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

ON PRINCIPAL BLOCKS OF p-CONSTRAINED GROUPS

MARTIN HERTWECK AND WOLFGANG KIMMERLE

Abstract. A theorem of K. W. Roggenkamp and L. L. Scott [23] shows that for a finite group G with a normal p-subgroup containing its own centralizer, each two group bases of the integral group ring ℤG are conjugate in the units of ℤ_{p}G. Though the theorem presents itself in the work of others and appears to be needed (cf. [10, 12, 13, 27, 28]), there is no published account. There seems to be a flaw in the proof, because a ‘theorem’ appearing in the surveys [16, 17], where the main ingredients to a proof are given, is false. In this paper, it is shown how to close this gap, at least if one is only interested in the conclusion mentioned above. Therefore, some consequences of the results of A. Weiss on permutation modules [30, 31] are stated. The basic steps of which any proof should consist of are discussed in some detail. In doing so, a complete, yet short proof of the theorem is given in the case that G has a normal Sylow p-subgroup.

1. Introduction

This paper is a substantially extended version of the note [7]. Let G be a finite group, and denote its integral group ring by ℤG. There are many interesting questions dealing with the unit group U(ℤG) of ℤG. Then, it is obviously sufficient to consider the normalized unit group V(ℤG), which consists of the units of ℤG with augmentation 1. Some of the questions concern the finite subgroups of V(ℤG). For example, is any finite subgroup H of V(ℤG) isomorphic to a subgroup of G? If H is a group basis of ℤG, which means that ℤG = ℤH as augmented algebras, this question has become known as the “isomorphism problem for integral group rings” (a negative answer to this problem has been given recently in [5, 6]). Over the last twenty years, the following conjecture of H. Zassenhaus has been in the centre of research. (ZC) For any group basis H of ℤG there is a central automorphism α of ℤG with

H = Gα. A survey about (ZC) and related questions is given in [11]. By definition, a central automorphism fixes the center element-wise, so with α as above, G and Gα are rationally conjugate, that is, conjugated in the units of the rational group ring ℚG, by the Noether-Skolem theorem. In particular, (ZC) does not only imply a positive answer to the isomorphism problem, it also makes a strong statement about the embedding of group bases in ℤG. Though it is known for some time that (ZC) is not true in general, it has given a conceptual point of view to many problems related to the integral group ring. The conjecture is closely connected with the names of K. W. Roggenkamp and L. L. Scott, who gave the first counter-examples [24] and proved in the fundamental paper [22] that for a finite p-group G, each two group bases of ℤ_{p}G are p-adically conjugate. (At this place one should remark that all definitions made above also make sense when the coefficient ring ℤ is replaced by an arbitrary commutative ring.)

Date: July 13, 2000.

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

The research was supported in part by the Deutsche Forschungsgemeinschaft.

This paper deals with another theorem due to Roggenkamp and Scott (see [23, 27], [21, VI.1]), henceforth called the “F^{∗}-Theorem”. In order to state it, let R be a p-adic ring, that is, the integral closure of the p-adic integers ℤ_{p} in a finite extension field of the p-adic field ℚ_{p}. (Then R is a complete discrete valuation ring.)

If N is a normal subgroup of G, then I_{R}(N)G denotes the kernel of the natural map RG → RG/N.

F^{∗}-Theorem. Let G be a finite group with a normal p-subgroup N containing its centralizer, C_{G}(N) ⊆ N. Then for any augmented automorphism α of RG which stabilizes I_{R}(N)G, the groups Gα and G are conjugate in the units of RG.

It should be remarked that Roggenkamp and Scott have stated this theorem only for unramified extensions R of ℤ_{p} (however, see [17, (4.2) Remarks] and [27, p. 266]).

The hypothesis on G is that G is p-constrained with O_{p′}(G) = 1, or, equivalently, that the generalized Fitting subgroup F^{∗}(G) equals O_{p}(G) (whence the name of the theorem).

Though the theorem presents itself in the work of others and appears to be needed (cf. [10, 12, 13, 27, 28]), there is no published account. Some ingredients to the proof are given in [15–19]. There seems to be a flaw in the proof, because a ‘theorem’ appearing in the surveys [16,17] is false, see Section 4, where it is shown how the use of it can be avoided if one is only interested in automorphisms of ℤG. Also a complete, yet short proof of the F^{∗}-Theorem is given in the case that G has a normal Sylow p-subgroup (Theorem 4.6; then, the assumption on the stabilization of I_{R}(N)G becomes superfluous).

It will become clear that the strong results of A. Weiss on permutation modules [30, 31] constitute the basis for any proof of the F^{∗}-Theorem. In Section 3, some consequences of these results are recorded, which fit well in the given context. Section 2 contains some preparatory results.

It should be remarked that it is not known whether the stabilization of the ideal I_{R}(N)G is automatic for N = O_{p}(G). This seems to be a fundamental question in representation theory. This problem does not occur in the ‘global’ situation, when a “class sum correspondence” is available (see [26], [21, IV]). More precisely, let S be a G-adapted ring, that is, an integral domain of characteristic 0 in which no prime divisor of |G| is invertible (a basic example is ℤ_{π(G)}, the intersection of the localizations ℤ_{(p)} with p in π(G), the set of prime divisors of |G|.) Then any augmented automorphism α of SG automatically stabilizes I_{S}(O_{p}(G))G. Since questions about isomorphisms tend to reduce to questions about automorphisms (see [9, 5.3], [22, p. 597]), the following theorem, which includes a positive answer to the isomorphism problem for G, is a consequence of the F^{∗}-Theorem.

