The Fitting Submodule

U. Meierfrankenfeld B. Stellmacher

Department of Mathematics, Michigan State University, East Lansing MI 48840

meier@math.msu.edu

Mathematisches Seminar, Christian-Albrechts- Universität, D24098 Kiel stellmacher@math.uni-kiel.de

Abstract

Let H be a finite group, 𝔽 a field and V a finite dimensional 𝔽H-module. We introduce the Fitting submodule F_{V} (H), an 𝔽H submodule of V which has properties similar to the generalized Fitting subgroup of a finite group.

1 Introduction

Throughout this paper 𝔽 is a field of characteristic p, p a prime, H is a finite group, and V is a finite dimensional 𝔽H-module.

We will use the concept of the generalized Fitting subgroup of a finite group as a model for our definition of the Fitting submodule F_{V} (H) of V. In particular, F_{V} (H) will be defined by means of components which in turn resemble components of finite groups.

Our first result can be stated without mentioning the Fitting submodule:

Theorem 1.1 Suppose that V is faithful and O_{p}(H) = 1. Then there exists an 𝔽H-section of V that is faithful and semisimple.

In fact 1.1 is a corollary of 1.3 and 1.4 below, which show that F_{V} (H)/rad_{F_{V} (H)}(H) has the desired properties.

To introduce the concept of a Fitting submodule we need a few basic definitions, some of them inspired by corresponding definitions in finite group theory.

Definition 1.2 (a) H acts nilpotently on V if [W, H] < W for all non-zero 𝔽H-submodules W of V.

(b) C_{H}^{∗}(V) is the largest normal subgroup of H acting nilpotently on V. It is elementary to show that C_{H}^{∗}(V)/C_{H}(V) = O_{p}(H/C_{H}(V)) and that C_{H}^{∗}(V) is the largest subnormal subgroup of H acting nilpotently on V.

(c) V is H-reduced if C_{H}^{∗}(V) = C_{H}(V) (that is if any normal subgroup of H which acts nilpotently on V already centralizes V).

(d) C_{V}^{∗} (H) is largest 𝔽H-submodule of V on which H acts nilpotently (so C_{V}^{∗} (H) = C_{V} (O^{p}(H));

(e) rad_{V} (H) is the intersection of the maximal 𝔽H-submodules of V (so rad_{V} (H) is the smallest 𝔽H-submodule with semisimple quotient).

Mathematics Subject Classification (2000) 20C20

(f) Let W be an 𝔽H submodule of V and N ⊴ H. Then W is N-quasisimple if W is H-reduced, W/rad_{W} (H) is simple, W = [W, N] and N acts nilpotently on rad_{W} (H). If N = H we often write quasisimple rather than H-quasisimple.

(g) S_{V} (H) is the sum of all simple 𝔽H-submodules of V, and E_{H}(V) := C_{F^{∗}(H)}(S_{V} (H)).

(h) W is a component of V (or an H-component if we want to emphasize the dependence on H) if either W is a simple 𝔽H-submodule with [W, F^{∗}(H)] ≠ 0 or W is an E_{H}(V)-quasisimple 𝔽H-submodule. The Fitting submodule F_{V} (H) of V is the sum of all components of V.

(i) R_{V} (H) := ∑ rad_{W} (H), where the sum runs over all the components of V, and \overline{F_{V} (H)} := F_{V} (H)/R_{V} (H).

Theorem 1.3 The Fitting submodule F_{V} (H) is H-reduced and R_{V} (H) is a semisimple 𝔽F^{∗}(H)- module. Moreover, R_{V} (H) = rad_{F_{V} (H)}(H); in particular F_{V} (H)/R_{V} (H) is semisimple.

Theorem 1.4 Let V be faithful and H-reduced. Then also F_{V} (H) and F_{V} (H)/R_{V} (H) are faithful and H-reduced.

In section 3 we will discuss the relation between the Fitting submodule F_{V} (H) and the Fitting submodule F_{V} (N), where N is a subnormal subgroup of H. Finally, in section 4 the structure of the F^{∗}(H)-components of V is given in the case where 𝔽 is a finite or algebraically closed field.

2 The Fitting Submodule

We will frequently use the following well-known and elementary properties of the generalized Fitting subgroup F^{∗}(H) of H, see for example [1]:

Lemma 2.1 Let E ≤ H.

(a) If E ⊴ H, then F^{∗}(E) = E ∩ F^{∗}(H).

(b) If E = F^{∗}(E), then E = O_{p}(E)O^{p}(E) and [O^{p}(E), O_{p}(E)] = 1.

(c) If E is a product of components of H, then F^{∗}(H) = EC_{F^{∗}(H)}(E).

(d) If E is a component of H and N ⊴ H, then either E ≤ N or [E, N] = 1.

□

Lemma 2.2 Let E and P be normal subgroups of H with [E, P] ≤ C_{H}(V). Suppose that V = [V, E] and V/C_{V}^{∗} (E) is a simple 𝔽H-module. Then the following hold:

(a) rad_{V} (H) = C_{V}^{∗} (E).

