Units of p-power order in principal blocks of p-constrained groups

Martin Hertweck^{*}

Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany E-mail: hertweck@mathematik.uni-stuttgart.de

Let G be a finite group which has a normal p-subgroup N with C_{G}(N) ≤ N, and let R be a p-adic ring. Let V(RG) denote the group of units in RG of augmentation 1. It is shown that any finite p-group in V(RG) which normalizes N is conjugate to a subgroup of G in the units of RG, and that any finite pgroup in V(RG) which centralizes N is contained in N.

1. INTRODUCTION

This paper is motivated by the attempt to understand more fully parts of a theorem of K. W. Roggenkamp and L. L. Scott (cf. [9, Theorem 6], [10, Theorem 19], [16, Theorem], [17]), and, more generally, certain finite p-subgroups in the unit group of a p-adic group ring.

Let G be a finite group, and let S be a G-adapted ring, that is, an integral domain of characteristic 0 in which no prime divisor of the order of G is invertible. There are many interesting questions dealing with the unit group of the group ring SG; some of them concern the finite subgroups. Then, it is obviously sufficient to consider the normalized unit group V(SG), which consists of the units of SG with augmentation 1. We list some known results, which are due to A. I. Saksonov ([15]; see also [7, ch. 3.2]). Recall that a unit u of SG is said to be trivial if u = sg for some s ∈ S, g ∈ G, and that the trace of u is its 1-coefficient.

(1) a non-trivial unit of SG of finite order has trace zero;

(2) the order of a finite subgroup of V(SG) divides the order of G;

(3) a central unit of finite order is trivial.

* The research was supported by the Deutsche Forschungsgemeinschaft Date: August 14, 2000

One might ask what happens if S is replaced by the ring ℤ_{p} of p-adic integers, for some prime p, and attention is turned to finite p-subgroups of V(ℤ_{p}G). However, it is easily seen that (1) and (2) does not hold in this situation, even not if RG consists of a single block only (cf. [12, XIV]).

Thus, some additional conditions have to be imposed, and then one might also ask about the way a p-subgroup P is embedded in V(ℤ_{p}G). One possibility is to assume that P is a subgroup of a group basis H of ℤ_{p}G, i.e. a subgroup H ≤ V(ℤ_{p}G) of the same order as G with ℤ_{p}G = ℤ_{p}H.

If we concentrate on the principal block B of ℤ_{p}G, and assume that P is the projection of a Sylow p-subgroup of a group basis on B, then the question whether P is conjugate by a unit of B to the projection of a Sylow p-subgroup of G on B has become known as the “defect group (conjugacy) question” [17]. A first positive answer for p-groups was given by Roggenkamp and Scott in their fundamental paper [13].

The aim of this paper is to prove the following two theorems. Let N be a normal p-subgroup of G with C_{G}(N) ≤ N, and 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.) Note that the hypothesis on G is that G is p-constrained with O_{p′}(G) = 1 (see [5, VII 13.3]), and that RG consists of a single (principal) block only (see [5, VII 13.5]).

Theorem A. Any finite p-group in V(RG) which normalizes N is conjugate to a subgroup of G in the units of RG.

Theorem B. Any finite p-group in V(RG) which centralizes N is contained in N.

The proof is somewhat complicated by the fact that we do not know in advance that RG is free for the “multiplication action” of the finite p-group under consideration (taking this for granted, Theorem A should be part of the theorem of Roggenkamp and Scott mentioned at the beginning). The p = 2 case deserves special attention (cf. also [3]), and Section 2 contains some preparatory results. Then, Theorems A and B are proved in Section 3. The bimodule arguments used there are inspired by [9, p. 231]. The proof depends heavily on the strong result of A. Weiss on p-permutation lattices ([19]; generalized to arbitrary R by Roggenkamp [11]). The following lemma makes its application possible. (It is an extension of a result of D. B. Coleman. In its present form, it is due to Saksonov [15, 1.14]; see also [6, 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).

UNITS OF P -POWER ORDER IN PRINCIPAL BLOCKS 3

A special case of Theorem B deserves to be mentioned explicitely:

Corollary. If G is p-constrained with O_{p′}(G) = 1, then any central unit of (finite) p-power order in V(RG) is trivial, i.e. contained in Z(G).