ON PRINCIPAL BLOCKS OF p-CONSTRAINED GROUPS 3

Theorem 1.1. Let G be a finite group with a normal p-subgroup N containing its centralizer, C_{G}(N) ⊆ N. Then (ZC) holds for ℤ_{π(G)}G, that is, any two group bases of ℤ_{π(G)}G are rationally conjugate.

The following theorems emphasize the impact of the F^{∗}-Theorem on the structure of unit groups of integral group rings. As a general reference, we refer to the books [21, 29]. The first theorem has been proved in [13], and depends heavily on Theorem 1.1.

Theorem 1.2. Let G be a finite solvable group and H a group basis of ℤG. Then for all p, Sylow p-subgroups of H and G are conjugate in the units of ℚG.

Sometimes statements about (the automorphisms of) ℤG can be deduced from knowledge about the automorphisms of ℤG/N_{i}, where the 1 ≠ N_{i} ⊴ G are such that G naturally embeds into ∏_{i} G/N_{i}. In inductive proofs, the F^{∗}-Theorem may guaranty the existence of such normal subgroups. This strategy has been successfully applied in [10] to give the following theorem.

Theorem 1.3. The isomorphism problem for integral group rings has a positive solution for finite nilpotent by abelian groups.

In [12], the representation of a (solvable) group G as a subdirect product of various factor groups (as above), together with the corresponding projective limits of group rings, has been systematically studied. However, the resulting theory leads to effective results only if Theorem 1.1 holds.

2. Preparatory Results

In this section, some well-known results are given which are needed for the proof of the F^{∗}-Theorem, along with some additional information. In the first step of its proof, it is shown that the normal p-subgroups N and Nα are conjugate in the units of RG (cf. Lemma 4.1); the following lemma explains to some extent why conjugacy of p-groups is so powerful. It is an extension of a result of D. B. Coleman (see [21, II Lemma 2.1]; in its present form, it is due to A. I. Saksonov [26, 1.14]).

Throughout this section, G denotes a finite group.

Lemma 2.1. Let P be a p-subgroup of G, and R a commutative ring with pR ≠ R. Then N_{V} (P) = N_{G}(P) · C_{V} (P), where V = V(RG).

The following proposition is based on this lemma and an observation of F. Gross [4, Corollary 2.4]. It shows that a “conjugating unit” occurring in the F^{∗}-Theorem is unique up to central units and group elements.

Proposition 2.2. Let N be a normal p-subgroup of G containing its own centralizer, C_{G}(N) ⊆ N. Let R be a commutative ring with pR ≠ R, and let V = V(RG). Then N_{V} (G) = G · C_{V} (G).

Proof. Take any u ∈ N_{V} (G) and define φ ∈ Aut(G) by gφ = g^{u} for all g ∈ G. We have to show that φ ∈ Inn(G). Let S be a Sylow p-subgroup of G. By Lemma 2.1, we may assume that every element of S is fixed by φ. In particular, all elements of N are fixed by φ, so g^{−1}(gφ) ∈ C_{G}(N) = Z(N) =: A for all g ∈ G. Hence φ fixes every element of A and every coset of A, so there is a 1-cocycle δ ∈ Z^{1}(G/A, A), defined by δ(gA) = g^{−1}(gφ) for all g ∈ G. It suffices to show that δ is a 1-coboundary (see [8, I 17.1]). Since A is a p-group, the restriction map H^{1}(G/A, A) → H^{1}(S/A, A) is injective [8, I 16.21], and the image of δ under this map is the zero element since every element of S is fixed by φ.

The following criteria on indecomposability of permutation modules plays a crucial role in the proof of the F^{∗}-Theorem (see Lemma 4.1 below). It is proved in [15, p. 231, Claim 1]. Here, a different proof is given, thereby pointing out an important detail.

Let R be either a complete discrete valuation ring with residue class field of characteristic p > 0, or a field of characteristic p > 0.

Lemma 2.3. Let N be a normal p-subgroup of G with C_{G}(N) ⊆ N. Let α be either a group automorphism of N or an augmented R-algebra automorphism of RG. Consider M = RG as R(G × N)-module, the action given by v · (g, n) = g^{−1}v(nα) for all n ∈ N, g ∈ G and v ∈ M. Then M is absolutely indecomposable.

Proof. Let E = End_{R(G×N)}(M); we show that E/rad(E) ≅ R/rad(R) (cf. [3, §30B]). Let φ ∈ E. Then φ is given by right multiplication with 1φ, and 1φ is centralized by Nα. In this way, we may identify E with C_{RG}(Nα). If α is a group automorphism of N, then it follows from C_{G}(N) ⊆ N that

1φ ∈ RC_{G}(N) + I_{R}(N)G + pRG ⊆ RN + I_{R}(N)G + pRG
⊆ R + rad(RG)

(for the last step, see [3, 5.26]), and if α is an augmented R-algebra automorphism of RG, then we obtain similarly

1φ ∈ RC_{Gα}(Nα) + I_{R}(Nα)Gα + pRG ⊆ R + rad(RG).

