Author: Dr. M. Hertweck, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany Fax: (49 711)685-5304 E-mail: hertweck@mathematik.uni-stuttgart.de

AUTOMORPHISMS OF PRINCIPAL BLOCKS STABILIZING SYLOW SUBGROUPS

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.

1. 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 1).

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 [10, 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 element-wise; for that part, an extremely short argument of ‘Maschke type’ is given (see (10) next section). So the difficulty is really to show the existence of the group automorphism ρ. Therefor, 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) = 1 (cf. [2, VII 13.3]), or, equivalently, that the generalized Fitting subgroup F^{∗}(G) equals O_{p}(G) (cf. [3, X 13.12, X 13.13]). Also note that RG consists of a single (principal) block only (cf. [2, VII 13.5]).

Date: July 20, 2000.

2000 Mathematics Subject Classification. Primary 20C05; Secondary 16U60.

The research was supported by the Deutsche Forschungsgemeinschaft.

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, 1.14]; 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 − 1, 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 1, then x maps to 1 under the natural map RG → (R/pR)G/N, and x ∈ 1 + rad(RG), so x is a unit in RG.

Proof. Note that an R-basis of C_{RG}(N) is given by the ‘N-conjugacy classes’ ∑_{x∈g^{N}} x, 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 [1, 5.7, 5.26]). If x ∈ C_{RG}(N) has augmentation 1, then x maps to 1 under the natural map RG → (R/pR)G/N, and x ∈ 1 + rad(RG), so x is a unit in RG by [1, 5.10].

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.

(1) Nα = N.

Proof. Note that a subgroup H of G is a normal subgroup if and only if ∑_{h∈H} h lies in the center of RG. Since ∑_{g∈Nα} g = (∑_{n∈N} n)α lies in the center of RG, and Nα is a subgroup 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 (1), Aα = C_{P α}(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 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, 1.1.8]).

(2) The automorphism α induces a group automorphism on (R/pR)G/N. There is a map h : G → G such that

h(g)^{−1}(gα) ∈ C_{V(RG)}(N) for 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 (1), 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) = 1. Then for any a_{1}, …, a_{n} ∈ A and x_{1}, …, x_{n} ∈ RG with all x_{i} of augmentation 1, there is a ∈ A such that

∑_{i=1}^{n} a_{i}x_{i} ≡ a ∑_{i=1}^{n} x_{i} mod I_{R}(A)I_{R}(G).

Proof. Note that from the identity xy − 1 = (x − 1) + (y − 1) + (x − 1)(y − 1) it follows that

∑_{i=1}^{n}(a_{i} − 1) ≡ (∏_{i=1}^{n} a_{i}) − 1 mod I_{R}(A)^{2}.

Let s = ∑_{i=1}^{n} x_{i}, b = ∏_{i=1}^{n} a_{i} and a ∈ A with a^{n} = b. Then

∑_{i=1}^{n} a_{i}x_{i} ≡ s + ∑_{i=1}^{n}(a_{i} − 1)x_{i} ≡ s + ∑_{i=1}^{n}(a_{i} − 1) ≡ s + (b − 1)

≡ s + n(a − 1) ≡ s + (a − 1)s ≡ a ∑_{i=1}^{n} x_{i} mod I_{R}(A)I_{R}(G).

For a set T of right coset representatives of P in G, define

u(T, α) = \frac{1}{|G : P |} ∑_{t∈T} h(t)^{−1}(tα) ∈ RG.

Since u(T, α) has augmentation 1 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. For each t ∈ T there is x_{t} ∈ P such that x_{t}t ∈ S, and

u(S, α) = \frac{1}{|G : P |} ∑_{t∈T} h(x_{t}t)^{−1}(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)^{−1} · h(x_{t})^{−1} = h(x_{t}t)^{−1}. Since h(x) = xα for all x ∈ P, it follows that

u(S, α) = \frac{1}{|G : P |} ∑_{t∈T} a_{t} · h(t)^{−1}(tα),

and by (3), there is a ∈ A such that

u(S, α) ≡ a\frac{1}{|G : P |} ∑_{t∈T} h(t)^{−1}(tα) ≡ a · u(T, α) mod I_{R}(A)I_{R}(G).

(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)^{−1} · u(T, α) · gα ≡ a(g) · u(T, α) mod I_{R}(A)I_{R}(G).

Proof. Fix g ∈ G. It follows from (2) that for each t ∈ T, there is a_{t} ∈ A such that h(g)^{−1} · h(t)^{−1} = a_{t} · h(tg)^{−1}, and by (3), there is b ∈ A such that

h(g)^{−1} · ∑_{t∈T} h(t)^{−1}(tα) · gα = ∑_{t∈T} a_{t} · h(tg)^{−1}(tgα)

≡ b · ∑_{t∈T} h(tg)^{−1}(tgα) mod I_{R}(A)I_{R}(G).