It is not known whether a corresponding result holds for the principal block of an arbitrary group. Nevertheless, it would be desirable to have more knowledge about. (This might have structural consequences for finite groups, through G. R. Robinson’s work [8] on odd analogs of the “Z^{∗}- Theorem”.)

There is some hope that the corollary also contributes to the case of an arbitrary group G. For example, if G has abelian p-subgroups, then M. Broué’s “perfect isometry conjecture” [1], which has already been checked on a long series of cases, would imply that central units of p-power order in the principal block of G are trivial.

2. MODULARLY CONJUGATE INVOLUTIONS

This section contains some results which are crucial for the proof of the theorems in the p = 2 case.

Let R be a commutative ring, and G a finite group. For a subgroup H of G, we denote by I_{R}(H) the augmentation ideal of RH, i.e. the R-span of the elements h − 1 (h ∈ H). If H is a normal subgroup, I_{R}(H)G denotes the kernel of the natural map RG → RG/H.

Motivated by Coleman’s Lemma, we take a closer look at C_{V(RG)}(N), for G and N as in Theorems A and B.

Proposition 2.1. Let N be a normal p-subgroup of G with C_{G}(N) ≤ N, and R a commutative ring. Then the following holds.

(i) C_{RG}(N) ⊆ R + I_{R}(N)G + pRG;

(ii) If p = 2 and T ⊆ G is chosen such that \overline{T} is the set of involutions in \overline{G} = G/O_{2}(G), then C_{RG}(N) ⊆ R + I_{R}(O_{2}(G))G + 4RG + 2R[T].

Proof. Note that an R-basis of C_{RG}(N) is given by the ‘N-class sums’ K_{N}(g) = ∑_{n∈C_{N} (g)\N} g^{n}, where g ranges over the elements of G. If C_{N}(g) = N, then K_{N}(g) = g ∈ N since C_{G}(N) ≤ N, and if |N : C_{N}(g)| = p^{a} > 1, then K_{N}(g) ∈ I_{R}(N)G + p^{a}RG. Hence (i) holds. The group G acts via conjugation on the quotient N^{∗} = N/Φ(N) of N by its Frattini subgroup Φ(N). Since N is not centralized by p′-elements of G\{1}, the kernel K of this operation is a p-group [4, III 3.18], whence contained in O_{p}(G). Now let p = 2 and g ∈ G with |N : C_{N}(g)| = 2. Then |N^{∗} : C_{N^{∗}}(g)| ≤ 2, so g^{2} ∈ K ≤ O_{2}(G) and K_{N}(g) ∈ R+I_{R}(O_{2}(G))G+2R[T], which completes the proof of (ii).

In particular, we are led to consider units u of order p in some group ring RG which map to 1 in (R/pR)G. The following lemma, which is a special case of [20, Lemma (53.3)], shows that this is actually only a matter for p = 2.

Lemma 2.2. Let p be an odd prime, and assume that ⋂_{i=1}^{∞} p^{i}R = 0 and that p is no zero divisor in R. Then the group of units in 1 + pRG has no p-torsion.

Proof. Suppose that u ∈ 1 + pRG is a unit of order p, and write u = 1+p^{a}x with a ∈ ℕ, x ∈ RG\pRG. Writing out the binomial expansion for (1+p^{a}x)^{p} = 1 and solving for x, we get p·p^{a}x = − ∑_{n=2}^{p} (\begin{matrix}p \\ n \end{matrix})p^{an}x^{n}. But the right hand side has a factor p·p^{2a} since p > 2. This contradiction proves the lemma.

The reader might wish to convince himself that this is no longer true if p = 2, even if u is assumed to be normalized.

For the rest of this section, let R be an integral domain of characteristic 0, with quotient field K. As usual, [RG, RG] denotes the additive commutator of RG, defined by

[RG, RG] = _{R}⟨g^{h} − g : g, h ∈ G⟩ = {xy − yx : x, y ∈ RG}.

Let p be a rational prime. The following formulas are well-known (see [18, Lemma (7.1)]). Let a_{1}, …, a_{l} ∈ RG and n ∈ ℕ.