If 1φ ∈ rad(RG), then 1φ ∈ rad(C_{RG}(Nα)) (see [3, 5.6, §5B]). This completes the proof.

For later use (Proposition 3.12, Remark 4.4), the observation on the centralizer of N just made is recorded in a separate lemma. Let R be a discrete valuation ring with residue class field of characteristic p > 0, choose π ∈ R with rad(R) = πR, and let t ∈ ℕ such that pR ⊆ π^{t}R.

Lemma 2.4. Let N be a normal p-subgroup of G with C_{G}(N) ⊆ N, and let R, π^{t} as above. Let C_{t} ⊆ RG consist of those x ∈ RG such that xn ≡ nx mod π^{t}RG for all n ∈ N. Then C_{t} is contained in R + I_{R}(N)G + π^{t}RG. In particular, elements of C_{t} of augmentation 1 are units in RG.

Finally, let us recall the following useful criteria of when two finite subgroups of V(RG) are conjugate; the arguments work for any commutative ring R. Let H be a finite group, and Λ = εRG, for some central idempotent ε of RG. We are interested in group homomorphisms α : H → Λ^{×}. Two such homomorphisms α, β are called R-equivalent if there is a unit u ∈ Λ^{×} such that hα = u^{−1} · hβ · u for all h ∈ H. Given α, we turn Λ into a right R(G × H)-module by linear extension of the group action m · (g, h) = g^{−1}m(hα) for all g ∈ G, h ∈ H and m ∈ Λ. This module is denoted by _{1}Λ_{α}.

Lemma 2.5. Group homomorphisms α, β : H → Λ^{×} are R-equivalent if and only if _{1}Λ_{α} ≅ _{1}Λ_{β} as R(G × H)-modules.

Proof. If there is u ∈ Λ^{×} with hβ = u^{−1} · hα · u for all h ∈ H, then right multiplication with u is an isomorphism _{1}Λ_{α} → _{1}Λ_{β} of R(G × H)-modules. Conversely, let θ : _{1}Λ_{α} → _{1}Λ_{β} be an R(G × H)-isomorphism. Then θ is given by multiplication with u = εθ from the right, hα · u = u · hβ for all h ∈ H, and Λu = Λ implies that u is a unit in Λ.

The next lemma shows how results on permutation lattices come into play.

Lemma 2.6. A homomorphism α : H → V(RG) is R-equivalent to a homomorphism H → G (followed by the inclusion G ↪ V(RG)) if and only if _{1}RG_{α} is a permutation lattice for G × H over R.

Proof. Suppose that _{1}RG_{α} is a permutation lattice for G × H over R. Since the restriction of _{1}RG_{α} to G×1 is a transitive permutation lattice, it follows that there is u ∈ RG such that Gu is a R-basis of RG whose elements are permuted under the action of G × H. It follows that u ∈ U(RG) and Hα ≤ G^{u}. The converse is obvious.

3. Some Applications of Weiss’ Results

The paper [30] of A. Weiss contains results which are of fundamental interest for the representation theory of p-group permutation modules. The proof of the F^{∗}-Theorem relies heavily on them, respectively their generalization (to arbitrary R), first given by K. W. Roggenkamp [20], and then by A. Weiss in [31].

In this section, some useful conclusions of these results are recorded which stand in direct relationship to the F^{∗}-Theorem.

Let us recall the following definitions from [31]. Let G be a finite group and R a p-adic ring. An R-representation of G is an RG-module which is free of finite rank as R-module. A monomial lattice for G over R is an RG-module which is isomorphic to a direct sum of modules of the form φ↑_{H}^{G}, for some homomorphism φ from the subgroup H into the group R^{×} of units of R. Here φ↑_{H}^{G} stands for “R” ⊗_{RH} RG with H acting on “R” via φ : H → R^{×}, and if H acts trivial on “R”, 1↑_{H}^{G} is written instead of φ↑_{H}^{G}. A permutation lattice for G over R is an RG-module which is isomorphic to a direct sum of modules of the form 1↑_{H}^{G}.

Throughout this section, ζ denotes a primitive p^{th} root of unity.

For convenience of the reader, we state Weiss’ results as given in [31]. A short proof of the first theorem can be found in [14, Appendix 1].

Theorem 3.1. Let M an R-representation of the finite p-group G. Suppose N is a normal subgroup of G so that:

(a) The N-module M_{N} is a free RN-module;

(b) The fixed-point module M^{N} is a permutation lattice for G/N over R. Then M is a permutation lattice for G over R.

The discussion of the F^{∗}-Theorem in the next section will show that this theorem provides the basic tool for proving conjugacy of Sylow p-subgroups.

Theorem 3.2. Assume that ζ ∈ R, and let M be an R-representation of the finite p-group G. Suppose M/(1 − ζ)M is the reduction mod (1 − ζ) of some monomial lattice of G over R. Then M itself is a monomial lattice of G over R.

Lemma 3.3. Let G be a finite p-group.

(a) If M is a permutation lattice for G over R then H^{1}(G, M) = 0.

