ of G is defined by gσ = g b(g) for all g G, and that σ β agrees with conjugation with u(t, β) on s R (G, A), so (v) holds by (4). (7) Let g G. Then the subgroup U g ={x P : x g P } P is normalized by g (gβ), and there is y A with [y g (gβ), U g ] =. Proof. Let x U g. By (6.iv) and (6.iii), there is a A such that x g = (x g β)a, and x g (gβ) = (xβ)a gβ = x x (xβ) a g xa. Hence g (gβ) normalizes U g, and by the Coleman Lemma, there is y G such that [y g (gβ), U g ] =. Since N U g,it follows from (6.iii) that y C G (N) = A. Some notation needs to be fixed. Choose W G such that G = PwP (disjoint union). w W For all w W, choose X w P such that PwP = Pwx (disjoint union). x X w For all w W, choose a w A with [a w w (wβ), U w ] =, cf. (7). Put a w,x = x a w x(x β)x and s w,x = a w,x (wx) (wxβ) for all w W, x X w. Note that a w,x A and s w,x C V(RG) (N), and that s w, = a w w (wβ). (8) For any w W, P operates on {s w,x : x X w } transitively via conjugation. Proof. Letw W and g P. Then there is x X w with Pwg = Pwx. Observe that gx U w,so g s w, g = g (a w w (wβ)) g by definition of a w = x (a w w (wβ)) x = x a w x (wx) (wxβ) (xβ) x = x a w x (xβ) x (wx) (wxβ) by (6.iii) and (6.iv) = s w,x. In particular, x s w, x = s w,x for all x X w,sop operates on {s w,x : x X w } transitively.

6 6 Martin Hertweck arch. math. (9) There is a A such that xσ β = x a for all x P. Proof. Put u = w W x X w s w,x = w W x X w a w,x (wx) (wxβ). Note that S ={wx : w W and x X w } is a system of right coset representatives of P in G. By (3), there is b A such that bu u(s, β) mod I R (A)I R (G) (the corresponding map h for β is chosen to be the identity map). By (6.v), there is c A such that σ β and conjugation with c u(s, β) agree on s R (G, A). By (8), u commutes with the elements of P. Hence for x P, xσ β x c u(s,β) x cbu x cb Put a = cb A. Since Pσ β = P, it follows that xσ β = x a for all x P. With a as in (8), and γ Aut(G) being conjugation with aα G, it follows that xα = xγτσ for all x P. Therefore, the proof of the theorem is completed by the following observation. (0) Assume that α fixes the elements of the Sylow p-subgroup P element-wise. Then α is an inner automorphism. Proof. LetT be a set of right coset representatives for P in G, and put u = t (tα). Note that the definition of u does not depend on the particular choice of T,sog u(gα) = u for all g G, and u is a unit by the proposition. Hence α is conjugation with u. References [] C. W. Curtis and I. Reiner, Methods of Representation Theory Vol. I (John Wiley, New York 98). [2] B. Huppert and N. Blackburn, Endliche Gruppen II (Springer, Berlin 982). [3] B. Huppert and N. Blackburn, Endliche Gruppen III (Springer, Berlin 982). [4] S. Jackowski and Z. Marciniak, Group automorphisms inducing the identity map on cohomology. J. Pure Appl. Algebra 44, (987). [5] K. W. Roggenkamp, The Isomorphism Problem for Integral Group Rings of Finite Groups, Proceedings of the Int. Congress of Mathematicians, Kyoto 990 (Springer, Berlin, 99) pp [6] K. W. Roggenkamp, The isomorphism problem for integral group rings of finite groups. Progr. Math. Vol. 95 (Birkhäuser, Basel, 99), pp [7] K. W. Roggenkamp and L. L. Scott, Isomorphisms of p-adic group rings. Ann. of Math. 26, (987).

7 Vol. 00, 2002 Automorphisms of principal blocks stabilizing Sylow subgroups 7 [8] K. W. Roggenkamp and L. L. Scott, A strong answer to the isomorphism problem for finite p-solvable groups with a normal p-subgroup containing its centralizer, manuscript (987). [9] A. I. Saksonov, On the group ring of finite groups I (Russian). Publ. Math. Debrecen 8, (97). [0] L. L. Scott, Recent Progress on the Isomorphism Problem. Representations of finite groups, Proc. Conf., Arcata/Calif Proc. Symp. Pure Math. 47, (987). M. Hertweck Mathematisches Institut B Universität Stuttgart Pfaffenwaldring 57 D Stuttgart Germany hertweck@mathematik.uni-stuttgart.de Received: 26 July 2000