[KG, KG] ∩ RG = [RG, RG]  (1a)
[RG, RG]^{p} ⊆ pRG + [RG, RG]  (1b)
(a_{1} + ⋯ + a_{l})^{p^{n}} ≡ a_{1}^{p^{n}} + ⋯ + a_{l}^{p^{n}} mod pRG + [RG, RG] (1c)

For any subset T of G, let R[T] denote the R-span of the elements of T. Let G_{p′} denote the set of the p′-elements of G. It follows immediately from (1b) and (1c) that for any x ∈ RG and large enough n ∈ ℕ (p^{n} ≥ |G|_{p} suffices),

x^{p^{n}} ∈ pRG + [RG, RG] + R[G_{p′}]. (1d)

Fix some c ∈ G, and put C = ⟨c⟩. We define

[RG]^{1−c} = _{R}⟨g − g^{c} : g ∈ G⟩ = {x − x^{c} : x ∈ RG}.

If RG is considered as a C-module for the “conjugation action” m · c =

c^{−1}mc (m ∈ RG), then [RG]^{1−c} = im(RG ⟶^{1−c} RG). Since H^{1}(C, RG) = 0

(see [13, (1.4.1) Proposition]), also [RG]^{1−c} = ker(RG \underrightarrow{Ĉ} RG), with Ĉ the sum of the elements in C. In particular, it follows that

[KG]^{1−c} ∩ RG = [RG]^{1−c}. (2)

We remark that [RG]^{1−c} is invariant under left (and right) multiplication with c. Also, if c^{2} = 1, then ([RG]^{c−1})^{2} ⊆ 2RG + [RG]^{c−1}, the analog to (1b), holds since (g−g^{c})(h−h^{c}) = gh+(gh)^{c} −g^{c}h−(g^{c}h)^{c} for all g, h ∈ G.

Lemma 2.3. Let c ∈ G and u ∈ V(RG) be 2-elements. Assume that c^{2} = u^{2} and c^{−1}u = 1 + 2f for some f ∈ RG. Then

(i) f + f^{2} ∈ 2RG + [RG]^{1−c};

(ii) f + f^{2^{m}} ∈ 2RG + [RG, RG] for all m ∈ ℕ;

(iii) f ∈ 2RG + [RG, RG] + R[G_{2′}].

Proof. Squaring the equation c^{−1}u = 1 + 2f yields 1 + 4(f + f^{2}) = (c^{−1}u)^{2}, so 4(f + f^{2})u^{−1}c = c^{−1}u − u^{−1}c. Now c^{−1}u = cu^{−1} since c^{2} = u^{2}, and it follows that 4(f +f^{2})u^{−1}c = cu^{−1} −(cu^{−1})^{c} ∈ [RG]^{1−c}. By (2), and since c^{−1}u ∈ 1+2RG, (i) follows. Now (ii) follows inductively from (1b) and f +f^{2^{m}} = (f +f^{2})+(f +f^{2^{m−1}})^{2} −2(f^{2} +f^{2^{m−1}+1}). Part (iii) follows from (ii) and (1d).

Let T_{2} ⊆ G be the set of involutions of G.

Lemma 2.4. If x ∈ R[T_{2}] ∩ I_{R}(G), then x^{2} ∈ (R[G \ T_{2}] ∩ [RG, RG]) + 2RG.

Proof. Write x = ∑_{s∈T_{2}} r_{s}(s − 1), with r_{s} ∈ R for all s ∈ T_{2}. Then

x^{2} = ∑_{s∈T_{2}} r_{s}^{2}(s − 1)^{2} + ∑_{\begin{matrix}\{s, t\} ⊆ T_{2} \\ s ≠ t \end{matrix}} r_{s}r_{t}((s − 1)(t − 1) + (t − 1)(s − 1)).

Let s, t ∈ T_{2}. Then (s − 1)^{2} = 2(1 − s) ∈ 2RG and

(s − 1)(t − 1) + (t − 1)(s − 1) = 2(1 − s − t + st) − st + ts

∈ \{ \begin{matrix}2RG & if [s, t] = 1, \\ 2RG + (R[G \ T_{2}] ∩ [RG, RG]) & otherwise. \end{matrix}

This proves the lemma.

Proposition 2.5. Let R be a 2-adic ring, and let c ∈ G and u ∈ V(RG) be 2-elements with c^{2} = u^{2} and c^{−1}u = 1+2f for some f ∈ 2RG+R[T_{2}]+R. Then c and u are conjugate in the units of RG.

