On blocks stably equivalent to a quantum complete intersection of dimension 9 in characteristic 3 and a case of the abelian defect group conjecture

Using a stable equivalence due to Rouquier, we prove that Broue's abelian defect group conjecture holds for 3-blocks of defect 2 whose Brauer correspondent has a unique isomorphism class of simple modules. The proof makes use of the fact, also due to Rouquier, that a stable equivalence of Morita type between self-injective algebras induces an isomorphism between the connected components of the outer automorphism groups of the algebras

On perfect isometries for tame blocks

Any 2-block of a finite group G with a quaternion defect group Q8 is Morita equivalent to the corresponding block of the centraliser H of the unique involution of Q8 in G; this answers positively an earlier question raised by M. Broué

A block theoretic analogue of a theorem of Glauberman and Thompson

If p is an odd prime, G a finite group and P a Sylow-p-subgroup of G, a theorem of Glauberman and Thompson states that G is p-nilpotent if and only if NG(Z(J(P))) is p-nilpotent, where J(P) is the Thompson subgroup of P generated by all abelian subgroups of P of maximal order. Following a suggestion of G. R. Robinson, we prove a block-theoretic analogue of this theorem

On two theorems of Flavell

We extend two theorems due to P. Flavell [6] to arbitrary fusion systems

ZJ-theorems for fusion systems

For p an odd prime, we generalise the Glauberman-Thompson p-nilpotency theorem [5, Ch. 8, Theorem 3.1] to arbitrary fusion systems. We define a notion of Qd(p)- free fusion systems and show that if F is a Qd(p)-free fusion system on some finite p-group P then F is controlled by W(P) for any Glauberman functor W, generalising Glauberman’s ZJ-theorem [3] to arbitrary fusion systems

Symmetric groups, wreath products, Morita equivalences, and Broue's abelian defect group conjecture

It is shown that for any prime p, and any non-negative integer w less than p, there exist p-blocks of symmetric groups of defect w, which are Morita equivalent to the principal p-block of the group Sp [rmoust ] Sw. Combined with work of J. Rickard, this proves that Broué's abelian defect group conjecture holds for p-blocks of symmetric groups of defect at most 5

The graded center of the stable category of a Brauer tree algebra

We calculate the graded center of the stable category of a Brauer tree algebra. The canonical map from the Tate analogue of Hochschild cohomology to the graded center of the stable category is shown to induce an isomorphism module taking quotients by suitable nilpotent ideals. More precisely, we show that this map is surjective with nilpotent kernel in even degrees, while this map need not be surjective in odd degrees in general

On blocks of strongly p-solvable groups

We prove that a block of a finite strongly p-solvable group G with defect group P is Morita equivalent to its corresponding block of NG(Z(J(P))) via a bimodule with endo-permutation source

On isotypies between Galois conjugate blocks

We show that between any pair of Galois conjugate blocks of finite group algebras, there exists an isotypy with all signs positive

The structure of blocks with a Klein four defect group

We prove Erdmann’s conjecture [16] stating that every block with a Klein four defect group has a simple module with trivial source, and deduce from this that Puig’s finiteness conjecture holds for source algebras of blocks with a Klein four defect group. The proof uses the classification of finite simple groups