Broué's abelian defect group conjecture for arbitrary finite groups and its missing reduction to quasi-simple groups

Broué's abelian defect group conjecture (circa 1990) predicts that whenever a $p$-block $A$ of a finite group $G$ has an abelian defect group $P$, then $A$ is derived equivalent to its Brauer correspondent block $B$ of the normalizer $N_G(P)$; Rickard's stronger splendid equivalence form predicts a splendid derived equivalence. The source paper (Koshitani–Müller–Noeske, 2013/2015) verified both statements for the 3-blocks of the sporadic Conway group $Co_1$ and flagged two still-open directions: the general conjecture is wide open, with no systematic reduction of it to finite quasi-simple groups, and the decomposition-matrix data needed for further verifications (Modular Atlas project) is incomplete. The 2014–2026 literature confirms the source's assessment: the conjecture continues to be verified only class-by-class (most recently for $2$-blocks with abelian defect groups and inertial quotient of prime order, April 2026), no reduction theorem to quasi-simple groups has been found, and no general proof or counterexample exists. The problem is recorded as open at full generality: prove or refute Broué's conjecture for all finite groups, with the reduction to quasi-simple groups as the source-flagged strategic sub-question.

Background

Modular representation theory studies finite groups over a complete $p$-modular system $(K,\mathcal{O},k)$, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic $0$, $k$ is its residue field of prime characteristic $p$ (taken algebraically closed), and $K$ is the field of fractions. The group algebra $kG$ (or $\mathcal{O}G$) decomposes as a direct product of indecomposable two-sided ideals called blocks, each corresponding to a primitive central idempotent; a block is the correct organizational unit for the irreducible modular (Brauer) characters and the decomposition numbers that express ordinary irreducible characters as sums of Brauer characters. To each block $A$ is attached a conjugacy class of defect groups, which are $p$-subgroups $P$ of $G$ measuring the failure of the block to be semisimple; by Brauer's main theorems there is a Brauer correspondent block $B$ of the group algebra of the normalizer $N_G(P)$, and the block theory of $G$ is to a large extent controlled by this purely local, $p$-subgroup-level data. A derived equivalence between two algebras is an equivalence of their bounded derived categories of modules, a notion from Rickard's derived Morita theory that is coarse enough to be tractable yet fine enough to preserve numerical invariants such as decomposition matrices and Cartan matrices. Around 1990 Michel Broué conjectured, by analogy with what is provable for blocks with cyclic (i.e., generated by one element) defect groups, that whenever the defect group $P$ is abelian the block $A$ of $\mathcal{O}G$ and its Brauer correspondent $B$ of $\mathcal{O}N_G(P)$ are derived equivalent. Rickard strengthened this in 1996 to a splendid equivalence: the equivalence should be induced by an explicit complex of bimodules whose terms are direct sums of permutation modules with vertex the diagonal $\Delta P$, a condition tying the equivalence tightly to the local subgroup structure. Despite a substantial verified record — blocks with cyclic defect groups (classical), principal 3-blocks with elementary abelian defect group of order $9$ (Koshitani–Kunugi, 2002), all blocks of symmetric groups via $\mathfrak{sl}_2$-categorification (Chuang–Rouquier, 2008), blocks of $p$-solvable groups (Harris–Linckelmann's reduction framework; refined by Miller, 2024), many sporadic groups one group at a time (e.g., the Held and sporadic Suzuki groups, the Harada–Norton group, and the Conway groups $Co_3$ and $Co_1$), $\mathrm{SL}(2,p^n)$ and $\mathrm{GL}(2,p^n)$ (Huang–Li–Zhang, 2023), and, in April 2026, all $2$-blocks whose defect group is abelian and whose inertial quotient (the quotient $N_G(P,e_P)/P\mathrm{C}_G(P)$ controlling non-nilpotent local action) has prime order (Zhou–Zhang) — the conjecture remains open in general. A key structural gap, flagged in the source paper, is that unlike the situation for counting conjectures such as Alperin's weight conjecture or the McKay conjecture, no reduction theorem is known that would deduce Broué's conjecture for all finite groups from its verification on the finite list of quasi-simple groups (perfect groups $G$ with $G/Z(G)$ simple, the building blocks supplied by the classification of finite simple groups); all known verifications proceed class-by-class using structure special to each class, for instance perverse equivalences (a type of derived equivalence compatible with a filtration, developed by Craven–Rouquier) for groups of Lie type and computational modular character theory for sporadic groups, the latter depending on decomposition-matrix data whose complete determination for all simple groups is itself the still-unfinished Modular Atlas project.

Problem Statement

Let $(K,\mathcal{O},k)$ be a complete $p$-modular system with $k$ algebraically closed of characteristic $p$, let $G$ be an arbitrary finite group, let $A$ be a block algebra of $\mathcal{O}G$ with abelian defect group $P$, and let $B$ be the Brauer correspondent block of $\mathcal{O}N_G(P)$, where $N_G(P)$ is the normalizer of $P$ in $G$: by Brauer's first main theorem there is a conjugacy-class-preserving bijection between the blocks of $G$ with defect group $P$ and the blocks of $N_G(P)$ with defect group $P$, and $B$ denotes the block of $\mathcal{O}N_G(P)$ paired with $A$ under this bijection. Broué's abelian defect group conjecture asserts that $A$ and $B$ are derived equivalent, i.e. that there exists a triangulated equivalence between the bounded derived categories of finitely generated modules, $D^b(\mathrm{mod}\,A)\cong D^b(\mathrm{mod}\,B)$; Rickard's splendid equivalence strengthening asserts, in addition, that the equivalence is induced by a splendid complex of $(A,B)$-bimodules whose terms are sums of permutation bimodules with vertex the diagonal subgroup $\Delta P=\{(x,x):x\in P\}$. The problem is to settle this statement in full generality — either prove it for every prime $p$ and every block of every finite group with abelian defect group (in the derived form, and ideally in Rickard's splendid form), or refute it by an explicit block with abelian defect group that is not derived equivalent to its Brauer correspondent. As a source-flagged strategic sub-question, for which no such theorem is currently known in the literature: does there exist a reduction theorem deducing the conjecture for all finite groups from its verification for finite quasi-simple groups (perfect groups $G$ with $G/Z(G)$ simple, the building blocks supplied by the classification of finite simple groups), analogous to the known reductions for the McKay and Alperin weight conjectures? Completion of the main objective is a proof or refutation of the general statement; a reduction theorem alone would constitute decisive progress but not closure.

References

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