THE PROJECTIVE COVER OF THE TRIVIAL REPRESENTATION FOR A FINITE GROUP OF LIE TYPE IN DEFINING CHARACTERISTIC

SHIGEO KOSHITANI AND JÜRGEN MÜLLER

Abstract. We give a lower bound of the Loewy length of the projective cover of the trivial module for the group algebras kG of a finite group G of Lie type defined over a finite field of odd characteristic p, where k is an arbitrary field of characteristic p; the proof uses Auslander-Reiten theory. We point out that in one of the main results of a recent paper by Lübeck–Malle [2016] there is a gap, implying that contrary to the claim made there a question asked in Koshitani–Külshammer–Sambale [2014] is still open.

1. Introduction

One of the main problems in the modular representation theory of finite groups is obtaining classes of finite groups G such that the group algebras kG have specific ring-theoretical properties, where k is a field of positive characteristic p, just as Brauer stated in [4, Problem 16]. Examples of such properties are the Loewy length LL(P (k_{G})) of the projective cover P (k_{G}) of the trivial kG-module k_{G}, and the ‘first’ Cartan invariant c_{11}(G) := [P (k_{G}): k_{G}] of the group algebra kG, that is the multiplicity of k_{G} as a composition factor of P (k_{G}).

In the present paper, we are interested in the questions for representations of finite groups of Lie type in their defining characteristic; see also for example [12, (2.6)] and [13, Section 11.4]. In even characteristic we are in a good shape indeed: Let G be a simple finite group of Lie type defined over a finite field of characteristic p = 2. Then it follows from results of Okuyama [35] and Erdmann [8] that LL(P (k_{G})) = 3 if and only if G = SL_{3}(2), see [22, Theorem 3.3]; and by [22, Theorem 1.2] we always have LL(P (k_{G})) ≠ 4. Moreover, by [23, Lemma 4.5] we have c_{11}(G) = 2 if and only if G = SL_{3}(2). Hence in conclusion for this class of groups we have LL(P (k_{G})) ≥ 5 if and only if c_{11}(G) ≥ 3 if and only if G≅ S̸L_{3}(2).

Here we are now interested in the odd characteristic case, and the purpose of this paper is to present the following theorem.

Theorem 1.1. Assume that p is an odd prime, and G is a simple or an almost simple finite group of Lie type (in the sense of 2.1) defined over a finite field of characteristic p, such that the Sylow p-subgroups of G are non-cyclic, that is G ∉ \{SL_{2}(p),^{2}G_{2}(\sqrt{3})′\}. Then the projective cover P (k_{G}) of the trivial kG-module has Loewy length LL(P (k_{G})) ≥ 5.

As it turns out, this sheds new light on a couple of our earlier results in [24]. Firstly, we get the following, providing an alternative proof of [24, Proposition 4.12]:

Date: September 26, 2016.

2000 Mathematics Subject Classification. 20C20, 20C05.

Corollary 1.2. Under the same assumptions as in Theorem 1.1, the principal block algebra B_{0}(G) of kG has the Loewy length LL(B_{0}(G)) ≥ 5.

Secondly, our method of proof of Theorem 1.1 also allows to generalize [24, Proposition 4.10] (which was proved under the additional assumption p ≥ 5):

Theorem 1.3. Let p be an odd prime, let G be a finite group such that O_{p′}(G) = {1}, and assume that the principal block algebra B_{0}(G) of kG has Loewy length LL(B_{0}(G)) = 4. Then F^{∗}(G) = E(G) = O^{p′}(G) is a non-abelian simple group.

1.4. Loewy lengths vs. Cartan invariants. Let p be odd and G a finite group.

(a) Our line of reasoning in proving the above results can be summarized as follows: Assume that LL(B_{0}(G)) ≤ 4, thus we have LL(P (k_{G})) ≤ 4 as well, from which by Proposition 3.2 we conclude that the heart H(k_{G}) := rad(P (k_{G}))/soc(P (k_{G})) of P (k_{G}) actually is a simple kG-module S, where apart from a trivial exceptional case we have S ≅ k̸_{G}. In order to prove Theorem 1.1 we then apply the Kawata–Michler– Uno Theorem 2.7, which in turn is a specific application of Auslander-Reiten theory to finite groups of Lie type. Note that this approach stays entirely in the realm of modular representations, no results of ordinary representation theory are needed.