(b) V is a semisimple 𝔽P -module, and every simple 𝔽P -submodule of V is isomorphic to a simple 𝔽P -submodule of V/rad_{H}V. In particular, if I_{1} and I_{2} are simple 𝔽P -submodules of V, then there exists h ∈ H such that I_{1} and I_{2}^{h} are isomorphic as 𝔽P -modules.

(c) If E = F^{∗}(E), then C_{H}(V/rad_{V} (H)) = C_{H}(V), V is H-reduced and V is E-quasisimple as an 𝔽H-module.

(d) If E = F^{∗}(E), then E centralizes rad_{V} (H) and rad_{V} (H) is semisimple as an 𝔽EP -module.

(e) Either [V, P] = 0, or C_{V} (P) = 0 and [V, P] = V.

Proof: (a): Let Y be a maximal 𝔽H-submodule of V. If C_{V}^{∗} (E) ≰ Y then V = Y + C_{V}^{∗} (E). So E acts nilpotently on V/Y, contradicting V = [V, E]. Thus C_{V}^{∗} (E) ≤ Y and so C_{V}^{∗} (E) ≤ rad_{V} (H). The other inclusion follows from the simplicity of V/C_{V}^{∗} (E).

(b): Since V/rad_{V} (H) is a simple H-module, Clifford Theory shows that V/rad_{V} (H) is a semisimple 𝔽P -module and that any two simple 𝔽P -submodules of V/rad_{V} H are isomorphic up to conjugacy under H. Let U be the sum of all 𝔽P -submodules of V that are isomorphic to some simple 𝔽P -submodule of V/rad_{V} (H). It remains to show that V = U.

Let x ∈ O^{p}(E). Then

ϕ : V → [V, x] with v ↦ [v, x]

is an 𝔽P -module homomorphism since [E, P] ≤ C_{H}(V), so [V, x] ≅_{𝔽P} V/C_{V} (x). As rad_{V} (H) ≤ C_{V} (x) and V/rad_{V} (H) is a semisimple 𝔽P -module, [V, x] ≤ U. Hence also

V = [V, E] = [V, O^{p}(E)] ≤ U

and thus V = U.

(c): Put C := C_{H}(V/rad_{V} (H)). By (a) C ∩ E acts nilpotently on V/rad_{V} (H) and on rad_{V} (H). Thus O^{p}(C ∩ E) centralizes V. By 2.1(a), C ∩ E = F^{∗}(C ∩ E), and by 2.1(b),

C ∩ E = O_{p}(C ∩ E)O^{p}(C ∩ E) ≤ O_{p}(E)C_{E}(V) and [O_{p}(E), O^{p}(E)] = 1.

Thus [O^{p}(E), C ∩ E] ≤ C_{E}(V). Since

[V, C, O^{p}(E)] ≤ [rad_{V} (H), O^{p}(E)] = 0 and V = [V, O^{p}(E)],

the Three Subgroups Lemma gives C ∩ E ≤ C_{H}(V) and then that C ≤ C_{H}(V). Clearly C_{H}(V) ≤ C and so C = C_{H}(V). Since C_{H}^{∗}(V) acts trivially on every simple 𝔽H-section of V, C_{H}^{∗}(V) ≤ C = C_{H}(V) and so V is H-reduced. Together with (a) we see that V fulfills all the conditions of an E-quasisimple 𝔽H-module.

(d): According to (c), [rad_{V} (H), O^{p}(E)] = 0 and [V, O_{p}(E)] = 0. Hence E centralizes rad_{V} (H) since E = F^{∗}(E) = O_{p}(E)O^{p}(E).

By (b) V and so also rad_{V} (H) is a semisimple 𝔽P -module. Thus rad_{V} (H) is a semisimple 𝔽EP -module.

(e): This is an immediate consequence of (b). □

Lemma 2.3 Let N ⊴ H with N ≤ E_{H}(V) and N ≰ C_{H}^{∗}(V). Let W be an 𝔽H-submodule of V that is minimal with respect to N ≰ C_{H}^{∗}(W). Then

(a) W is a H-component of V.

(b) W = [W, N] and C_{W}^{∗} (N) = rad_{W} (H) = C_{W} (E_{H}(V)) ≠ 0.

(c) There exists a unique normal subgroup M of H minimal with respect to M ≤ E_{H}(V) and [W, M] ≠ 0.

(d) M ≤ N, W = [W, M] and M is a product of components of H transitively permuted by H.

(e) C_{E_{H} (V )}(M) = C_{E_{H} (V )}(W).

Proof: The minimality of W implies that C_{W}^{∗} (N) is the unique maximal H-submodule of W. Thus

1^{∘} C_{W}^{∗} (N) = rad_{W} (H) and W/rad_{W} (H) is a simple 𝔽H-module,

If [W, N] ≤ C_{W}^{∗} (N), then N acts nilpotently on W, a contradiction. Thus [W, N] ≰ C_{W}^{∗} (N) and the minimality of W gives

2^{∘} W = [W, N].

Observe that by 2.1 N = F^{∗}(N). Hence by 2.2(c) applied with N and W in place of E and V, respectively: 3^{∘} W is a H-reduced.

Since [W, N] ≠ 0 and N ≤ E_{H}(V), W is not a simple H-module. Thus 4^{∘} rad_{W} (H) ≠ 0.