(b) If ζ ∈ R and M is a monomial lattice for G over R then 1 − ζ annihilates H^{1}(G, M).

For the rest of this section, G always denotes an arbitrary finite group. We fix π ∈ R with rad(R) = πR and denote by e the ramification index of K/ℚ_{p}, where K is the quotient field of R. Thus pR = π^{e}R. We also fix a p-adic ring S which contains R and ζ.

The following corollaries of Theorem 3.2 and Lemma 3.3 extend their scope considerably. We begin with an easy lemma.

Lemma 3.4. Let t ∈ ℕ. Then π^{t}S ⊆ (1−ζ)S if and only if π^{t(p−1)}R ⊆ pR = π^{e}R, and if one inclusion is proper, then so is the other.

Proof. Since (1 − ζ)^{p−1}/p ∈ S^{×}, it follows that π^{t}S ⊆ (1 − ζ)S if and only if π^{t(p−1)}S ⊆ pS. In that case, π^{t(p−1)}/p is integral, so π^{t(p−1)} = pr for some r ∈ R. If one of the inclusions is proper, then r lies in rad(R). This proves the lemma.

In particular, it follows that if t > e/(p − 1), then the roots of unity of K map injectively into (R/π^{t}R)^{×}, which is also proved in [1, Lemma 4.3].

The following corollary is a consequence of Lemma 3.3, but we like to mention that a proof which avoids this lemma has been given by R. Boltje and B. Külshammer [1, Theorem 4.5(e)].

Corollary 3.5. Let M and N be monomial lattices for G over R, and let t ∈ ℕ. If t > e/(p−1) and M/π^{t}M ≅ N/π^{t}N as R/π^{t}R-representations for G, then M ≅ N.

Proof. The reductions mod π^{t} of the S-representations M′ = S ⊗_{R} M and N′ = S ⊗_{R} N of G are isomorphic.

By [31, Lemma (53.1)(a)], Hom_{S}(M′, N′) is a monomial lattice for G over S, and by Lemma 3.3 and [8, I 16.18], Ext_{SG}^{1}(M′, N′), which is isomorphic to H^{1}(G, Hom_{S}(M′, N′)), is annihilated by 1 − ζ.

Since π^{t}S ⊂ (1 − ζ)S by Lemma 3.4, it follows as in [3, 30.14] (a result due to Maranda), that a monomial lattice for G over S is uniquely determined by its reduction mod π^{t}, and application of the Noether-Deuring Theorem [3, 30.25] concludes the proof.

Using Theorem 3.2, this result can be strengthened to give the following corollary.

Corollary 3.6. Let M be an R-representation of G, and N a monomial lattice for G over R. Suppose that t ∈ ℕ is chosen such that t > e/(p − 1). If M/π^{t}M ≅ N/π^{t}N as R/π^{t}R-representations for G, then M ≅ N.

Proof. The reductions mod π^{t} of the S-representations M′ = S ⊗_{R} M and N′ = S⊗_{R}N of G are isomorphic. Let P be a Sylow p-subgroup of G. Since π^{t}S ⊂ (1−ζ)S by Lemma 3.4, it follows from Theorem 3.2 that with N′_{P}, the restriction of N′ to P, also the restriction M′_{P} is a monomial lattice for G over S. By Lemma 3.3 and [8, I 16.18] it follows that 1 − ζ annihilates H^{1}(G, Hom_{S}(N′, M′)). Again, it follows as in [3, 30.14] that M′ ≅ N′, and therefore M ≅ N by [3, 30.25].

The difference between π^{t}S and (1 − ζ)S is crucial. The example given in [20, p. 343], [21, VI.1.4] doesn’t illustrate this since there, π^{t(p−1)}R ≠ pR for all t. The following example was communicated to the first author by G. Nebe.

Example 3.7. Let R = ℤ_{3}[\sqrt{3}], i = \sqrt{−1}, and ζ = \frac{−1+i\sqrt{3}}{2}, a primitive 3^{rd} root of unity. Then S = R[ζ] = ℤ_{3}[\sqrt{3}, i] is the integral closure of R in the quotient field of R[ζ], rad(R) = πR with π = \sqrt{3} and πS = (1 − ζ)S.

Let A = [ \begin{matrix}1 & \sqrt{3} \\ −\sqrt{3} & −2 \end{matrix} ]. Then A is a unit of order 3 in Mat_{2}(R), with A ≡ 1 mod \sqrt{3}

and T^{−1}AT = [ \begin{matrix}ζ & 0 \\ 0 & ζ^{2} \end{matrix} ], where T = [ \begin{matrix}i & i \\ ζ^{2} & −ζ \end{matrix} ] ∈ GL_{2}(S). So if G = ⟨A⟩ acts naturally on M := R ⊕ R by multiplication, then G acts trivial on M/πM, and S ⊗_{R} M is a monomial lattice for G over S, but M itself is not a monomial lattice.

The following proposition improves Proposition 3.1 of [7]. (Since these notes were written, the work [2] of Cliff and Weiss has come to our attention, where the same argument – counting fixed point dimensions – is used.) Let K be the quotient field of R.