(b) Thus, apart from the above-mentioned exception, we conclude that in general the condition LL(P (k_{G})) ≤ 4 implies that the Cartan invariant c_{11}(G) = 2. Hence, in particular in view of the comments on the even characteristic case made above, the question arises whether the condition LL(P (k_{G})) ≥ 5 implies c_{11}(G) ≥ 3. But this is not true in general, in other words c_{11}(G) = 2 is a strictly weaker condition than LL(P (k_{G})) ≤ 4, at least as far as the class of all finite groups is concerned:

One of the smallest counter-examples is G := C_{3}^{2} : Q_{8} for p = 3, where C_{3} and Q_{8} are the cyclic group of order 3 and the quaternion group of order 8, respectively, and the action of Q_{8} on the elementary abelian group C_{3}^{2} is regular. Then we have c_{11}(G) = 2, although LL(P (k_{G})) = 5; actually, all projective indecomposable kGmodules in B_{0}(G) have Loewy length 5, so that LL(B_{0}(G)) = 5 as well; see also [25, Lemma (4.2) and Lemma (4.3)], where infinitely many such examples are given.

Of course, the picture might change, if we restrict ourselves to the class of quasisimple finite groups of Lie type in odd characteristic. But so far we are not able to prove this. We come back to this in Remark 4.1, where we point out a gap related to this in the recent paper [31] by Lübeck–Malle, implying that a question raised in [24] is still open, contrary to the claim made in [31].

1.5. Notation and terminology. We shall in particular use the following notation. Let k be an arbitrary field of positive characteristic p, and let A be a finite-dimensional k-algebra. Unless stated otherwise we mean by an A-module a finitely generated right A-module. Let M, N be A-modules. We write N | M, if N is isomorphic to a direct summand of M as A-modules. We write rad(M) for the Jacobson radical of M, and P (M) for the projective cover of M. We say that n = LL(M) is the Loewy length of M, if n is smallest with rad^{n}(M) = {0}.

Moreover, if G is a finite group, we write Z(G) for the center of G, and F (G) for the Fitting subgroup of G. We let E(G) be the layer of G, that is the central product of the components of G, where a component of G is a subnormal quasi-simple subgroup, and we write F^{∗}(G) = F (G)E(G) for the generalized Fitting subgroup of G. We denote by k_{G} the trivial kG-module, and by B_{0}(G) := B_{0}(kG) the principal block algebra of kG. Given a kG-module M, we write M^{∗} := Hom_{k}(M, k) for the dual module of M, which also becomes a right kG-module, and M is called self-dual if M ≅ M^{∗} as kG-modules.

For other general notation and terminology we refer [34] and [38], as far as representation theory and finite group theory are concerned, respectively. Moreover, for the necessary background on finite groups of Lie type and Auslander-Reiten theory, we refer to [32] and [2], respectively.

This paper now is organized as follows: In Section 2 we collect the necessary prerequisites for finite groups of Lie type and their representation theory in defining characteristic. In Section 3 we give proofs of the above-mentioned results. In Section 4 we give a few concluding remarks.

2. Groups of Lie type in defining characteristic

We collect the facts needed from the theory of finite groups of Lie type and their representations.

2.1. Tits’s Theorem. Let G be a simply-connected simple linear algebraic group over the algebraic closure \overline{𝔽}_{p} of the field 𝔽_{p} with p elements. Let F: G → G be a Steinberg endomorphism, see [32, Ch.21]. Let T be an F -stable maximal torus of G, contained in an F -stable Borel subgroup B of G, and let U be the unipotent radical of B. Let q be the absolute value of the eigenvalues of F for its action on the character group of T. Then q is an integral power of p, except in the ‘very-twisted cases’ giving rise to the Suzuki and Ree groups, where it is an half-integral power. The associated set of fixed points G(q) := G^{F} is called a finite group of Lie type.

Now Tits’s Theorem, see [32, Theorem 24.17], says that except in the cases

SL_{2}(2), SL_{2}(3), SU_{3}(2), Sp_{4}(2), G_{2}(2), ^{2}B_{2}(\sqrt{2}), ^{2}G_{2}(\sqrt{3}), ^{2}F_{4}(\sqrt{2})

the group G := G(q) is perfect, implying that G is quasi-simple, that is G/Z(G) is non-abelian simple; note that Z(G) always is a p′-group. In this case, G is called a quasi-simple finite group of Lie type.

The non-solvable groups amongst the above exceptions, that is

Sp_{4}(2), G_{2}(2), ^{2}G_{2}(\sqrt{3}), ^{2}F_{4}(\sqrt{2}),

all turn out to have trivial center, and their derived subgroups have index p and are non-abelian simple. These groups are called the almost simple finite groups of Lie type. In both the quasi-simple and the almost simple cases the group (G/Z(G))′ is called a simple group of Lie type; note that this encompasses the Tits group 2F_{4}(\sqrt{2})′, which does not occur elsewhere in the classification of finite simple groups.

Proposition 2.2. Finite and tame cases; see also [13, Section 8.9]. Let G be a simple or an almost simple finite group of Lie type, and let U ≤ G be a Sylow p-subgroup of G. Then the following holds:

(a) If p is arbitrary, then U is cyclic if and only if one of the following holds: ∘ G ≅ SL_{2}(p), in which case U ≅ C_{p} has order p; ∘ G ≅ ^{2}G_{2}(\sqrt{3})′, in which case U ≅ C_{9} has order 9.