Choose M ⊴ H minimal in N with [W, M] ≰ rad_{W} (H). Then 5^{∘} W = [W, M] = [W, O^{p}(M)], and so by the minimality of M and 2.1, M = O^{p}(M) = F^{∗}(M). Hence M is a p′-group or a product of components transitively permuted by H. As a subgroup of N, M acts nilpotently on rad_{W} (H), so 6^{∘} [rad_{W} (H), M] = 0.

Assume that M is a p′-group. Then Maschke’s Theorem implies W = C_{W} (M) ⊕ [W, M] and rad_{W} (M) = C_{W} (M) = 0, which contradicts (4^{∘}). Thus 7^{∘} M is a the product of components of H transitively permuted by H.

By 2.1(c)

(∗)  E_{H}(V) = MC_{E_{H} (V )}(M).

We now apply 2.2(e) with E := M, P := C_{E_{H} (V )}(M) and W in place of V. Since P ≤ E_{H}(V), P centralizes all simple 𝔽H-submodules of W. Thus C_{W} (P) ≠ 0 and 2.2(e) implies [W, P] = 0. Hence by (6^{∘}) 8^{∘} [W, P] = 0, and E_{H}(V) = MP centralizes rad_{W} (H).

Since N ≤ E_{H}(V), (2^{∘}) implies 9^{∘} W = [W, E_{H}(V)].

If M_{1} is any normal subgroup of H with M_{1} ≤ E_{H}(V) and M_{1} ≰ P, then 1 ≠ [M, M_{1}] ≤ M ∩ M_{1}, and (7^{∘}) shows that M = [M, M_{1}] ≤ M_{1}. In particular,

[W, M_{1}] ≠ 0 and so by (8^{∘}) 10^{∘} C_{E_{H} (V )}(W) = P = C_{E_{H} (V )}(M).

Hence M_{1} is an arbitrary normal subgroup of E_{H}(V) not centralizing W. Thus M ≤ M_{1} implies (c). By (1^{∘}),(4^{∘}) (8^{∘}) and (9^{∘}) W is E_{H}(V)-quasisimple and so (a) holds. Moreover, (b) follows from (1^{∘}),(2^{∘}), (3^{∘}) and (8^{∘}), and (d) follows from (5^{∘}) and (7^{∘}). Finally (e) is (10^{∘}).□

Lemma 2.4 (a) \overline{F_{V} (H)} is a semisimple H-module and [F_{V} (H), F^{∗}(H)] = F_{V} (H).

(b) F_{V} (H) is H-reduced.

(c) R_{V} (H) is a semisimple 𝔽F^{∗}(H)-module. Proof: Let W be a component of V. Note that either W ≤ R_{V} (H) or W ∩R_{V} (H) = rad_{W} (H).

(a): Using the above observation, (a) is an immediate consequence of the definition of F_{V} (H). (b): By definition all components are either simple or E_{H}(V)-quasisimple, so they are re-

duced. Clearly sums of reduced modules are reduced and so (b) holds.

(c): Let W be a non-simple component of V. Then by 2.2(d) applied with E := E_{H}(V), P := C_{F^{∗}(H)}(E) and W in place of V, rad_{W} H is a semisimple 𝔽EP -module. By 2.1 F^{∗}(H) = EP and so (c) holds. □

Lemma 2.5 The following hold:

(a) C_{F_{V} (H)}^{∗}(E_{H}(V)) = [S_{V} (H), F^{∗}(H)] + R_{V} (H) = C_{F_{V} (H)}(E_{H}(V))

(b) C_{F^{∗}(H)}^{∗}(V) = C_{F^{∗}(H)}(F_{V} (H)) = O_{p}(H)C_{F^{∗}(H)}(V) ≤ E_{H}(V).

(c) F(H) ∩ E_{H}(V) = C_{F(H)}^{∗}(V) = C_{F(H)}(F_{V} (H)) = O_{p}(H)C_{F(H)}(V).

(d) If V is faithful and H-reduced, then E_{H}(V) is the direct product of perfect simple groups.

Proof:

(a): Let W be a component of V. Then either radW = 0 and W ≤ [S_{V} (H), F^{∗}(H)] or radW ≠ 0, W = [W, E_{H}(V)] and W/rad_{W} (H) is simple. Thus U := F_{V} (H)/[S_{V} (H), F^{∗}(H)] + R_{V} (H) is a sum of simple 𝔽H-module that are not centralized by E_{H}(V). So C_{U}(E_{H}(V)) = 0 and C_{F_{V} (H)}^{∗}(E_{H}(V)) ≤ [S_{V} (H), F^{∗}(H)] + R_{V} (H).

Let W be a component of V with rad_{W} (H) ≠ 0. Then by 2.3(b), applied with N := E_{H}(V), [rad_{V} (H), E_{H}(V)] = 0. The definition of E_{H}(V) shows that [S_{V} (H), E_{H}(V)] = 0 and so

[S_{V} (H), F^{∗}(H)] + R_{V} (H) ≤ C_{F_{V} (H)}(E_{H}(V)).

Clearly C_{F_{V} (H)}(E_{H}(V)) ≤ C_{F_{V} (H)}^{∗}(E_{H}(V)) and so (a) holds.