Proposition 3.8. Suppose that M is a monomial lattice for G over R such that M/πM is the reduction mod π of some permutation lattice N for G over R, and K ⊗_{R} M_{P} ≅ K ⊗_{R} N_{P} as KP -modules for some Sylow p-subgroup P of G. Then M ≅ N.

Proof. Let M = ⨁_{i=1}^{n} φ_{i}↑_{H_{i}}^{G}. Suppose that G is a p-group; then M/πM is the reduction of the permutation lattice L = ⨁_{i=1}^{n} 1↑_{H_{i}}^{G}, so L ≅ N (see [3, 81.17]). The fixed point dimension d = dim_{K} (K ⊗_{R} M)^{G} can be computed from the character ξ of M: it is the number of times the trivial character occurs in ξ, which is by Frobenius reciprocity the number of homomorphisms φ_{i} which are trivial. Since K ⊗_{R} M ≅ K ⊗_{R} L, the same argument shows that d = n, whence M ≅ L ≅ N.

Let P be a Sylow p-subgroup of G. We have just proved that M_{P} is a permutation lattice for P over R. Hence M is a direct summand of a permutation lattice for G over R. But M/πM lifts uniquely (up to isomorphism) to a direct summand of a permutation lattice for G over R [3, 81.17], and it follows that M ≅ N.

For t ∈ ℕ, let κ_{t} : RG → (R/π^{t}R)G denote reduction mod π^{t}. The next theorem deals with groups of units in RG which, mod π^{t}, are conjugate to subgroups of G.

Theorem 3.9. Let H be a finite group, and φ, σ : H → U(RG) group homomorphisms with Hσ ⊆ G and φκ_{t} = σκ_{t} for some t ∈ ℕ such that t ≥ e/(p − 1). If the latter is an equality, assume further that φ|_{P} and σ|_{P} are K-equivalent, for some Sylow p-subgroup P of H. Then φ is R-equivalent to σ. More precisely, there is a unit u ∈ 1 + π^{t}RG such that xφ = xσ^{u} for all x ∈ H.

Proof. Let M = _{1}RG_{φ} and N = _{1}RG_{σ} as R-representations of G × H. By assumption, M/π^{t}M is the reduction mod π^{t} of the permutation lattice N. It is shown that M ≅ N. If t > e/(p−1), this follows from Corollary 3.6. So let us assume that φ|_{P} and σ|_{P} are K-equivalent, for some Sylow p-subgroup P of H. By Theorem 3.2 and Lemma 3.4, M′ := S ⊗_{R} M is a monomial lattice for G × H over S. Clearly M′/π^{t}M′ is the reduction mod π^{t} of the permutation lattice N′ := S ⊗_{R} N for G × H over S, and by Lemma 2.5, L ⊗_{S} M′_{P} ≅ L ⊗_{S} N′_{P}, where L is the quotient field of S. Hence M′ ≅ N′ by Proposition 3.8, and M ≅ N by the Noether-Deuring Theorem [3, 30.25]. By Lemma 2.5, it follows that φ and σ are R-equivalent.

Let us make the connection with cohomology explicit. Let V = RG, considered as R-representation of H, the action given by v ·x = (xσ)^{−1}v(xφ) for all x ∈ H and v ∈ V. A 1-coboundary θ ∈ B^{1}(H, V) is given by θ(x) = 1·(1−x) = 1−(xσ)^{−1}(xφ) for all x ∈ H. Since φκ_{t} = σκ_{t}, there is β ∈ Z^{1}(H, V) with θ = π^{t} · β. We already know that V is a permutation lattice for H over R. By Lemma 3.3, and [8, I 16.18], β is a coboundary, so there is v ∈ V with

1 − (xσ)^{−1}(xφ) = π^{t}(v − (xσ)^{−1}v(xφ)) for all x ∈ H.

With u = 1 − π^{t}v, it follows that (xσ)u = u(xφ) for all x ∈ H, and u is a unit in RG since u ∈ 1 + rad(RG). The theorem is proven.

Remark 3.10. A similar statement holds for homomorphisms φ, σ : H → U(εRG) with Hσ ⊆ εG, where ε is a central idempotent of RG. Then, it suffices to consider the case where G → εG is an isomorphism, so that via H →^{σ} εG → G an action of H on RG can be defined. One may consider the R-representations M = _{1}εRG_{φ} ⊕ 1(1 − ε)RG_{σ} and N = _{1}RG_{σ} of G × H, and apply Krull-Schmidt after showing that M ≅ N.

The following example shows that in the situation of Theorem 3.9, some additional assumptions besides t ≥ e/(p − 1) are necessary.

Example 3.11. Let G = ⟨a : a^{2}⟩ × ⟨b : b^{n}⟩ for some odd number n > 1, so G is a cyclic group of order 2n. Then ε = \frac{1}{n} ∑_{i=1}^{n} b^{i} is a central idempotent in ℤ_{2}G, and a unit of order 2 is given by c = (ε − (1 − ε))a = (−1 + 2ε)a.

A normalized automorphism φ of ℤ_{2}G is defined by aφ = c and bφ = b. Though φ induces the identity mod 2, it is not an inner automorphism.

However, note the following remarkable instance.