(b) If p = 2, then U is dihedral, semi-dihedral or generalized quaternion if and only if one of the following holds:

∘ G ≅ SL_{2}(4), in which case we have U ≅ C_{2}^{2}, the Klein 4-group;

∘ G ≅ SL_{3}(2), in which case we have U ≅ D_{8}, the dihedral group of order 8; ∘ G ≅ Sp_{4}(2)′, in which case we have U ≅ D_{8}, the dihedral group of order 8.

Proof. Although there is a thorough discussion of the structure of U in [11, Section 3.3], we choose a direct approach, tailored for our purposes. The almost simple groups and their derived subgroups being discussed in 2.8 below, we may assume that G = G^{F} is quasi-simple, and that U := U^{F}, by [32, Corollary 24.11].

(a) Assume that U is cyclic. Then it follows from [32, Proposition 23.7, Corollary 23.9] that G has twisted Lie rank 1, thus U is a root subgroup. For the structure of the root subgroups occurring we refer to [32, Example 23.10] and [5, Proposition 13.6.3, Proposition 13.6.4]. We now consider the various types:

For type A_{1} we have G ≅ SL_{2}(q) and U ≅ 𝔽_{q}^{+}, hence U is cyclic if and only if q = p. For type ^{2}A_{2} we have G ≅ SU_{3}(q) and U/U′ ≅ 𝔽_{q^{2}}^{+}, hence U is not cyclic.

For types ^{2}B_{2} and ^{2}G_{2} we have q = \sqrt{2}^{2f+1} and q = \sqrt{3}^{2f+1}, respectively, for some f ≥ 1, and U/U′ ≅ 𝔽_{q^{2}}^{+}, hence U is not cyclic either.

(b) Let p = 2, and assume that U is dihedral, semi-dihedral or generalized quaternion. Then, by [14, Satz I.14.9(b)], we have U/U′ ≅ C_{p}^{2}. Thus the Chevalley commutator formula, see [32, Proposition 23.11], shows that G has has twisted Lie rank at most 2. We first consider the various types of Lie rank 1:

For type A_{1} the subgroup U ≅ 𝔽_{q}^{+} has the desired shape if and only if q = 4. For type ^{2}A_{2} we have U/U′ ≅ 𝔽_{q^{2}}^{+}, where q > 2, hence U does not have the desired shape. For type ^{2}B_{2} we have U/U′ ≅ 𝔽_{q^{2}}^{+}, where q^{2} = 2^{2f+1} > 2, hence U does not have the desired shape either.

If G has twisted Lie rank 2, then U/U′ has a quotient isomorphic to (U_{α}/U′_{α}) × (U_{β}/U′_{β}), where α, β denote the fundamental roots, and we assume α to be the long one if there are two distinct root lengths. Hence we have U_{α}/U′_{α} ≅ C_{p} ≅ U_{β}/U′_{β}, implying that U_{α} ≅ C_{p} ≅ U_{β}. We again consider the various types:

For type A_{2} we have G ≅ SL_{3}(q) and U_{α} ≅ 𝔽_{q}^{+} ≅ U_{β}, hence q = 2, in which case from U′ = Z(U) = U_{α+β} ≅ 𝔽_{q}^{+} we get U ≅ D_{8}. For types B_{2} and G_{2}, where B_{2}(q) ≅ Sp_{4}(q), we have U_{α} ≅ 𝔽_{q}^{+} ≅ U_{β}, where q > 2, hence U does not have the desired shape. For type ^{2}A_{3} we have G ≅ SU_{4}(q), where U_{α} ≅ 𝔽_{q}^{+} and U_{β} ≅ 𝔽_{q^{2}}^{+}. For type ^{2}A_{4} we have G ≅ SU_{5}(q), where U_{α} ≅ 𝔽_{q^{2}}^{+} and U_{β}/U′_{β} ≅ 𝔽_{q^{2}}^{+}. For type

3D_{4} we have U_{α} ≅ 𝔽_{q}^{+} and U_{β} ≅ 𝔽_{q^{3}}^{+}. For type ^{2}F_{4} we have q = \sqrt{2}^{2f+1}, for some f ≥ 1, and U_{α} ≅ 𝔽_{q^{2}}^{+} and U_{β}/U′_{β} ≅ 𝔽_{q^{2}}^{+}. □

2.3. Blocks. Given a finite group G of Lie type, then the p-blocks of kG are wellunderstood; see for example [13, Chapter 8] or [7, Corollary]: There is a unique p-block of defect 0, its ordinary character being the Steinberg character. All other p-blocks have maximal defect, and if k contains |Z(G)|-th primitive roots of unity, there are precisely |Z(G)| of them. In particular, the principal p-block is the only block of positive defect of a simple or an almost simple finite group of Lie type.