As T g is also a system of right coset representatives of P in G, it follows from (4) that there is c ∈ A such that

∑_{t∈T} h(tg)^{−1}(tgα) ≡ c · ∑_{t∈T} h(t)^{−1}(tα) mod I_{R}(A)I_{R}(G).

Putting a(g) = bc, the result follows.

(6) There are σ, τ ∈ Aut(G) such that for β = τ^{−1}α the following hold:

(i) P β = P; (ii) nβ = n for all n ∈ N; (iii) g^{−1}(gβ) ∈ C_{V(RG)}(N) for all g ∈ G; (iv) x^{−1}(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^{−1}β agrees with conjugation with a · u(S, β) on s_{R}(G, A).

Proof. Keep the notation 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^{−1}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^{−1} · u(T, β) · gβ ≡ b(g) · u(T, β) mod I_{R}(A)I_{R}(G). Again, it follows that an automorphism σ of G is defined by gσ = g · b(g) for all g ∈ G, and that σ^{−1}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^{−1}} ∈ P } ≤ P is normalized by g^{−1}(gβ), and there is y ∈ A with [y · g^{−1}(gβ), U_{g}] = 1.

Proof. Let x ∈ U_{g}. By (6.iv) and (6.iii), there is a ∈ A such that x^{g^{−1}} = (x^{g^{−1}}β)a, and x^{g^{−1}(gβ)} = (xβ)a^{gβ} = x · x^{−1}(xβ) · a^{g} ∈ xA. Hence g^{−1}(gβ) normalizes U_{g}, and by the Coleman Lemma, there is y ∈ G such that [y · g^{−1}(gβ), U_{g}] = 1. Since N ≤ U_{g}, it follows from (6.iii) that y ∈ C_{G}(N) = A.

Some notation needs to be fixed.

• Choose 1 ∈ W ⊆ G such that G = ⋃_{w∈W} P wP (disjoint union).

• For all w ∈ W, choose 1 ∈ X_{w} ⊆ P such that P wP = ⋃_{x∈X_{w}} P wx (disjoint union).

• For all w ∈ W, choose a_{w} ∈ A with [a_{w} · w^{−1}(wβ), U_{w}] = 1, cf. (7).

• Put a_{w,x} = x^{−1}a_{w}x(x^{−1}β)x and s_{w,x} = a_{w,x} · (wx)^{−1}(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,1} = a_{w} · w^{−1}(wβ).

(8) For any w ∈ W, P operates on {s_{w,x} : x ∈ X_{w}} transitively via conjugation.

Proof. Let w ∈ W and g ∈ P. Then there is x ∈ X_{w} with P wg = P wx. Observe that gx^{−1} ∈ U_{w}, so

g^{−1} · s_{w,1} · g = g^{−1} · (a_{w} · w^{−1}(wβ)) · g  by definition of a_{w}
= x^{−1} · (a_{w} · w^{−1}(wβ)) · x
= x^{−1} · a_{w} · x · (wx)^{−1}(wxβ) · (xβ)^{−1}x
= x^{−1} · a_{w} · x · (xβ)^{−1}x · (wx)^{−1}(wxβ) by (6.iii) and (6.iv)
= s_{w,x}.

In particular, x^{−1} · s_{w,1} · x = s_{w,x} for all x ∈ X_{w}, so P operates on {s_{w,x} : x ∈ X_{w}} transitively.

(9) There is a ∈ A such that xσ^{−1}β = x^{a} for all x ∈ P.

Proof. Put

u = \frac{1}{|G : P |} ∑_{w∈W} ∑_{x∈X_{w}} s_{w,x} = \frac{1}{|G : P |} ∑_{w∈W} ∑_{x∈X_{w}} a_{w,x} · (wx)^{−1}(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^{−1}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^{−1}β ≡ x^{c·u(S,β)} ≡ x^{cbu} ≡ x^{cb} mod I_{R}(A)I_{R}(G).

Put a = cb ∈ A. Since P σ^{−1}β = P, it follows that xσ^{−1}β = x^{a} for all x ∈ P.

With a as in (8), and γ ∈ Aut(G) being conjugation with aα^{−1} ∈ G, it follows that xα = xγτσ for all x ∈ P. Therefore, the proof of the theorem is completed by the following observation.

(10) Assume that α fixes the elements of the Sylow p-subgroup P element-wise. Then α is an inner automorphism.

Proof. Let T be a set of right coset representatives for P in G, and put

u = \frac{1}{|G : P |} ∑_{t∈T} t^{−1}(tα).

Note that the definition of u does not depend on the particular choice of T, so g^{−1}u(gα) = u for all g ∈ G, and u is a unit by the proposition. Hence α is conjugation with u.

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Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany

E-mail address: hertweck@mathematik.uni-stuttgart.de