Proposition 3.12. Assume that G has a normal p-subgroup N with C_{G}(N) ⊆ N. Let H be a finite group, P a Sylow p-subgroup of H, and φ, σ : H → U(RG) group homomorphisms with N ⊆ Hσ ⊆ G and φ|_{P} · κ_{t} = σ|_{P} · κ_{t} for some t ∈ ℕ with t ≥ e/(p − 1). Then φ is R-equivalent to σ.

Proof. Consider the augmentation ideal V = I_{S}(G) as S-representation of H, the action of H given by v · x = (xσ)^{−1}v(xφ) for all x ∈ H and v ∈ V. A 1-cocycle θ ∈ Z^{1}(H, V) is given by θ(x) = 1 − (xσ)^{−1}(xφ) for all x ∈ H.

We shall show that θ is a coboundary. By [8, I 16.18], it suffices to show that θ|_{P} is a coboundary. By assumption, and Theorem 3.2, V_{P} is a monomial lattice for P over S. There is β ∈ Z^{1}(P, V) with θ|_{P} = π^{t} · β. Since π^{t} ∈ (1 − ζ)S, it follows from Lemma 3.3 that θ|_{P} is a coboundary.

Hence there is v ∈ V with

1 − (xσ)^{−1}(xφ) = v − (xσ)^{−1}v(xφ) for all x ∈ H.

With u = 1 − v, it follows that (xσ)u = u(xφ) for all x ∈ H, and u is a unit in RG by Lemma 2.4.

By Lemma 2.5, _{1}SG_{φ} ≅ _{1}SG_{σ}, and application of the Noether-Deuring Theorem concludes the proof.

We finish this section with two corollaries concerning automorphisms of group rings.

Corollary 3.13. Let t ∈ ℕ with t > e/(p − 1). Let α be an automorphism of RG which induces the identity on (R/π^{t}R)G. Then α is an inner automorphism of RG. In other words, the homomorphism Picent(RG) → Picent((R/π^{t}R)G) is injective.

Proof. This follows from Theorem 3.9. As to the reformulation, see [3, remark after (55.23)].

Corollary 3.14. Let A = ℤ_{π(G)}, and α an automorphism of AG which induces the identity on ℤ/pℤ G, for some prime p ∈ π(G). Then α is given by conjugation with a unit from ℤ_{(p)}G.

Proof. Since A is G-adapted, the class sum correspondence is available (see [21, IV]). Since α fixes all class sums of G, it is given by conjugation with a rational unit (for this kind of reasoning, see [29, 37.6, 41.4]). It follows from Theorem 3.9 that α is given by conjugation with a unit from ℤ_{p}G, and hence also with a unit from ℤ_{(p)}G (see [29, 41.18]).

4. To the proof of the F^{∗}-Theorem

The intention of this section is to give at least an idea of how a proof of the F^{∗}-Theorem should work. Emphasis is put on the additional difficulties that arise when p = 2. Some ingredients to a proof are given in [15–17]. Additionally in this section, a short proof for Proposition 4.3 below is presented, which enables one to avoid [17, (4.8) Lemma]. Also, Lemma 4.2 below is proved, so that a complete, yet short proof of [15, Theorem 6] (the case of a normal Sylow p-subgroup) can be given.

Let RG, N and α satisfy the assumptions of the F^{∗}-Theorem. Essentially, any proof of it should consist of the following steps.

Step 1: (a) Modify α by an inner automorphism such that Nα = N; (b) Modify

α by an inner automorphism such that P α = P. Step 2: Modify α by a group automorphism of G such that α fixes P element-

wise. Step 3: Show that α is an inner automorphism.

There might be some overlap between the first two steps, for example if the proof involves an induction on the group order, using calculations in the ‘small group ring’.

4.1. To Step 1(a). This step has been sketched in [15, p. 231]. For convenience of the reader, the missing details are given below.

As before, M = RG is considered as R(G×G)-module, the action of G×G given by m·(x, y) = x^{−1}m(yα) for all (x, y) ∈ G×G. The proof of the next lemma makes use of the elementary theory of vertices and sources, the Krull-Schmidt Theorem and the Mackey decomposition. As general reference we give [3].

Lemma 4.1. Assume that α fixes I_{R}(N)G. Then the groups N and Nα are conjugate in the units of RG.

Proof. The fixed point module M^{N×1} is the annihilator in RG of I := I_{R}(N)G, and the fixed point module M^{1×N} is the annihilator in RG of I_{R}(Nα)Gα = Iα. Therefore M^{N×1} = M^{1×N}, and M^{1×N} is a permutation lattice for P × 1 over R. By Theorem 3.1,

M_{P ×N} ≅ ⨁_{j=1}^{n} 1↑_{U_{j}}^{P ×N}. (∗)

The number of summands equals the R-rank of M^{P ×N}. As M^{P ×N} = M^{P ×1}, it follows that n = |G : P |. Now

M_{P ×1} ≅ ⨁_{j=1}^{|G:P |} ⨁_{U_{j}\a/P ×1} 1↑_{U_{j}^{a}∩(P ×1)}^{P ×1},

and similarly for M_{1×N}. Since both modules are free, it follows that U_{j} ∩(P ×1) = 1 and U_{j} ∩ (1 × N) = 1 for all j. In particular, |U_{j}| ≤ |N| for all j, and comparing ranks in (∗) gives |U_{j}| = |N| for all j. Restricting to N × N in (∗) gives

M_{N×N} ≅ ⨁_{j=1}^{|G:P |} ⨁_{U_{j}(N×N)\a} 1↑_{U_{j}^{a}∩(N×N)}^{N×N}.