Moreover, excluding the cases explicitly mentioned in Proposition 2.2, it follows from the comments in 2.4 below that any block algebra of kG of positive defect has wild representation type.

2.4. Auslander-Reiten theory. We recall the necessary facts from Auslander-Reiten theory. Let G be any finite group, and let B be a p-block algebra of kG, where k is assumed to be algebraically closed. Then it is well-known, see for example [3, Theorem 4.4.4], that B has wild representation type if and only if the defect groups of B are neither cyclic, dihedral, semi-dihedral nor generalized quaternion.

In this case, by [9, Theorem 1], any connected component of the stable Auslander-Reiten quiver of B has tree class A_{∞}. Hence it makes sense to discuss whether or not a non-projective indecomposable B-module M lies at the end of its connected component in the stable Auslander-Reiten quiver; see for example [18, Section 2.2] or [19, Introduction]. As the Heller operator Ω induces an automorphism of the stable Auslander-Reiten quiver of B, the module M lies at the end of its connected component in the stable Auslander-Reiten quiver, if and only if any and hence all of its Heller translates Ω^{i}(M), for i ∈ ℤ, have this property.

In particular, if S is a simple B-module, then the Auslander-Reiten sequence ending in Ω^{−1}(S) is the standard short exact sequence

{0} → Ω(S) → P (S) ⊕ H(S) → Ω^{−1}(S) → {0},

where H(S) := rad(P (S))/soc(P (S)) is the heart of P (S). Hence S lies at the end of its connected component in the stable Auslander-Reiten quiver, if and only if H(S) is indecomposable.

Another sufficient condition to ensure that S lies at the end of its connected component in the stable Auslander-Reiten quiver is given in the following Proposition 2.5. Actually, it is the key ingredient in the proof of Theorem 2.7, but its immediate Corollary 2.6 is used in 2.8 below:

Proposition 2.5. [20, Proposition 2.1]. Let G be a finite group, and let U ≤ G be a Sylow p-subgroup, which is neither cyclic, dihedral, semi-dihedral nor generalized quaternion. Let S be a simple kG-module, such that U is a vertex of S, and

dim_{k}(Hom_{kU}(k_{U}, S↓_{U})) = 1 = dim_{k}(Hom_{kU}(S↓_{U}, k_{U})).

Then S lies at the end of its connected component in the stable Auslander-Reiten quiver of kG.

Proof. By assumption, kU has wild representation type. Moreover, the second condition on S implies that S↓_{U} is indecomposable. Now the assertion follows from the argument in [20, Section 3] in conjunction with [20, Proposition 1.3]. □

Corollary 2.6. Keeping the assumptions on G and U, let S be a simple kG-module.

(a) Let Φ(U) be the Frattini subgroup of U. Suppose that S↓_{Φ(U)} is indecomposable, and that

dim_{k}(Hom_{kU}(k_{U}, S↓_{U})) = 1 = dim_{k}(Hom_{kU}(S↓_{U}, k_{U})).

Then S lies at the end of its connected component in the stable Auslander-Reiten quiver of kG.

(b) Let H be a finite group containing G as a normal subgroup, such that S extends to a (simple) kH-module T. Moreover, assume that S fulfills the conditions in (a). Then T lies at the end of its connected component in the stable Auslander-Reiten quiver of kH.

Proof. (a) It follows from the second condition that S↓_{U} is indecomposable. Assume there is a maximal subgroup X < U such that S↓_{U} is relatively X-projective. Then we have S↓_{U} | (S↓_{X})↑^{U}. From Φ(U) < X we get that S↓_{X} is indecomposable, hence by Green’s indecomposability theorem we infer that (S↓_{X})↑^{U} is indecomposable as well. Thus we have S↓_{U} ≅ (S↓_{X})↑^{U} as kU-modules, a contradiction. Hence U is a vertex of S, and the assertion follows from Proposition 2.5.

(b) Let V be a Sylow p-subgroup of H, where we may assume that U = G ∩ V. Then we have U ⊴V, and thus by [14, Hilfssatz III.3.3(b)] we get Φ(U)⊴Φ(V), implying that T ↓_{Φ(V)} is indecomposable. Moreover, since Hom_{kV} (k_{V} , T ↓_{V}) ≠ {0} ≠ Hom_{kV} (T ↓_{V} , k_{V}) anyway, from the assumptions on S we infer that

dim_{k}(Hom_{kV} (k_{V} , T ↓_{V})) = 1 = dim_{k}(Hom_{kV} (T ↓_{V} , k_{V})).

Hence the assertion follows from (a), applied to the kH-module T. □

The key to prove our main theorem will be the following result:

Theorem 2.7. Kawata–Michler–Uno [20, Theorem]. Let G be a quasi-simple, an almost-simple or a simple finite group of Lie type (in the sense of 2.1), defined over a finite field of characteristic p, such that its Sylow p-subgroups are neither cyclic, dihedral, semi-dihedral nor generalized quaternion, that is

G ∉ \{SL_{2}(p), SL_{2}(4), SL_{3}(2), Sp_{4}(2)′,^{2}G_{2}(\sqrt{3})′\}.