Proof. From Lemma 2.3(iii) we know that f ∈ 2RG+[RG, RG]+R[G_{2′}]. Compared with f ∈ 2RG + R[T_{2}] + R, f ∈ 2RG + (R[T_{2}] ∩ [RG, RG]) + R follows. Taking augmentation shows that f ∈ 2RG + (R[T_{2}] ∩ [RG, RG]). Hence f^{2} ∈ 2RG+(R[G\T_{2}]∩[RG, RG]) by Lemma 2.4. By Lemma 2.3(i), f + f^{2} ∈ 2RG + [RG]^{1−c}. The last three equations show that

f ∈ 2RG + [RG]^{1−c}. (∗)

Let C = ⟨x⟩ be a cyclic group whose order is the maximum of the orders of c and u. Consider RG as an RC-lattice, the action of x given by m · x = c^{−1}mu for all m ∈ RG. A 1-cocycle β : C → RG is defined by β(x) = f = (c^{−1}u − 1)/2. Notice that c and u are conjugate in the units of RG if β is a 1-coboundary (if β(x) = m − c^{−1}mu for some m ∈ RG, then v = 1 + 2m is a unit in RG by [2, 5.10], and u = c^{v}). The exact sequence 0 ⟶ RG ⟶^{2} RG ⟶^{−} \overline{RG} ⟶ 0 has cohomology exact sequence

… ⟶ H^{1}(C, RG) ⟶^{2} H^{1}(C, RG) ⟶^{−} H^{1}(C, \overline{RG}) ⟶ …

Now we are in a position to apply Weiss’ results on permutation modules. By [20, Theorem (50.2)], RG is a monomial lattice for C over R, so 2 annihilates H^{1}(C, RG) by [20, Lemma (53.1)]. Hence the cohomology exact sequence implies that H^{1}(C, RG) ⟶^{−} H^{1}(C, \overline{RG}) is injective. By (∗), the class of β maps to zero under this map; hence β is a 1-coboundary, and c and u are conjugate in the units of RG.

3. PROOF OF THE THEOREMS

Throughout this section, the following notation is fixed. Let G be a finite group which has a normal p-subgroup N containing its own centralizer, C_{G}(N) ≤ N, and let R be a p-adic ring. Let Q be a (finite) p-subgroup of V(RG) containing N as a normal subgroup. We consider M = RG as R(G × Q)-module, the action given by m · r(g, x) = g^{−1}rmx for all x ∈ Q, g ∈ G, r ∈ R and m ∈ M.

We will need the following indecomposability criteria. It is proved in greater generality in [9, p. 231, Claim 1]. For the convenience of the reader, we include the (short) proof.

Lemma 3.1. As R(G × N)-module, M = RG is indecomposable.

Proof. An endomorphism of M_{G×1} is given by right multiplication with some element of RG, and therefore E = End_{R(G×N)}(M) = C_{RG}(N). Now C_{RG}(N)/rad(C_{RG}(N)) ≅ C_{kG}(N)/rad(C_{kG}(N)) ≅ k; the first isomorphism follows from [2, (5.22)], and the second from Proposition 2.1(i) and [2, (5.26)] (I_{k}(N)G is nilpotent). Hence E is local, and therefore M is indecomposable [2, (6.10)].

By Coleman’s Lemma, we may fix (and do) for each x ∈ Q some g_{x} ∈ G such that g_{x}^{−1}x ∈ C_{V(RG)}(N). Put F = O_{p}(G) (the Fitting subgroup of G), and let − denote the natural map RG → RG/F (so that \overline{G} = G/F and \overline{R} = R). Let U = ⟨g_{x} : x ∈ Q⟩. Since C_{G}(N) ≤ N, it follows that x ↦ g_{x}N defines a homomorphism Q → UN/N.

Lemma 3.2. Let x ∈ Q with g_{x} ∈ F. Then \overline{x} = 1.

Proof. Let x ∈ Q. Since x is augmented, \overline{x} ∈ \overline{g_{x}} + p\overline{RG} by Proposition 2.1(i). Now assume that g_{x} ∈ F, and, by the way of contradiction, that \overline{x} has order p. Then p = 2 by Lemma 2.2, and by Proposition 2.1(ii), \overline{x} = \overline{g_{x}^{−1}x} ∈ R + 4R\overline{G} + 2R[\overline{T}], where \overline{T} is the set of involutions in \overline{G}. Hence we may apply Proposition 2.5 (with c = 1 and u = \overline{x}) to conclude that \overline{x} = 1, a contradiction, and the lemma is proved.