(b): Since F_{V} (H) is H-reduced by 2.4(b) , C_{F^{∗}(H)}^{∗}(V) ≤ C_{F^{∗}(H)}(F_{V} (H)) =: N. Then N ≤ E_{H}(V) since

S_{V} (H) = (S_{V} (H) ∩ F_{V} (H)) + C_{V} (F^{∗}(H)).

If N does not act nilpotently on V, then 2.3 gives a component W of V with [W, N] ≠ 0, which contradicts [F_{V} (H), N] = 0. Thus N acts nilpotently on V and so O^{p}(N) ≤ C_{N}(V). By 2.1(b) N = O_{p}(N)O^{p}(N) = O_{p}(N)C_{N}(V) ≤ O_{p}(H)C_{F^{∗}(H)}(V).

Clearly O_{p}(H)C_{F^{∗}(H)}(V) ≤ C_{F^{∗}(H)}^{∗}(V) and so (b) holds.

(c): By (b), C_{F(H)}^{∗}(V) = C_{F(H)}(F_{V} (H)) = O_{p}(H)C_{F(H)}(V) ≤ F(H) ∩ E_{H}(V). Suppose F(H) ∩ E_{H}(V) ≰ C_{F(H)}^{∗}(V). Then by 2.3(d), F(H) ∩ E_{H}(V) is not nilpotent, a contradiction.

(d) If V is faithful and H-reduced, C_{H}^{∗}(V) = 1. So by (c), F(E_{H}(V)) = F(H) ∩ E_{H}(V) = 1. Since E_{H}(V) is the central product of nilpotent and quasisimple groups, (d) holds. □

Lemma 2.6 Let W be an H-submodule of V such that W/C_{W}^{∗} (E_{H}(V)) is a simple H-module. Choose an 𝔽H-submodule Y in W minimal with Y ≰ C_{W}^{∗} (E_{H}(V)). Then

Y = [W, O^{p}(E_{H}(V))] and W = Y + C_{W}^{∗} (E_{H}(V)),

and Y is a component of V.

Proof: Since W/C_{W}^{∗} (E_{H}(V)) is simple, W = Y + C_{W}^{∗} (E_{H}(V)). Thus [W, O^{p}(E_{H}(V))] ≤ Y. Since [W, O^{p}(E_{H}(V))] ≰ C_{W}^{∗} (H), the minimality of Y implies Y = [W, O^{p}(E_{H}(V))]. By 2.3(a), Y is a component. □

Lemma 2.7 Put C := \{\overline{W} | W component of V, \overline{W} ≠ 0\}. Then every 𝔽H-submodule of \overline{F_{V} (H)} is the direct sum of elements of C.

Proof: It suffices to show the assertion for simple submodules since \overline{F_{V} (H)} is semisimple by 2.4(a).

Let U/R_{V} (H) be a simple 𝔽H-submodule of \overline{F_{V} (H)}. We need to show that U = Y + R_{V} (H) for some component Y of V. Since by 2.5(a) R_{V} (H) ≤ C_{U}(E_{H}(V)) either C_{U}^{∗}(E_{H}(V)) = R_{V} (H) or C_{U}^{∗}(E_{H}(V)) = U. In the first case the claim for U follows from 2.6.

In the second case 2.5(a) implies U ≤ [S_{V} (H), F^{∗}(H)] + R_{V} (H) and so

U = (U ∩ [S_{V} (H), F^{∗}(H)]) + R_{V} (H).

Since [S_{V} (H), F^{∗}(H)] is the sum of simple H-components of V, so is U ∩ [S_{V} (H), F^{∗}(H)]. Thus, also in this case the claim holds for U. □

Lemma 2.8 Let N ⊴ H. Then

C\overline{F_{V} (H)}(N) = \overline{C_{F_{V} (H)}(N)}.

Proof: Let C be as in 2.7. Then by 2.7

C\overline{F_{V} (H)}(N) = ⟨\overline{W} ∈ C | [\overline{W} , N] = 0⟩.

Let W be a H-component of V such that \overline{W} ≠ 0 and [\overline{W} , N] = 0. Since \overline{W} is semisimple and rad_{W} (H) is the unique maximal 𝔽H-submodule of W, rad_{W} (H) = W ∩R_{V} (H). If rad_{W} (H) = 0, this shows that [W, N] = 0. If rad_{W} (H) ≠ 0, then W is E-quasisimple for E := E_{H}(V). In this case 2.2(c) with W in place of V implies that [W, N] = 0, so the lemma holds. □

Lemma 2.9 Let W be a set of 𝔽H-submodules of V. Then

rad_{∑_{W ∈W} W} (H) = ∑_{W ∈W} rad_{W} (H).

Proof: Clearly ∑_{W ∈W} W/ ∑_{W ∈W} rad_{W} (H) is semisimple and so

rad_{∑_{W ∈W} W} (H) ≤ ∑_{W ∈W} rad_{W} (H).

On the other hand, for W ∈ W, W + rad_{∑_{W ∈W} W} (H)/rad_{∑_{W ∈W} W} (H) semisimple, so

rad_{W} (H) ≤ rad_{∑_{W ∈W} W} (H),

and the reverse inequality holds . □

Lemma 2.10 R_{V} (H) = rad_{F_{V} (H)}(H).

