Loewy lengths of blocks with abelian defect groups

By Charles W. Eaton and Michael Livesey

Abstract

We consider -blocks with abelian defect groups and in the first part prove a relationship between its Loewy length and that for blocks of normal subgroups of index . Using this, we show that if is a -block of a finite group with abelian defect group , where for all and , then , where . When the upper bound can be improved to . Together these give sharp upper bounds for every isomorphism type of . A consequence is that when is an abelian -group the Loewy length is bounded above by except when is a Klein-four group and is Morita equivalent to the principal block of . We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal -blocks.

1. Introduction

Let be a finite group and be an algebraically closed field of characteristic . For a block of , write for the Loewy length of , that is, the smallest such that and write for the block idempotent corresponding to . We are interested in upper and lower bounds for in terms of the isomorphism type of the defect groups. For -solvable groups it is proved in Reference 10, p. 141 that the Loewy length is at most the order of a defect group and in Reference 11 that it is strictly greater than , where is the defect of . The upper bound does not hold when we remove the -solvability hypothesis, as by Reference 7, Theorem 2 the principal -block of (with Klein-four defect groups) has Loewy length , although it is tempting to think that blocks Morita equivalent to are the only counterexamples, as we will see is indeed the case for . As remarked in Reference 14 the lower bound also does not hold when we remove the -solvability hypothesis, although could still be a lower bound. In this paper we restrict our attention to blocks with abelian defect groups, but investigate bounds on the Loewy length of a block for arbitrary finite groups.

Using the results of Reference 6, which depends on the classification of finite simple groups, we prove our main result (see Theorem 4.1).

Theorem.

Let be a -block with abelian defect group . Suppose , where for all and . Write .

Then . If , then .

The upper bounds in the above theorem are sharp for every isomorphism type of : take to be the principal block of (see Reference 1). The lower bound was suggested in Reference 14, where there is an excellent discussion of lower bounds on Loewy lengths. Note that for blocks of -solvable groups with abelian defect groups, we have . We also note that in Reference 19 it is proved that for principal -blocks with abelian defect groups, the Loewy length is bounded above by the maximum of and .

A crucial element in establishing the above bounds is the consideration of the case where there is such that . For elementary abelian we may apply the main result of Reference 13, which says that if is a split extension of (by a direct factor of ), then behaves as if were a direct factor of . Theorem 2.1 generalizes this result and we use it to show how for arbitrary abelian we may compare the Loewy lengths of and the block of covered by . This works for all primes, and we hope Theorem 2.1 is of wider interest.

The paper is structured as follows. In Section 2 we give the generalization of the theorem of Reference 13 and the theorem concerning Loewy length of blocks when there is a normal subgroup of index . In Section 3 we give some preliminary results needed for the proof of the bounds, which we give in Section 4. In Section 5 we consider similar bounds for odd primes and make a conjecture.

2. Normal subgroups of index

We first prove a generalization of a theorem of Koshitani and Külshammer Reference 13.

Theorem 2.1.

Let be a finite group and a block of with abelian defect group with primary decomposition . Let such that for some , for all and . Let be a block of covered by . Then and there exists an element of multiplicative order dividing such that .

Proof.

We follow the proof of Reference 13, where it is assumed that . Let be a root of in and the stabilizer of in . As in Reference 13, , is a crossed product of over and is a -group.

Now acts on both and by conjugation. Consider the subgroup generated by and all its -conjugates. This group has exponent and so by Reference 9, , Theorem 2.2 , where and are -invariant, is homocyclic of exponent and has exponent strictly less than . As we have , where is cyclic of order and is homocyclic of exponent . Again by Reference 9, , Theorem 2.2 we can assume and are both -invariant and therefore by Reference 9, , Theorem 3.2 must have an -invariant complement in . So is cyclic, -invariant, and .

Now and the natural homomorphism is surjective and so any non-trivial -element of acts non-trivially on . However, and so every element of acts trivially on . Therefore, commutes with .