Then any non-projective simple kG-module, where k is algebraically closed, lies at the end of its connected component in the stable Auslander-Reiten quiver.

Proof. Some care in applying in [20, Section 3] has to be excercised: The strategy of proof is to ensure the properties listed in Proposition 2.5. These in turn follow from the results, including their proofs, given in [6, 7]. Now checking the assumptions made there it turns out that the admissible groups are precisely the quasi-simple groups (in the sense of 2.1). Hence this covers the quasi-simple cases, and it remains to consider the almost simple groups and their derived subgroups, which is done in 2.8 below. □

2.8. The lost cases. We discuss the cases excluded in Theorem 1.1 and the Kawata–Michler–Uno Theorem 2.7, and show that the statements do not hold in these cases. Moreover, we complete the proof of Theorem 2.7 by considering explicitly constructed projective indecomposable modules for the relevant almost simple groups and their derived subgroups.

In order to proceed, we assume k to be algebraically closed. Given a finite group G, by [39, Theorem E] the heart H(k_{G}) of P (k_{G}) is decomposable if and only if p = 2 and the Sylow p-subgroups of G are dihedral, including the Klein 4-group. In this case, there are three simple kG-modules belonging to the principal 2-block, and the structure of the associated projective indecomposable modules has been determined in [8, Theorems 2 and 4]. If the Sylow p-subgroups of G are cyclic, then the theory of blocks cyclic defect applies, and the projective indecomposable kG-modules are described in terms of the Brauer trees of kG; see [17] or [28].

In contrast to this bright picture, for block algebras of wild representation type general theory does not provide too much insight. But the projective indecomposable modules of interest here are straightforwardly constructed explicitly and analysed, using the computer algebra systems GAP [10] and C-MeatAxe [37], and the databases compiled in the framework of the ModularAtlas project [16]. The only exception is the group ^{2}F_{4}(\sqrt{2}) and its derived subgroup, where the projective indecomposable modules are not easily computationally tractable due to sheer size. Here we instead apply Corollary 2.6 to the simple modules in question. Note that restriction induces a bijection between the simple modules in the principal p-blocks of an almost simple group and its derived subgroup, respectively.

We now consider the various cases:

∘ For G = SL_{2}(p), where p ≥ 5, the Sylow p-subgroups are cyclic of order p. There are \frac{p−1}{2} simple modules in each of the two p-blocks of positive defect. Their Brauer trees are straight lines having the exceptional vertex of multiplicity 2 at the end, see [13, Section 16.9]. Thus there are two simple modules, amongst them k_{G}, such that the heart of the associated projective cover is indecomposable, and p − 3 simple modules not having this property. Moreover, we have LL(P (k_{G})) = 3.

∘ For G = ^{2}G_{2}(\sqrt{3}) ≅ PSL_{2}(8): 3 the Sylow 3-subgroups are isomorphic to the extra-special group 3_{−}^{1+2}, those of G′ = ^{2}G_{2}(3)′ ≅ SL_{2}(8) ≅ PSL_{2}(8) are isomorphic to C_{9}. There are two simple modules {k_{G}, 7} in the principal 3-block of G′. The assocoated Brauer tree is a straight line having the exceptional vertex of multiplicity 4 at the end. Thus H(k_{G′}) is indecomposable and LL(P (k_{G′})) = 3, while H(7) is decomposable and LL(P (7)) = 5. The principal 3-block of G has wild representation type, and H(S) turns out to be indecomposable for both its simple modules S ∈ {k_{G}, 7}, where LL(P (k_{G})) = 5 and LL(P (7)) = 7.

∘ For G = SL_{2}(4) = PSL_{2}(4) ≅ PSL_{2}(5) and for G = SL_{3}(2) = PSL_{3}(2) ≅ PSL_{2}(7) the Sylow 2-subgroups are dihedral, so that H(k_{G}) is decomposable.

∘ For G = Sp_{4}(2) ≅ 𝔖_{6} the Sylow 2-subgroups are isomorphic to D_{8} × C_{2}, those of G′ = Sp_{4}(2)′ ≅ 𝔄_{6} ≅ PSL_{2}(9) are isomorphic to D_{8}. Hence P (k_{G′}) is decomposable. The principal 2-block of G has wild representation type, and for all its simple modules S ∈ {k_{G}, 4, 4′} the heart H(S) turns out to be indecomposable, where LL(P (S)) = 10.