Proof: This follows immediately from 2.9 and the definition of R_{V} (H). □

Lemma 2.11 C_{H}(F_{V} (H)) = C_{H}(\overline{F_{V} (H)})

Proof: Let N = C_{H}(F_{V} (H)). Then by 2.8, F_{V} (H) = C_{F_{V} (H)}(N) + R_{V} (H), and by 2.10, F_{V} (H) = C_{F_{V} (H)}(N). □

Theorem 2.12 Suppose that V is faithful and H-reduced. Then F_{V} (H) and F_{V} (H)/R_{H}(V) are faithful and H-reduced 𝔽H-modules.

Proof: By 2.4(b) F_{V} (H) is reduced. Moreover \overline{F_{V} (H)} is semisimple and thus also H-reduced. From 2.5(b) we get C_{F^{∗}(H)}(F_{V} (H)) ≤ C_{H}^{∗}(V) = 1. Hence F^{∗}(C_{H}(F_{V} (H))) = 1 and so C_{H}(F_{V} (H)) = 1. Thus F_{V} (H) is faithful. Now 2.11 implies that also \overline{F_{V} (H)} is faithful. □

The proof of the Theorems 1.1, 1.3, and 1.4: Theorem 1.3 is 2.4(a), (c) and 2.10, while Theorem 1.4 is 2.12. Moreover, Theorem 1.1 is a direct consequence of these two theorems.

3 The Fitting Submodule for Normal Subgroups

In this section we investigate the relationship between the Fitting submodules for H and for normal subgroups.

Lemma 3.1 Let N ⊴ ⊴ H. Then S_{V} (H) ≤ S_{V} (N) and E_{H}(V) ∩ N = E_{N}(V).

Proof: Using induction on the subnormal defect of N in H, it suffices to treat the case N ⊴H.

Let W be a simple H-submodule. Then by Clifford Theory W is a semisimple N-module and so S_{V} (H) ≤ S_{V} (N). Thus together with 2.1 E_{N}(V) ≤ E_{H}(V) ∩ N.

Conversely, let Y be a simple N-submodule of V and let U be the sum of all N-submodule of V isomorphic to some H-conjugate of Y. Then U is an 𝔽H-submodule of V and we can choose a simple H-submodule W of U. As an 𝔽N-module, W is a direct sum of modules isomorphic to H-conjugates of Y, with each H-conjugate appearing at least once. Hence E_{H}(V) ∩ N ≤ C_{F^{∗}(N)}(W) ≤ C_{F^{∗}(N)}(Y). Intersecting over all the possible Y gives E_{H}(V) ∩ N ≤ E_{N}(V). □

Lemma 3.2 Let W be a H-component of V and N ⊴ ⊴ H. Then there exists a 𝔽N-submodule Y ≤ W with W = ⟨Y^{H}⟩ such that either Y is an N-component of V or Y is simple and [W, F^{∗}(N)] = 1. In particular, F_{V} (H) ≤ S_{V} (N) + F_{V} (N).

Proof: Again we use induction on the defect d_{H}(N) of N in H. Assume that d_{H}(N) > 1. Then there exists N ≤ N_{1} ⊴ H so that d_{N_{1}}(N) < d_{H}(N). By induction there exists an 𝔽N_{1}- submodule Y_{1} of W with W = ⟨Y_{1}^{H}⟩ such that either Y_{1} is simple and [W, F^{∗}(N_{1})] = 0 or Y_{1} is a N_{1}-component of V. In the first case by 2.1 also [W, F^{∗}(N)] = 0 and we can choose a simple 𝔽N-submodule Y of Y_{1}. In the second case induction applies to every H-conjugate of Y_{1} in W, so either [W, F^{∗}(N)] = 0 and Y_{1} = ⟨Y^{N_{1}}⟩ for a simple 𝔽N-submodule Y of Y_{1} or there exists h ∈ H and an N-component Y of V with W^{h} = ⟨Y^{N_{1}}⟩. In any case W = ⟨Y^{H}⟩ and the lemma holds.

Thus it remains to treat the case N ⊴ H. Let Y ≤ W be a N-submodule minimal with Y ≰ rad_{W} (H). Then Y/Y ∩ rad_{W} (H) is a simple N-module and W = ⟨Y^{H}⟩.

Suppose that Y is a semisimple 𝔽N-module, then the minimality of Y shows that Y is simple. If [Y, F^{∗}(N)] = 0, then also [W, F^{∗}(N)] = 0; and if [Y, F^{∗}(N)] ≠ 0, then Y is a N-component of V.

Suppose that Y is not a semisimple 𝔽N-module. Then Y ∩ rad_{W} (H) ≠ 0 and also W is not a semisimple 𝔽N-module. By the definition of a component, W is E_{H}(V)-quasisimple. Now 2.2(b), with (N, E_{H}(V), W) in place of (P, E, V), shows that R := [E_{H}(V), N] ≰ C_{H}(W).