Let . By Watanabe’s result Reference 22, Theorem 2(ii), the map

is an isomorphism of -algebras. Note that since commutes with we have that and also . Therefore and hence are -graded. Recall that is also -graded and since , respects these gradings. Setting , where , proves the theorem.

We now use Theorem 2.1 to prove upper and lower bounds on the Loewy length of in terms of the Loewy length of . Note that using Reference 5, Theorem 3.4(I) one can derive an upper bound of . However, we require a much more precise bound.

Corollary 2.2.

Assume we are in the setting of Theorem 2.1. Then

(i) ;

(ii) .

Proof.

(i) Certainly and are nilpotent ideals of and so

Then

where the last isomorphism is given by , where . Therefore

is semisimple and so .

(ii) If with , then is a non-zero product of elements of , and so .

Next suppose with with . Then by (i) there exist such that and each is either or is an element of . Say of the ’s are ’s. Then clearly and since we also have that . As increasing can only decrease the left hand side of this inequality, it is also true that and so giving that and the theorem is proved.

3. Further preliminary results

The following result is well known, but we are not aware of an explicit reference.

Lemma 3.1.

Let be a finite group and be a -block of with defect group . Let and let be a block of covered by . If , then .

Proof.

The proof given in Reference 15, Lemma 4.1 for the case is a -group carries through in the general situation, but we include a simple argument suggested by Markus Linckelmann.

Let be the stabilizer of in . Since there is a block of covering and Morita equivalent to , we may assume that . Then . Since , every -module is relatively -projective, and so in particular and every quotient module is. Now the restriction of to is a semisimple -module, so it follows that is semisimple as a -module. Hence , and so . Again since is -stable, we have .

Lemma 3.2.

Let be finite groups and . For each let be a block of and let be the block of covering each . Let be a defect group of , so is a defect group of . Then .

In particular, suppose , and , where each . If for each , then . If, in addition, for at least one , then .

Proof.

The first part is Reference 3, 1.1 and the second part is just a direct application of the first.

Before proceeding we recall the definition and some properties of the generalized Fitting subgroup of a finite group . Details may be found in Reference 4.

A component of is a subnormal quasisimple subgroup of . The components of commute, and we define the layer of to be the normal subgroup of generated by the components. It is a central product of the components. The Fitting subgroup is the largest nilpotent normal subgroup of , and this is the direct product of for all primes dividing . The generalized Fitting subgroup is . A crucial property of is that , so in particular may be viewed as a subgroup of .

Lemma 3.3.

Let be a block of a finite group with abelian defect group . Then there is a group and a block of with defect group such that and the following are satisfied:

(i) is quasiprimitive, that is, for every normal subgroup of , every block of covered by is -stable.

(ii) is generated by the defect groups of .

(iii) If and covers a nilpotent block of , then . In particular, .

(iv) .

(v) Every component of is normal in .

(vi) If is the principal block of , then is the principal block of .

Proof.

Consider pairs with the lexigraphic ordering. There are three processes, labelled (a), (b), (c), which will be applied repeatedly and in various combinations. We describe these processes and show that they strictly reduce when applied non-trivially, so that repeated application of (a), (b) and (c) terminates.

(a) Let and let be a block of covered by . Write for the stabilizer of in , and for the Fong-Reynolds correspondent. Now is Morita equivalent to and they have isomorphic defect groups. Clearly , and if , then . Process (a) involves replacing by .

(b) is the replacement of by and by any block of covered by , as in Lemma 3.1. In this case and if . Since defect groups of are intersections of defect groups of with (see Reference 2, 15.1), it follows that and share a defect group.

(c) Let and suppose that covers a nilpotent block of such that . Let be a block of covered by and covering . By performing (a) first we may assume that is -stable. Further must also be nilpotent. Using the results of Reference 17, as outlined in Reference 6, Proposition 2.2, is Morita equivalent to a block of a central extension of a finite group by a -group such that there is an with