∘ For G = G_{2}(2) ≅ PSU_{3}(3): 2 the Sylow 2-subgroups have order 64, those of G′ ≅ G_{2}(2)′ ≅ PSU_{3}(3) ≅ SU_{3}(3) have order 32. In both cases they have nilpotency class 3, thus by [14, Satz III.11.9(b)] they are neither cyclic, dihedral, semi-dihedral nor generalized quaternion. Hence both the principal 2-blocks of G′ and G have wild representation type. For all the simple kG′-modules S ∈ {k_{G′}, 6, 14} the heart H(S) turns out to be indecomposable, where LL(P (S)) = 19, and similarly for all the simple kG-modules S ∈ {k_{G}, 6, 14} the heart H(S) turns out to be indecomposable, where LL(P (S)) = 20.

∘ For G ≅ ^{2}F_{4}(\sqrt{2}) the Sylow 2-subgroups have order 4096, those of ^{2}F_{4}(\sqrt{2})′ have order 2048. In both cases they have nilpotency class 8, thus they are neither cyclic, dihedral, semi-dihedral nor generalized quaternion. Hence both the principal 2-blocks of G′ and G have wild representation type. Due to sheer size, in these cases the projective indecomposable modules are not easily computationally tractable, so that here we revert to Corollary 2.6. By 2.6(b) it suffices to check the conditions of 2.6(a) for the simple kG′-modules, which is straightforwardly done explicitly. This shows that for all the simple kG′-modules or kG-modules S the heart H(S) is indecomposable, but does not provide any further information on the Loewy lengths of the associated projective indecomposable modules. Actually, with considerable computational effort it is possible to show that for the simple kG′-modules {k_{G′}, 26, 246} we have LL(P (k_{G′})) = 34 and LL(P (26)) = 40 = LL(P (246)), and for the simple kG-modules S ∈ {k_{G}, 26, 246} we similarly have LL(P (k_{G′})) = 35 and LL(P (26)) = 41 = LL(P (246)).

3. Proofs

We are now prepared to prove our results. To do so, we will need an easy lemma, which probably is well-known, but for convenience we present it:

Lemma 3.1. Let ι: N → M be an embedding of A-modules, such that

Hom_{A}( N/rad(N), M/(ι(N) + rad(M)) ) = {0},

that is the heads of N and M/ι(N) do not have any composition factors in common. Then any epimorphism π: M → N is split, and ι is a splitting map.

Proof. Assume that ι(N) + ker(π) ⪇ M. Then there is a maximal A-submodule L ⪇ M containing ι(N)+ker(π). Hence the simple A-module M/L is an epimorphic image both of M/ι(N) and of M/ker(π) ≅ N, contradicting the assumption on composition factors. Hence we have ι(N) + ker(π) = M. Now we get dim_{k}(ι(N)) + dim_{k}(ker(π)) = dim_{k}(N) + (dim_{k}(M) − dim_{k}(N)) = dim_{k}(M), implying ι(N) ∩ ker(π) = {0}, and thus M = ι(N) ⊕ ker(π) as A-modules. □

Our standard application of Lemma 3.1 will be the following: Let G be a finite group, and let N ≤ M be self-dual kG-modules, such that even N and M/N do not have any composition factors in common. Then dualising the natural inclusion map ι: N → M gives rise to an epimorphism π: M ≅ M^{∗} → N^{∗} ≅ N, and hence we have M ≅ N ⊕ (M/N) as kG-modules

The proofs of Theorems 1.1 and 1.3 will both be based on the following statement:

Proposition 3.2. Let p be odd, let G be a finite group such that p | |G|, and assume that LL(P (k_{G})) ≤ 4. Then we have LL(P (k_{G})) = 3, where the heart H(k_{G}) := rad(P (k_{G}))/soc(P (k_{G})) of P (k_{G}) is simple. Writing S := H(k_{G}) the following holds:

(i) If S ≅ k_{G}, then we have p = 3, and G is 3-nilpotent with Sylow 3-subgroups of order 3.

(ii) If S ≅ k̸_{G}, letting H(S) := rad(P (S))/soc(P (S)) be the heart of P (S), then ∘ either we have p = 3, and G is not 3-nilpotent, has Sylow 3-subgroups of order

3, and H(S) ≅ k_{G}, ∘ or H(S) is strictly decomposable such that k_{G} | H(S).

Proof. Since rad(k̃G) = rad(kG) ⊗_{k} k̃ for any separable field extension k ⊆ k̃, implying that LL(P (k_{G})) = LL(P (k̃_{G})), we can assume that k is algebraically closed; see also [23, Lemma 1.6 and its proof].

By Maschke’s Theorem, we have LL(P (k_{G})) ≥ 2. Suppose that LL(P (k_{G})) = 2, then k_{G} is the only simple kG-module in B_{0}(G), and from the Cartan invariant c_{11}(G) = [P (k_{G}): k_{G}] = 2 we conclude that p = 2, a contradiction. Similarly, it follows from [21, Corollary] that LL(P (k_{G})) ≠ 4.