Since W is reduced, R does not act nilpotently on W, so C_{W}^{∗} (E_{N}(V)) ≤ C_{W}^{∗} (R) ≤ rad_{W} (H). On the other hand E_{H}(V) ≤ C_{H}^{∗}(rad_{W} (H)) and so by 3.1 rad_{W} (H) ≤ C_{W}^{∗} (E_{N}(V)). This shows that C_{W}^{∗} (E_{N}(V)) = rad_{W} (H). Hence by 2.6, applied to (N, Y) in place of (H, W), and the minimality of Y show that Y is a N-component of V. □

Corollary 3.3 E_{F^{∗}(H)}(V) = E_{H}(V) and F_{V} (H) ≤ F_{V} (F^{∗}(H)).

Proof: Since E_{H}(V) ≤ F^{∗}(H) the first statement follows from 3.1 applied with N := F^{∗}(H).

Note that W = [W, F^{∗}(H)] for all components of V. The second statement then follows from 3.2 again with N := F^{∗}(H). □

Proposition 3.4 Let E and F be two distinct components of E_{H}(V). Then

[F_{V} (H)), E, F] = 0

Proof: By 3.3 we may assume that H = F^{∗}(H). Let W be an H-component of V with [W, E] ≠ 0. We can apply 2.3 with N := E_{H}(V). By part (c) and (d) of that lemma E is the unique normal subgroup of H minimal with [W, E] ≠ 0. Since [E, F] = 1, part (e) gives [W, F] = 0. Since F_{V} (H) is the sum of all the components of V the lemma holds. □

Recall that a Wedderburn-component for H on V is a maximal sum of isomorphic simple 𝔽H-submodules.

Proposition 3.5 Let W be an H-component of V, N ⊴ H and rad_{W} (H) ≤ Y_{1} ≤ W such that Y_{1}/rad_{W} (H) is a Wedderburn-component for N on W/rad_{W} (H). Put Y := Y_{1} if rad_{W} (H) = 0, and Y := [Y_{1}, E_{H}(V)] if rad_{W} (H) ≠ 0. Then E_{H}(V) ≤ N_{H}(Y_{1}), Y = Y_{1} + rad_{W} (H) and Y is a N_{H}(Y_{1})-component of V.

Proof: Set L := N_{H}(Y_{1}) and W̃ := W/rad_{W} (H). Let D ≤ H with [D, N] ≤ C_{N}(Y_{1}) and U be any simple N-submodule of Ỹ_{1}. Then [D, N] ≤ C_{N}(U), and for all d ∈ D the map

U → U^{d} with u ↦ u^{d} (u ∈ U)

is an 𝔽N-isomorphism. Thus U^{d} is in the Wedderburn-component Ỹ_{1}, and

(∗)  D ≤ L

By 2.5(c) and 2.1(d), E_{H}(V) = C_{E_{H} (V )}(F_{V} (H))(E_{H}(V) ∩ N)C_{E_{H} (V )}(N) and so by (∗), E_{H}(V) ≤ L. This is the first part of the claim.

Since W̃ is a simple 𝔽H-module, Clifford Theory implies that Ỹ_{1} is a simple 𝔽L-module. Suppose first that rad_{W} (H) ≠ 0. With E_{H}(V) in place of N 3.1 gives E_{H}(V) = E_{E_{H} (V )}(V). Then with (L, E_{H}(V)) in place of (H, N) 3.1 gives E_{H}(V) ≤ E_{L}(V). Observe that W = [W, E_{H}(V)] since W is E_{H}(V)-quasisimple, so Y = [Y, E_{H}(V)] ≠ 0. Hence 2.3(a) applied with (L, E_{H}(V), Y, V) in place of (H, N, W, V) shows that Y is a component for L.

Suppose now that rad_{W} (H) = 0. To show that Y is an L-component of V it suffices to show that [Y, F^{∗}(L)] ≠ 0. Observe that [F^{∗}(H), N] ≤ F^{∗}(N) ≤ F^{∗}(L). So if [F^{∗}(H), N] does not centralizes Y, we are done. If [F^{∗}(H), N] ≤ C_{N}(Y), then (∗) implies F^{∗}(H) ≤ L and so F^{∗}(H) ≤ F^{∗}(L). By the definition of a component, [W, F^{∗}(H)] ≠ 1, and since W is a simple 𝔽H-module also [Y, F^{∗}(H)] ≠ 1. Thus [Y, F^{∗}(L)] ≠ 1. □

Lemma 3.6 Let N ⊴ H. Then the following are equivalent:

(a) F_{V} (H) is a semisimple 𝔽N-module.

(b) If K is a component of E_{H}(V) with K ≤ N, then [V, K] = 0.

(c) E_{N}(V) = E_{H}(V) ∩ N ≤ C_{H}^{∗}(V).

(d) [N, E_{H}(V)] ≤ C_{H}(F_{V} (H)).

Proof: (a)⟹ (b): Let K be a component of E_{H}(V) with K ≤ N. By 3.1, K = E_{K}(V) and

so K centralizes all simple 𝔽K-submodules. Since F_{V} (H) is semisimple as an 𝔽N-module and so also as an 𝔽K-module, [F_{V} (H), K] = 1. Thus by 2.5(b), K centralizes V.