The number of summands in this decomposition is the R-rank of M^{N×N}. Since M^{N×N} = M^{N×1}, there are |G : N| summands. It follows that U_{j} ≤ N × N for all j. Hence U_{j} = {(x, xα_{j}) : x ∈ N} for some α_{j} ∈ Aut(N), and the R(G × N)- module 1↑_{U_{j}}^{G×N} is isomorphic to RG, with action m · (g, x) = g^{−1}m(xα_{j}^{−1}) for all g ∈ G, x ∈ N and m ∈ RG. By Lemma 2.3, the modules 1↑_{U_{j}}^{G×N} and M_{G×N} are indecomposable. Since M_{G×N} is relatively P × N-projective, it follows from (∗) that M_{G×N} ≅ 1↑_{U_{j}}^{G×N} for some j. Hence M is a permutation lattice for G×N over R, and Nα is conjugate to a subgroup of G in the units of RG, by Lemma 2.6. More precisely, the given isomorphism tells us that there is u ∈ M so that u · RG = RG and u · U_{j} = u, so u is a unit in RG and u^{−1}xu = (xα_{j})α for x ∈ N, that is, N^{u} = Nα.

Thus we assume from now on that N = Nα.

4.2. To Step 1(b). Aiming for this step, one can prove in the same spirit: if the fixed-point module M^{P ×1} is a permutation lattice for 1 × P over R, then P and P α are conjugate in the units of RG. Also, if M̃ = k ⊗_{R} M is a k(P × P)-permutation module, where k = R/rad(R), it follows in the very same way that P̃ and P̃ α are conjugate in the units of kG. However, the difficulty is to verify the assumption, that is, that M^{P ×1} is a permutation lattice for 1 × P over R, especially if one tries to avoid a complicated induction. (In this context, it should be noted that one might also try to prove the F^{∗}-Theorem under the assumption that α only stabilizes I_{k}(N)G, but the difficulties remain.)

To get a better understanding of the problem, lookup the following diagram, where R̄ = R/pR.

[figure: _{1}RG_{α}; _{1}RG/N_{α}; _{1}R̄G_{α}; _{1}R̄G/N_{α}]

Since N = Nα, Lemma 2.1 and Lemma 2.4 tells us that α induces on R̄G/N a group automorphism of G/N.

If this would imply that _{1}kG_{α} is a k(P × P)-permutation module, so that by the above considerations P̃ and P̃ α are conjugate in the units of kG, then in the e = 1 case, it would follow from Proposition 3.12 that P and P α are conjugate in the units of RG. This might have been the intention of [16, Theorem 26] and [17, Theorem (4.10)], where the following is claimed to be true (R denotes an unramified extension of ℤ_{p}, with residue field k = R/pR, and bars denote reduction mod p):

Suppose that P and Q are p-groups, and let M be an R(P × Q)- module (with the action (x, y) · m = x · m · y^{−1}, x ∈ P, y ∈ Q, m ∈ M), which is free of finite rank as RQ-module. If \overline{M/(M · I_{R}(Q))} is a permutation module for kP, then \overline{M} is a permutation module for k(P × Q).

Note that this would really give new informations only when p = 2. The following example shows, however, that for p = 2 the statement is not true in general.

Let G = ⟨x, y⟩ ≅ C_{2} × C_{2} be Klein’s four group, and let R = ℤ_{2}. An Rrepresentation of G is given by

x = [ \begin{matrix}1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} ] and y = [ \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} ] .

The underlying RG-module M is an absolutely indecomposable module with vertex G. Obviously, M is a free R⟨y⟩-module, and \overline{M/M · I_{R}(⟨y⟩)} = \overline{M}/\overline{M} · (y − 1) is as trivial k⟨x⟩-module of course a permutation module. However, it is easily seen that \overline{M} is not a permutation module for kG.

On the other hand, note that by Theorem 3.1, M^{P ×1} is a permutation lattice for 1 × P over R if _{1}RG/N_{α} is a permutation lattice for P × P over R. If p > 2, this follows from Corollary 3.6.

The automorphism which α induces on RG/N may be written as the product of the group automorphism which α induces on R̄G/N and an automorphism γ. Then, one might even try to prove that γ is an inner automorphism of RG/N. If p > 2, this follows from Theorem 3.9. For p = 2, one has to work harder. However, if a class sum correspondence is available, as is the case when Gα ⊂ ℤ_{π(G)}G, then γ, which induces the identity on R̄G/N, is a central automorphism of RG/N, whence given by conjugation with some unit of KG/N (K is the quotient field of R), and it follows from Theorem 3.9 that γ is an inner automorphism.

4.3. To Step 2. As to this step, suppose that P α = P. Then _{1}RG_{α} is a trivial source module with vertex {(x, xα^{−1}) : x ∈ P }. It would be desirable to have a proof which shows directly that _{1}RG_{α} is in fact a permutation module (so that Lemma 2.6 can be applied), but this seems to be a difficult task. However, the following lemma is easy to prove.