The next lemma is the place where we use Weiss’ theorem. Recall that we write Ĥ for the sum of the elements of a group H.

Lemma 3.3. M_{P ×Q} is a permutation lattice for P × Q over R.

Proof. Consider the module F̂M_{1×Q}, which is isomorphic to RG/F = R\overline{G}, with Q acting via Q → QF/F by right multiplication. Notice that for x ∈ Q, \overline{x} ∈ \overline{g_{x}} + p\overline{RG} by Proposition 2.1(i).

Write \overline{G} as a disjoint union \overline{G} = ⋃_{i} \overline{h_{i}} \overline{U}, with h_{i} ∈ G, and let B = ⋃_{i}\{\overline{h_{i}} \overline{x} : x ∈ Q\}. It follows from Lemma 3.2 that \overline{x} ↦ \overline{g_{x}} defines an isomorphism \overline{Q} → \overline{U}, so |B| = |\overline{G}|. Thus B is an R-basis of R\overline{G} by Nakayama’s Lemma, since B reduced modulo pR\overline{G} is the canonical basis \overline{G} of (R/pR)\overline{G}. Since B is permuted by the action of Q, it follows that F̂M_{1×Q} is a permutation lattice.

Next, we consider the module V = P̂M_{1×Q}. Write G as a disjoint union G = ⋃_{i} P k_{i}.

Assume that some x ∈ Q acts trivially on V/pV; we shall show that x already acts trivially on V. Since x acts on V/pV by right multiplication with g_{x}, it follows that P̂k_{i}g_{x} = P̂k_{i} (all i), and that the normal subgroup ⟨g_{x}^{g} : g ∈ G⟩ is contained in P. Thus g_{x} ∈ F, so x operates trivially on F̂M by Lemma 3.2, and therefore also on V.

Let P k_{1}, …, P k_{l} be orbit representatives of the action of U on the set of all cosets P k_{i}. As above, it follows that {P̂k_{j}x : 1 ≤ j ≤ l, x ∈ Q} is a basis of V which is permuted by Q.

The lemma now follows from Weiss’ theorem [20, Theorem (50.1)].

Now we are in a position to prove Theorem B. We shall make use of the elementary theory of vertices and sources, the Krull-Schmidt Theorem and the Mackey decomposition. As a general reference we give [2].

Theorem 3.4. If c ∈ V(RG) is of (finite) p-power order and [N, c] = 1, then c ∈ N.

Proof. Let C = ⟨c⟩, and consider M = RG as R(G × N × C)-module, the action given by m · (g, x, y) = g^{−1}mxy for all (g, x, y) ∈ G × N × C.

Notice that C acts trivially on V = N̂M. Indeed, V is a permutation module for C by Lemma 3.3, and C acts trivially on V/pV by Proposition 2.1(i).

It follows from Lemma 3.3 that

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

for some subgroups U_{j}. The number of summands equals the R-rank of the fixed point module M^{P ×N×C}, which coincides with M^{P ×1×1} (= P̂RG). Hence n = |G : P |. Since the modules M_{P ×1×1} and M_{1×N×1} are both free, it follows from (∗) and Mackey decomposition that U_{j} ∩ (P × 1 × 1) = 1 and U_{j} ∩ (1 × N × 1) = 1 for all j. In particular, |U_{j}| ≤ |N| · |C| for all j, and comparing ranks in (∗) gives |U_{j}| = |N| · |C| for all j. Restricting to N × N × C in (∗) gives

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

The number of summands in this decomposition is the R-rank of M^{N×N×C}. Since M^{N×N×C} = M^{N×1×1}, there are |G : N| summands. It follows that U_{j} ≤ N × N × C for all j. Hence U_{j} = {(x, (xα_{j})z_{j}^{−i}, c^{i}) : x ∈ N, i ∈ ℤ} for some z_{j} ∈ Z(N) and α_{j} ∈ Aut(N), and the R(G × N × 1)-module 1↑_{U_{j}}^{G×N×C} ↓_{G×N×1} is isomorphic to 1↑_{U_{j}∩(G×N×1)}^{G×N×1}, that is, to RG, the action given by m · (g, x, 1) = g^{−1}m(xα_{j}^{−1}) for all g ∈ G, x ∈ N and m ∈ RG. By Lemma 3.1, it follows that the modules 1↑_{U_{j}}^{G×N×C}, and M, are indecomposable. Since M is relatively P × N × C-projective, it follows from (∗) that

M | M_{P ×N×C} ↑^{G×N×C} ≅ ⨁_{j=1}^{|G:P |} 1↑_{U_{j}}^{G×N×C},

so M ≅ 1↑_{U_{j}}^{G×N×C} for some j. Since (G × 1 × 1) ∩ U_{j} = 1, this means that there is u ∈ M such that Gu is an R-basis of M, i.e. u is a unit in RG, and u = u · (1, z_{j}^{−1}, c) = uz_{j}^{−1}c. It follows that c = z_{j}.