(b)⟹ (c): By 2.5(c), F(E_{H}(V) ∩ N) ≤ C_{H}^{∗}(V) and by (b) any component of E_{H}(V) ∩ N is

contained in C_{H}(V). Hence E_{H}(V) ∩ N ≤ C_{H}^{∗}(V), and 3.1 shows that E_{N}(V) = E_{H}(V) ∩ N. (c)⟹ (d): By (c) [N, E_{H}(V)] ≤ C_{F^{∗}(H)}^{∗}(V) and by 2.5(c), C_{F^{∗}(H)}^{∗}(V) ≤ C_{H}(F_{V} (H)).

(d)⟹ (a): Let W be a component of V. If rad_{W} (H) = 0, W is a simple H-module and

so a semisimple N-module. Suppose that rad_{W} (H) ≠ 0. Then W = [W, E_{H}(V)] and by 2.2(b) with (W, N, E_{H}(V)) in place of (V, P, E), W is a semisimple N-module. □

Lemma 3.7 Let W be component of V and N ⊴H such that W is semisimple as an 𝔽N-module. Let Y be a Wedderburn-component for N on W. Then Y is an N_{H}(Y)-component of V.

Proof: Define W̃ := W/rad_{W} (H). Then since W is a semisimple N-module, Ỹ is a Wedderburn component for N on W̃. If rad_{W} (H) = 0 we are done by 3.5.

Suppose that rad_{W} (H) ≠ 0. Then W = [W, E_{H}(V)] and clearly W = F_{W} (H). Since 3.6(a) holds for W in place of V, 3.6(d) implies [N, E_{H}(W)] ≤ C_{H}(W). Thus also [N, E_{H}(V)] ≤ C_{H}(W) since E_{H}(V) ≤ E_{H}(W). This shows that E_{H}(V) normalizes every Wedderburn component for N on W.

Let Y_{1}, …, Y_{r} be the Wedderburn components of N on W. Then

W = Y_{1} ⊕ ⋯ ⊕ Y_{r} and [W, E_{H}(V)] = [Y_{1}, E_{H}(V)] ⊕ ⋯ ⊕ [Y_{r}, E_{H}(V)].

From W = [W, E_{H}(V)] we conclude that Y_{i} = [Y_{i}, E_{H}(V)] for i = 1, …, r. On the other hand by 2.2(d) and 3.5

[Y_{i} + rad_{W} (H), E_{H}(V)] = [Y_{i}, E_{H}(V)] = Y_{i}

is a N_{H}(Y_{i} + rad_{W} (H))-component of W. Then N_{H}(Y_{i} + rad_{W} (H)) = N_{H}(Y_{i}), and Y_{i} is a N_{H}(Y_{i})-component. □

4 The Structure of an F^{∗}(H)-component of V

In this section we determine the structure the F^{∗}(H)-components of V for the case that 𝔽 is finite or algebraically closed.

Lemma 4.1 Suppose 𝔽 is finite or algebraically closed. Let E, P ≤ H with [E, P] = 1. Suppose there exists a simple 𝔽P -module Y and n ∈ ℕ such that V ≅ Y^{n} has an 𝔽P -module. Put 𝕂 = End_{𝔽P}(Y). Then 𝕂 is a finite field extension of 𝔽 and there exists an 𝕂E-module X with dim_{𝕂} X = n such that

V ≅_{𝔽(E×P)} X ⊗_{𝕂} Y

Moreover, the following hold:

(a) If End_{𝔽EP}(V) = 𝔽, then 𝕂 = 𝔽.

(b) If V is a simple 𝔽EP -module, then X is a simple 𝕂E-module.

(c) If V is an E-quasisimple 𝔽EP -module, then X is a quasisimple 𝕂E-module.

Proof: By Schur’s Lemma 𝕂 is a division ring. Since V is finite dimensional, dim_{𝔽} 𝕂 is finite. More precisely, if 𝔽 is algebraically closed then 𝔽 = 𝕂; and if 𝔽 is finite then 𝕂 is finite. In any case 𝕂 is a field.

Let X = Hom_{𝔽P}(Y, V). Then X is a vector space over 𝕂 via

(kx)(y) := x(ky) for all k ∈ 𝕂, y ∈ Y and x ∈ X.

Moreover, E acts on X by (x^{e})(y) := x(y)^{e} and this action is 𝕂-linear. Thus X is a 𝕂Emodule. We now regard X ⊗_{𝕂} Y as an 𝔽(E × P)-module via (x ⊗ y)^{ea} = x^{e} ⊗ y^{a} for e ∈ E, a ∈ P. Let

ϕ : X ⊗_{𝕂} Y → V with x ⊗ y → x(y).

Then ϕ is well-defined since ϕ(kx ⊗ y) = (kx)(y) = x(ky) = ϕ(x ⊗ ky). Also if e ∈ E and a ∈ P, then

ϕ((x ⊗ y)^{ea}) = ϕ(x^{e} ⊗ y^{a}) = (x^{e})(y^{a}) = x((y^{a}))^{e} = (x(y)^{a})^{e} = x(y)^{ae} = x(y)^{ea} = ϕ(x ⊗ y)^{ea}.