Hence we have LL(P (k_{G})) = 3. Thus H(k_{G}) is semi-simple, hence is simple by [39, Theorem E], so we write S := H(k_{G}). If S ≅ k_{G}, then k_{G} is the only simple kG-module in B_{0}(G), hence by Brauer’s Theorem [34, Theorem V.8.3] we conclude that G is p-nilpotent, and from c_{11}(G) = [P (k_{G}): k_{G}] = 3 we conclude that p = 3 and that the Sylow p-subgroups of G have order 3.

Thus we may assume that S ≅ k̸_{G}, and let H(S) := rad(P (S))/soc(P (S)). Since 1 = c_{1,S}(G) = [P (k_{G}): S] = [P (S): k_{G}] = c_{S,1}(G), and since there is a uniserial kGmodule with composition factors S and k_{G}, Lemma 3.1 implies that H(S) = k_{G}⊕H′ for a kG-module H′.

Assume that H′ = {0}. Then {k_{G}, S} are the simple kG-modules in B_{0}(G), and thus the Cartan matrix C of B_{0}(G) is of the form

C = (\begin{matrix}2 & 1 \\ 1 & 2 \end{matrix}) .

Hence we have det(C) = 3, thus the Sylow p-subgroups of G have order 3. Moreover, it follows from the theory of blocks of cyclic defect that the inertial index of B_{0}(G) is 2, and thus G is not 3-nilpotent. Therefore we are left with the case H′ ≠ {0}, that is H(S) is decomposable of the desired form. □

Proof of Theorem 1.1. Assume to the contrary that LL(P (k_{G})) ≤ 4. Since by Proposition 2.2 we are assuming that G has non-cyclic Sylow p-subgroups, from Proposition 3.2 we infer the existence of a simple kG-module S≅k̸_{G}, such that H(S) is decomposable. By 2.4 this means that S does not lie at the end of its connected component in the stable Auslander-Reiten quiver, contradicting the Kawata– Michler–Uno Theorem 2.7. □

Actually, there are alternative ways to proceed, as soon as the existence of the simple kG-module S is granted, see Remark 4.2(b) below. As was remarked earlier we have chosen the approach presented here, since it stays entirely in the realm of modular representations, and does not need ordinary representation theory.

Proof of Corollary 1.2. This follows immediately from Theorem 1.1. □

Proof of Theorem 1.3. Set H := O^{p′}(G); note that p | |H|. Then we have O^{p′}(H) = H, and H ≤ G being a characteristic subgroup, from O_{p′}(G) = {1} we infer O_{p′}(H) = {1}.

By [26, Lemma 4.1] we have LL(B_{0}(H)) = 4, where B_{0}(H) is the principal block algebra of kH. Hence we have LL(P (k_{H})) ≤ 4, and thus from Proposition 3.2 we get that the heart S := H(k_{H}) of P (k_{H}) is simple; see also [24, Proposition 4.6]. Moreover, if S ≅ k_{H}, then from O_{p′}(H) = {1} we conclude that H ≅ C_{3}, hence LL(B_{0}(H)) = LL(P (k_{H})) = 3, a contradiction. Hence we have S≅k̸_{H}, so that the Cartan invariant c_{11}(H) = 2.

We are prepared to show that H is non-abelian simple: Suppose that there exists N such that {1} ≠ N ◁ H. Then from O_{p′}(H) = {1} and O^{p′}(H) = H we conclude that both p | |N| and p | |H/N|, contradicting [22, Lemma 2.5]. Moreover, if H were abelian, then H ≅ C_{p}, so that LL(B_{0}(H)) = LL(kH) = p, a contradiction.

Next we consider the Fitting subgroup F (G). Since by assumption we have O_{p′}(G) = {1}, it remains to consider O_{p}(G). Since H is non-abelian simple we have O_{p}(G) ∩ H = {1}. From this we get O_{p}(G) ≅ O_{p}(G)H/H ≤ G/H, where the latter is a p′-group, so that O_{p}(G) = {1}, hence F (G) = {1}, that is F^{∗}(G) = E(G).

Finally we consider the layer E(G). Clearly, as H is a component of G, we have H ≤ E(G). Hence in order to claim that H = E(G) we have to show that there is no other component of G. Suppose to the contrary that there is a component Q ≠ H of G; note that Q ≰ H. Then let N := ⟨Q^{g} | g ∈ G⟩ ⊴ G be the normal closure of Q in G. Hence we have N ∩ H = {1}, since otherwise, H being simple, we had N ∩ H = H, that is Q ≤ N ≤ H, a contradiction. From this we get N ≅ NH/H ≤ G/H, so that N is normal p′-subgroup of G, hence O_{p′}(G) = {1} implies Q ≤ N = {1}, a contradiction. □

4. Concluding remarks

We end this paper with a couple of concluding remarks.

