Loewy lengths of blocks with abelian defect groups
By Charles W. Eaton and Michael Livesey
Abstract
We consider $p$-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 $p$. Using this, we show that if $B$ is a $2$-block of a finite group with abelian defect group $D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s$, where $a_i > 1$ for all $i$ and $r \geq 0$, then $d < \operatorname {LL}(B) \leq 2^{a_1}+\cdots +2^{a_r}+2s-r+1$, where $|D|=2^d$. When $s=1$ the upper bound can be improved to $2^{a_1}+\cdots +2^{a_r}+2-r$. Together these give sharp upper bounds for every isomorphism type of $D$. A consequence is that when $D$ is an abelian $2$-group the Loewy length is bounded above by $|D|$ except when $D$ is a Klein-four group and $B$ is Morita equivalent to the principal block of $A_5$. We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal $3$-blocks.
1. Introduction
Let $G$ be a finite group and $k$ be an algebraically closed field of characteristic $p$. For a block $B$ of $kG$, write $\operatorname {LL}(B)$ for the Loewy length of $B$, that is, the smallest $n \in \mathbb{N}$ such that $\operatorname {Rad}(B)^n=0$ and write $e_B$ for the block idempotent corresponding to $B$. We are interested in upper and lower bounds for $\operatorname {LL}(B)$ in terms of the isomorphism type of the defect groups. For $p$-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 $(p-1)d$, where $d$ is the defect of $B$. The upper bound does not hold when we remove the $p$-solvability hypothesis, as by Reference 7, Theorem 2 the principal $2$-block$B_0(kA_5)$ of $A_5$ (with Klein-four defect groups) has Loewy length $5$, although it is tempting to think that blocks Morita equivalent to $B_0(kA_5)$ are the only counterexamples, as we will see is indeed the case for $p=2$. As remarked in Reference 14 the lower bound $(p-1)d$ also does not hold when we remove the $p$-solvability hypothesis, although $d$ 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).
The upper bounds in the above theorem are sharp for every isomorphism type of $D$: take $B$ to be the principal block of $k(C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times SL_2(2^s))$ (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 $p$-solvable groups with abelian defect groups, we have $\operatorname {LL}(B) = \operatorname {LL}(kD)$. We also note that in Reference 19 it is proved that for principal $2$-blocks with abelian defect groups, the Loewy length is bounded above by the maximum of $2d+1$ and $|D|$.
A crucial element in establishing the above bounds is the consideration of the case where there is $N \vartriangleleft G$ such that $G=ND$. For $D$ elementary abelian we may apply the main result of Reference 13, which says that if $G$ is a split extension of $N$ (by a direct factor of $D$), then $B$ behaves as if $N$ were a direct factor of $G$. Theorem 2.1 generalizes this result and we use it to show how for arbitrary abelian $D$ we may compare the Loewy lengths of $B$ and the block of $N$ covered by $B$. 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 $p$. 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 $p$
We first prove a generalization of a theorem of Koshitani and Külshammer Reference 13.
We now use Theorem 2.1 to prove upper and lower bounds on the Loewy length of $B$ in terms of the Loewy length of $b$. Note that using Reference 5, Theorem 3.4(I) one can derive an upper bound of $p\operatorname {LL}(b)$. However, we require a much more precise bound.
3. Further preliminary results
The following result is well known, but we are not aware of an explicit reference.
Before proceeding we recall the definition and some properties of the generalized Fitting subgroup $F^*(G)$ of a finite group $G$. Details may be found in Reference 4.
A component of $G$ is a subnormal quasisimple subgroup of $G$. The components of $G$ commute, and we define the layer$E(G)$ of $G$ to be the normal subgroup of $G$ generated by the components. It is a central product of the components. The Fitting subgroup$F(G)$ is the largest nilpotent normal subgroup of $G$, and this is the direct product of $O_r(G)$ for all primes $r$ dividing $|G|$. The generalized Fitting subgroup$F^*(G)$ is $E(G)F(G)$. A crucial property of $F^*(G)$ is that $C_G(F^*(G)) \leq F^*(G)$, so in particular $G/F^*(G)$ may be viewed as a subgroup of $\operatorname {Aut}(F^*(G))$.
4. Proof of the main result
5. Other primes
There is relatively little evidence for similar bounds for odd primes, but it is tempting to conjecture the following:
Note that when $D$ is cyclic $\operatorname {LL}(B)$ may be computed from the Brauer tree, and is bounded above by $\left( \frac{|D|-1}{e(B)} \right) e(B) +1 = |D|$, where $e(B)$ is the number of simple $B$-modules (here $\frac{|D|-1}{e(B)}$ is the multiplicity of an exceptional vertex and $e(B)$ is an upper bound for the number of edges emanating from a vertex). By Reference 14, 2.8$\operatorname {LL}(B) \geq \frac{|D|-1}{e(B)} + 1 > d$ since $e(B) \leq p-1$.
We discuss the case $p=3$ below, but mention now that the conjectured bound is achieved in this case by taking the principal block of $C_{3^{a_1}} \times \cdots \times C_{3^{a_r}} \times (M_{23})^{s/2}$ if $s$ is even or $C_{3^{a_1}} \times \cdots \times C_{3^{a_r}} \times (M_{23})^{(s-1)/2} \times C_3$ if $s$ is odd, where $M_{23}$ is the sporadic simple group of that name.
A key special case of Conjecture 5.1 is where $D \cong C_p \times C_p$. There is a relative scarcity of computed examples when $p>3$, and we do not know of any examples in this case which make the upper bound sharp for $s>1$. Hence we ask the following:
In Reference 12 Koshitani describes the finite groups whose Sylow $3$-subgroups are abelian, and in Reference 15 it is shown that any principal $3$-block with abelian defect groups of order $3^2$ has Loewy length $5$ or $7$. Based on this we show the following. Note that at present it is not realistic to calculate the Loewy length for the principal block of the sporadic simple group $O'N$ (with elementary abelian Sylow $3$-subgroups of order $3^4$) by computer or otherwise. Conjecture 5.1 predicts that the Loewy length lies between $5$ and $13$.
Finally we prove the result concerning blocks of $p$-solvable groups with abelian defect groups mentioned in the introduction, which in particular implies that such blocks satisfy sharper bounds than in general. Note that this does not use the classification of finite simple groups.
Acknowledgments
We thank Burkhard Külshammer for suggesting the application of Reference 6 to the determination of Loewy lengths. We also thank the referee for helpful comments. We thank Naoko Kunigi for spotting an error in an earlier version of the paper.
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Show rawAMSref\bib{3682626}{article}{
author={Eaton, Charles},
author={Livesey, Michael},
title={Loewy lengths of blocks with abelian defect groups},
journal={Proc. Amer. Math. Soc. Ser. B},
volume={4},
number={3},
date={2017},
pages={21-30},
issn={2330-1511},
review={3682626},
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}
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