So ϕ is an 𝔽(E × P)-module homomorphism. Note that for each submodule Z of V isomorphic to Y, there exists x ∈ X with x(Y) = Z and so ϕ(x ⊗ Y) = Z. Since V is the sum of such submodules, ϕ is surjective. As V ≅ Y^{n},

X = Hom_{𝔽P}(Y, V) ≅ Hom_{𝔽P}(Y, Y^{n}) ≅ End_{𝔽P}(Y)^{n} = 𝕂^{n}.

Hence the 𝔽-spaces X ⊗_{𝕂} Y and Y^{n} and thus also V have the same finite dimension. So ϕ is also injective and ϕ is an 𝔽(E × P)-isomorphism.

Observe that E × P acts 𝕂-linearly on X ⊗_{𝕂} Y. So if we view V as a 𝕂-space via

kϕ(u) = ϕ(ku) for all k ∈ 𝕂, u ∈ X ⊗_{𝕂} Y ,

then EP acts 𝕂-linearly on V. Hence if End_{𝔽EP}(V) = 𝔽 we conclude that 𝕂 = 𝔽. This proves (a).

Let X_{0} be a proper 𝕂E-submodule of X, then ϕ(X_{0} ⊗_{𝕂} Y) is a proper 𝕂EP -submodule of V. This gives (b). In a similar way [V, E] = V implies [X, E] = X.

Now assume that V is E-quasisimple, so rad_{V} (EP) is the unique maximal EP -submodule and rad_{V} (EP) = C_{V}^{∗} (E). Then

X_{0} ⊗ Y ≤ rad_{X⊗_{𝕂}Y} (EP) = C_{X⊗_{𝕂}Y}^{∗} (E) = C_{X}^{∗}(E) ⊗_{𝕂} Y.

This yields X_{0} ≤ C_{X}^{∗}(E). Since X_{0} was an arbitrary proper submodule, it also shows that C_{X}^{∗}(E) is the unique maximal 𝕂E-submodule of X and C_{X}^{∗}(E) = rad_{X}(E) and (c) follows. □

Proposition 4.2 Suppose 𝔽 is finite or algebraically closed. Let W be an F^{∗}(H)-component of V with rad_{W} (F^{∗}(H)) ≠ 0, and let E be the unique component of E_{H}(V) with [W, E] ≠ 0 (see 2.3). Put P := C_{F^{∗}(H)}(E). Then F^{∗}(H) = EP and there exists a finite field extension 𝕂 of 𝔽, a quasisimple 𝕂E-module X and an absolutely simple 𝕂P -module Y such that

V ≅_{𝔽F^{∗}(H)} X ⊗_{𝕂} Y

Proof: By 2.1(c), F^{∗}(H) = EP. Let Y be a simple 𝔽P -submodule of W. Since [E, P] = 1, any F^{∗}(H) conjugate of Y is isomorphic to Y. Thus by 2.2(b) applied to F^{∗}(H) in place of H, W ≅_{𝔽P} Y^{n} for some n. Now 4.1 shows that V ≅_{𝔽F^{∗}(H)} X ⊗_{𝕂}Y. Moreover, since 𝕂 = End_{𝔽P}(Y) and 𝕂 is commutative, 𝕂 = End_{𝕂P}(Y), and Y is an absolutely simple 𝕂P -module. Also since W is an E-quasisimple 𝔽EP -module, X is E-quasisimple. □

Corollary 4.3 Suppose 𝔽 is finite or algebraically closed. Let W be a F^{∗}(H)-component of V, and let K be the set consisting of all the components of H and all the O_{r}(H), r a prime divisor of |H|. Then there exists a finite field extension 𝕂 of 𝔽 and for each K ∈ K a 𝕂K-module W_{K} such that

W ≅_{𝔽F^{∗}(H)} ⨂_{K∈K} W_{K}.

Moreover, either 1. rad_{W} (F^{∗}(H)) = 0 and W_{K} is absolutely simple for every K ∈ K, or

2. rad_{W} (F^{∗}(H)) ≠ 0, W_{E} is E-quasisimple and W_{K} is absolutely simple for every K ∈ K \ {E}, where E is the unique component of E_{H}(V) with [W, E] ≠ 0.

Proof: We may assume that |K| > 1. If rad_{W} (H) = 0 put Y := W, 𝕂 := End_{𝔽F^{∗}(H)}(W) and K_{0} := K. Otherwise let X, Y and 𝕂 be as in 4.2 and put W_{E} := X and K_{0} := K \ {E}. Observe that then W_{E} is E-quasisimple.

In any case Y is an absolutely simple 𝕂P -module, where P := ⟨K_{0}⟩. If |K_{0}| = 1, we are done. In the other case pick K ∈ K_{0} and set K_{1} := K_{0} \ {K}. Then 4.1 applies with (𝕂, K, ⟨K_{1}⟩, Y) in place of (𝔽, E, P, V). Note that in addition 𝕂 = End_{𝕂K⟨K_{1}⟩}(Y), so 4.1(a) also applies. Now an easy induction finishes the proof. □

References

[1] H. Kurzweil, B. Stellmacher, The Theory of Finite Groups, An Introduction. Springer-Verlag, New York, (2004) 387pp.