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On Loewy lengths of blocks. (English) Zbl 1329.20009
From the introduction: We give a lower bound on the Loewy length of a \(p\)-block of a finite group in terms of its defect. We then discuss blocks with small Loewy length. Since blocks with Loewy length at most 3 are known, we focus on blocks of Loewy length 4 and provide a relatively short list of possible defect groups. It turns out that \(p\)-solvable groups can only admit blocks of Loewy length 4 if \(p=2\). However, we find (principal) blocks of simple groups with Loewy length 4 and defect 1 for all \(p\equiv 1\pmod 3\). We also consider sporadic, symmetric and simple groups of Lie type in defining characteristic. Finally, we give stronger conditions on the Loewy length of a block with cyclic defect group in terms of its Brauer tree.
Let \(F\) be an algebraically closed field of characteristic \(p>0\), and let \(B\) be a block of the group algebra \(FG\) of a finite group \(G\) over \(F\). Moreover, let \(D\) be a defect group of \(B\). We denote the inertial index of \(B\) by \(e(B)\).
For a finite-dimensional \(F\)-algebra \(A\), we denote by \(J(A)\) the Jacobson radical and by \(LL(A)\) the Loewy length of \(A\). Similarly, we denote by \(LL(M)\) the Loewy length of a finitely generated \(A\)-module \(M\). For a finite \(p\)-group \(P\), we denote by \(r(P)\) its rank and by \(\exp(P)\) its exponent.
One aim of this paper is to give a general lower bound on \(LL(B)\) in terms of the defect of \(B\). This is established in the next section by making use of work by S. Oppermann [Math. Z. 256, No. 3, 481-490 (2007; Zbl 1135.16007)] and B. Külshammer [J. Algebra 75, 59-69 (1982; Zbl 0488.16010)]. Since this inequality is usually very crude, we provide a different approach in terms of a certain fixed point algebra on \(Z(D)\). Here our result on the Loewy length of a fixed point algebra might be of general interest. Finally, for blocks with cyclic defect groups we express the Loewy length via Brauer trees.
The third section deals with blocks of small Loewy length. After stating the known result about Loewy length at most 3, we determine the possible defect groups for blocks with Loewy length 4. For fixed \(p\geq 5\) we get at most 12 isomorphism types of these groups. Since blocks of small Loewy length in solvable groups are well understood, we turn to blocks of (almost) (quasi)simple groups. Symmetric (and thus also alternating) groups can be completely handled, while for sporadic groups and simple groups of Lie type in defining characteristic we restrict to principal blocks. Here we develop general methods and reductions.

MSC:
20C20 Modular representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C30 Representations of finite symmetric groups
20C33 Representations of finite groups of Lie type
20C34 Representations of sporadic groups
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