Corollary 3.5. If x ∈ Q such that x ∈ 1 + I_{R}(N)G, then x ∈ N.

Proof. If x ∈ 1 + I_{R}(N)G, then g_{x} ∈ N by Proposition 2.1(i), and the corollary follows from Theorem 3.4, applied to c = g_{x}^{−1}x.

Lemma 3.6. M_{1×Q} is a free RQ-module.

Proof. By Lemma 3.3, M_{1×Q} is a permutation lattice for Q over R. By Corollary 3.5, the kernel of the natural map Q → V(RG/N) is N. Let V = N̂RG. By Proposition 2.1(i), x ∈ Q acts on V/pV by right multiplication with g_{x}. It follows that |Q| = |UN| (recall that U = ⟨g_{x} : x ∈ Q⟩), and that each orbit of the action of Q on a basis of M_{1×Q} which it permutes has length |Q|.

Finally, we prove Theorem A, following along general lines [9, p. 231].

Theorem 3.7. There is a unit u of RG with Q ≤ P^{u} and N = N^{u}.

Proof. By Lemma 3.3,

M_{P ×Q} ≅ ⨁_{k=1}^{l} 1↑_{V_{k}}^{P ×Q}

for for some subgroups V_{k}. Since the modules M_{P ×1} and M_{1×Q} are both free (the latter by Lemma 3.6), it follows from Mackey decomposition that V_{k} ∩ (P × 1) = 1 and V_{k} ∩ (1 × Q) = 1 for all k. We shall show that the induced modules 1↑_{V_{k}}^{G×Q} are indecomposable. Mackey decomposition gives

M_{P ×N} ≅ ⨁_{k=1}^{l} ⨁_{V_{k}\a/P ×N} 1↑_{V_{k}^{a}∩(P ×N)}^{P ×N}.

If g_{1}, …, g_{|G:P |} is a system of right coset representatives of P in G, then

M_{P ×N} ≅ ⨁_{j=1}^{|G:P |} 1↑_{U_{j}}^{P ×N}, U_{j} = {(n, n^{g_{j}}) : n ∈ N}.

It follows that for every k there is j = j(k) such that without lost of generality, V_{k} ∩ (P × N) = U_{j}. In particular, U_{j} ≤ V_{k}. Putting things together, we see that for each k, there is a subgroup Q_{k} of Q containing N so that V_{k} = {(xβ_{k}, x) : x ∈ Q_{k}} for some injective homomorphism β_{k} : Q_{k} → P. (Notice that nβ_{k} = n^{g} for some g ∈ G and all n ∈ N.) Hence the induced module 1↑_{V_{k}}^{G×Q_{k}} is isomorphic to the R(G×Q_{k})-module RG, the operation given by m · (g, x) = g^{−1}m(xβ_{k}) for all g ∈ G, x ∈ Q_{k} and m ∈ RG, and therefore is indecomposable by Lemma 3.1. Repeated application of Green’s Indecomposability Theorem [2, 19.22] yields that 1↑_{V_{k}}^{G×Q} is indecomposable. Since M_{G×Q} is relatively P × Q-projective, it follows that

M_{G×Q} | M_{P ×Q}↑^{G×Q} ≅ ⨁_{k=1}^{l} 1↑_{V_{k}}^{G×Q}.

The restricted module M_{G×Q} is also indecomposable by Lemma 3.1. Hence M_{G×Q} ≅ 1↑_{V_{k}}^{G×Q} for some k. Comparing ranks shows that Q_{k} = Q. Since V_{k} ∩(G×1) = 1, this means that there is u ∈ M such that Gu is an R-basis of M, i.e. u is a unit in RG, and u = u·(xβ_{k}, x) = (x^{−1}β_{k})ux for all x ∈ Q. In other words, Q ≤ P^{u} and N = N^{u}.

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