Lemma 4.2. If P α = P, then there is β ∈ Aut(N_{G}(P)) with β|_{P} = α|_{P}.

Proof. By the Schur-Zassenhaus Theorem, there is a complement K to P in N_{G}(P). By Lemma 2.1, there is a map f : K → N_{G}(P) with x^{kα} = x^{f(k)} for all x ∈ P. Let x ∈ P and h, k ∈ K. Then x^{f(h)f(k)} = x^{hα·kα} = x^{(hk)α} = x^{f(hk)}, so

f(h)f(k) ∈ Z(P) · f(hk) (∗)

since C_{G}(P) ⊆ Z(P). Let Q = ⟨Z(P), f(K)⟩. Then Z(P) ⊴ Q and Q/Z(P) ≅ K by the above. Again by Schur-Zassenhaus, there is a complement L to Z(P) in Q. Let f̄: K → L be defined by f̄(k) = g if f(k) = zg with z ∈ Z(P), g ∈ L. By (∗), f̄ is a homomorphism, which is clearly bijective. Define β : N_{G}(P) → N_{G}(P) by (xk)β = xα · f̄(k) for all x ∈ P, k ∈ K. Then β ∈ Aut(N_{G}(P)) as (x^{k})α = xα^{kα} = xα^{f̄(k)} for all x ∈ P, k ∈ K.

4.4. To Step 3. This step is unproblematic. Roggenkamp and Scott proved it using ‘Green correspondence of automorphisms’ (see [17, Lemma (4.8)]), but in case of the principal block, one can argue more directly.

Proposition 4.3. Let G be a finite group, α a normalized automorphism of B, the principal block of RG, and P a Sylow p-subgroup of G. Suppose that α fixes the image of P in B elementwise. Then α is an inner automorphism.

Proof. The ‘untwisted’ R(G × G)-module _{1}B_{1} is a trivial source module with vertex ΔP = {(x, x) : x ∈ P }. By assumption on α, this holds for the twisted R(G × G)-module _{1}B_{α} as well. Put L = N_{G×G}(ΔP), and let V be the RL-lattice corresponding to_{1}B_{α} under the Green correspondence. Since Green correspondents have the same vertices and sources, V is a direct summand of 1↑_{ΔP}^{L}, that is, V is an indecomposable projective module of RL/ΔP. We shall show that V is the projective cover of the trivial module. By Frobenius reciprocity ([3, 10.8]),

Hom_{RL}(V, 1_{L}) ≅ Hom_{R(G×G)}(V ↑_{L}^{G×G}, 1_{G×G}). (∗)

Since _{1}B_{α} and V are Green correspondents, _{1}B_{α} is a direct summand of V ↑_{L}^{G×G}. Moreover, B∩I_{R}(G) is a submodule of_{1}B_{α}, and the quotient is the trivial R(G×G)- module R since α is a normalized automorphism. Hence the right hand side of (∗) is not zero, and V has the trivial module as epimorphic image. It follows that _{1}B_{1} and _{1}B_{α} have isomorphic Green correspondents, so _{1}B_{1} ≅ _{1}B_{α}, and α is an inner automorphism by Lemma 2.5.

The shortest proof of Step 3 makes use of a Maschke argument.

Remark 4.4. If G has a normal p-subgroup N such that C_{G}(N) ⊆ N, then, in the situation of the above proposition, one can also give a Maschke argument.

Let X be a set of (right) coset representatives for P in G, and put

u = \frac{1}{|G : P |} ∑_{x∈X} x^{−1}(xα).

Note that the definition of u does not depend on the particular choice of X, so g^{−1}u(gα) = u for all g ∈ G. Moreover u has augmentation 1 and is contained in C_{RG}(N), so is a unit in RG by Lemma 2.4, and α is conjugation with u.

Remark 4.5. Proposition 4.3 is clearly a result for the principal block, as the following example shows. Let G = (⟨x : x^{3}⟩ × ⟨y : y^{5}⟩) ⋊ ⟨a : a^{2}⟩, the operation given by x^{a} = x^{−1} and y^{a} = y^{−1}. An automorphism α of G is defined by xα = x^{−1}, yα = y and aα = a. Since α does not preserve the conjugacy classes, it induces a noninner automorphism of ℤ_{5}G. But ℤ_{5}G decomposes into two blocks, so α induces a non-inner automorphism of the non-principal block, by the last proposition.

From the above discussion, the last theorem follows.

Theorem 4.6. Let G be a finite group with a normal Sylow p-subgroup P containing its own centralizer, C_{G}(P) ⊆ P. Then for any normalized automorphism α of RG, the groups Gα and G are conjugate in the units of RG.

Proof. The automorphism α stabilizes the induced augmentation ideal I_{R}(P)G since RG/P is the largest homomorphic image of RG, as R-algebra, which is torsion free and whose reduction mod π is semi-simple (cf. [16, p. 377]). So P and P α are conjugate in the units of RG, by Lemma 4.1, and we may assume that P α = P. By Lemma 4.2, there is β ∈ Aut(G) with β|_{P} = α|_{P}, so we may even assume that α|_{P} = id|_{P}. But then α is an inner automorphism by Proposition 4.3.

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

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