Remark 4.1. We point out a gap in the proof of [31, Theorem 7.1]. In consequence, contrary to the claim in [31, Theorem 3], this means that the question raised in [24], namely whether for any non-abelian finite simple group G having non-cyclic Sylow p-subgroups, where p is odd, the Cartan invariant c_{11}(G) ≥ 3, is still open. (The weaker assertion in [29, Theorem 1.2], saying that in the above situation the principal block algebra B_{0}(G) has Loewy length LL(B_{0}(G)) ≠ 4, by the line of reasoning given below fortunately turns out to hold true.)

Indeed, in [31, Theorem 7.1] it is said that, if G is a classical simple finite group of Lie type, and l is an odd prime such that the Sylow l-subgroups are non-cyclic, then the l-modular Cartan invariant c_{11}^{(l)}(G) ≥ 3. The proof is by way of contradiction, and in its first line it is asserted that the defining characteristic case l = p was settled in [24]. But, as far as we see, this is not quite true:

For a simple finite group G of Lie type it is only proved in [24, Proposition 4.12] that LL(B_{0}(G)) ≠ 4, by using results due to Külshammer [27] and Oppermann [36] on the Loewy length of the block algebra B_{0}(G) as a whole, but nothing comprehensive is said there about the projective indecomposable module P (k_{G}) or on c_{11}(G). Moreover, as we have discussed in 1.4, in general the condition c_{11}(G) = 2 is strictly weaker than LL(P (k_{G})) ≤ 4, so that Theorem 1.1 is not applicable here, at least not without some additional specific argument.

Remark 4.2. ‘Minimal’ projective indecomposable modules. (a) Given a finite group G and a simple kG-module S, let c(S) be the integer defined by

c(S) := \frac{dim_{k}(P (S))}{|G|_{p}},

where |G|_{p} is the largest power of p dividing |G|. Following [33], if c(S) = 1 then the projective indecomposable module P (S) is called minimal.

Now, given a normal subgroup N ⊴ G, Malle–Weigel [33, Proposition 2.2] show ‘supermultiplicativity’ c(k_{G}) ≥ c(k_{N}) · c(k_{G/N}), they give sufficient conditions as to when multiplicativity holds, and they ask whether this is possibly always fulfilled.

However, even more generally, given any simple kG-module S on which N acts trivially, and denoting its deflation to G/N by \overline{S}, it is an immediate consequence of the Alperin–Collins–Sibley Theorem, see [1, Corollary 1], that c(S) = c(k_{N}) · c(\overline{S}); see also [40, Lemma 2.6, Section 4] and [15, Theorem VII.14.2]. In particular, the above question has an affirmative answer.

(b) We just indicate two alternative ways to proceed in the final step in the proof of Theorem 1.1; they both need ordinary representation theory. Hence let G be as asserted, and recall that we know that S := H(P (k_{G})) is simple, where S≅k̸_{G}.

(i) Firstly, for the untwisted and twisted cases we have |U| = q^{N}, while for the very-twisted cases we have |U| = (q^{2})^{N}, where N is the number of positive roots in the root system associated with G. We have dim_{k}(S) = c(k_{G})·|U|−2, while by [13, Theorem 3.7] and a similar result for the very-twisted cases, we have dim_{k}(S) < |U|. This implies c(k_{G}) = 1, that is P (k_{G}) is minimal, contradicting [33, Theorem 5.8]; note that in proving the latter ordinary character theory of finite groups of Lie type is used as well.

(ii) Secondly, more straightforwardly, we observe that dim_{k}(S) = |U| − 2. Then, for the untwisted and twisted cases we may infer N = 1, proceeding similar to [13, Proof of Theorem 3.7], using Weyl’s character formula for ordinary Verma modules; see also [30, Proposition 5.1]. Thus we get G = SL_{2}(q), and hence dim_{k}(S) = q − 2. Whence Steinberg’s Tensor Product Theorem, see [13, Theorem 2.7], implies q = p, a contradiction. For the the very-twisted case ^{2}G_{2}(\sqrt{3}^{2f+1}) a similar dimension estimate yields a contradiction, see [13, Chapter 20].

Acknowledgments

A part of this work was done while the first author was visiting University of Jena several times in 2015–2016, while the second author was visiting University of Chiba in 2015, and also while the authors were both visiting the Centre Interfacultaire Bernoulli (CIB) at École Polytechnique Fédérale de Lausanne (EPFL) during the Semester ‘Local representation theory and simple groups’ in 2016. The authors are grateful to these institutions for their hospitality. The first author was partially supported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C)15K04776, 2015– 2018; and also by the CIB in EPFL. The second author was financially supported by the German Science Foundation (DFG) Scientific Priority Programme SPP-1478 ‘Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory’.

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Department of Mathematics, Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan.

   E-mail address: koshitan@math.s.chiba-u.ac.jp

Mathematisches Institut, Friedrich-Schiller-Universität, Jena D-07737, Germany.

   E-mail address: juergen.manfred.mueller@uni-jena.de