2-blocks with abelian defect groups and inertial quotient of prime order

Abstract: In this paper, we classify all $2$-blocks for which the defect groups are abelian and the inertial quotient has prime order. As a consequence, we prove that Broué's abelian defect group conjecture holds for all blocks under consideration here.

• The paper presents a comprehensive classification theorem for 2-blocks, detailing structural and Morita equivalence results.
• It employs case analyses including inertial, Klein four subgroup cases, and a unique scenario linked to A₁(2ᵃ) with Mersenne prime conditions.
• The results confirm Broué's abelian defect group conjecture and support algorithmic applications in modular representation theory.

Classification of 2-Blocks with Abelian Defect Groups and Inertial Quotient of Prime Order

Abstract and Context

The paper "2-blocks with abelian defect groups and inertial quotient of prime order" (2604.09974) provides a complete classification of 2-blocks of finite groups possessing abelian defect groups and an inertial quotient of prime order. The investigation yields precise structural and Morita equivalence results in these cases and, crucially, verifies Broué's abelian defect group conjecture for all such blocks.

Main Results

The central result is the comprehensive classification theorem for the specified 2-blocks. Let $b$ be a 2-block of a finite group $G$ with abelian defect group $P$ and inertial quotient $\mathbb{E}(b)$ of prime order. Then either:

1. $b$ is inertial;
2. the hyperfocal subgroup is a Klein four-group;
3. $b$ is basic Morita equivalent to the principal block of $A_1(2^a)\times R$, where $2^a-1$ is prime and $R$ is an abelian 2-group.

As a consequence, Broué’s abelian defect group conjecture is established for all blocks in this classification.

Theoretical Framework

The classification is built atop substantial prior results and structural theorems concerning reduced blocks, quasi-primitive blocks, and the local-global relationships in block theory. The paper utilizes the framework provided by:

• The classification of 2-blocks with abelian defect groups for quasi-simple groups [Eaton et al., Adv. Math. 254, 706-735 (2014)],
• Morita equivalence strategies for block algebras,
• Properties of inertial blocks and blocks with controlled hyperfocal subgroups,
• The structure theory of finite groups and the role of components and their centralizers.

The concept of inertial quotient, $\mathbb{E}(b)$, plays a pivotal role; it is the group of automorphisms of a defect group arising from the action of $G$0 on a maximal Brauer pair, modulo inner automorphisms by elements of the centralizer.

Methodology and Proof Strategy

The argument divides into cases according to the nature of the defect group and the possible influences from quasi-simple components. Key aspects are:

• If $G$1 is normal, $G$2 is inertial by general block theory.
• If $G$3 covers only nilpotent-covered blocks in components, $G$4 is again inertial by a combination of results on reduced blocks and inertial blocks.
• The analysis of non-nilpotent covered cases proceeds via the explicit classification of possible quasi-simple groups, showing that only types arising from $G$5 with $G$6 prime (i.e., Mersenne primes) can occur outside the inertial and Klein four cases.
• Outlier configurations, such as blocks of groups $G$7, $G$8, or $G$9, are eliminated by group-theoretic and character-theoretic arguments, including counting irreducible (Brauer) characters and leveraging the Alperin weight conjecture.
• The case of Klein four hyperfocal subgroup is resolved by linking the possible non-trivial actions via the inertial quotient and analyzing the consequences for block structure and Morita equivalence.

Explicit control of the structure—via normality of components, analysis of $P$0 over centralizer extensions, and subtleties in the action of the inertial quotient—yields the final trichotomy.

Numerical and Structural Highlights

• Sharpness of Classification: The cases are mutually exclusive and jointly exhaustive for blocks with the given local data (abelian defect, prime inertial quotient).
• Specificity: The identification of $P$1 with $P$2 prime as the only possible non-inertial, non-Klein-four scenario is a strong claim, highlighting the rigidity of block structure under these constraints.
• Verification of Broué's Conjecture: For all blocks classified, the Broué abelian defect group conjecture—derived equivalence of the block with its Brauer correspondent—holds.

Implications and Future Directions

Practical Implications

• Modular Representation Theory: These results provide explicit Morita equivalence and derived equivalence information for a broad, structurally significant class of 2-blocks, directly informing computational and theoretical work in modular representation theory.
• Algorithmic Applications: The classification aids practical algorithms in computational group theory (e.g., block recognition and identification in group algebras) by narrowing the field of possible block structures under observable local invariants.

Theoretical Directions

• Extension to Other Primes and Non-Abelian Defects: While focused on 2-blocks and abelian defect, the techniques and insights pave the way for further classification results at other primes or for blocks with more general defect structures, especially those with inertial quotients of prime—or otherwise restricted—order.
• Derived Equivalence and Local-Global Conjectures: The confirmation of Broué's conjecture strengthens evidence for related local-global conjectures at large and motivates efforts for their full resolution in broader classes.
• Connections with Finite Simple Group Theory: The exclusion of most quasi-simple types once the inertial quotient is prime adds further depth to the intricate relationship between local block invariants and the global structure of finite simple and quasi-simple groups.

Conclusion

The paper provides a thorough classification of all 2-blocks with abelian defect groups and inertial quotient of prime order, resolving their Morita and derived equivalence types and verifying Broué's abelian defect group conjecture in these settings. The careful blend of local block theory, classification of simple group blocks, and structural group analysis exemplifies a mature synthesis of modern modular representation theory and finite group theory, potentially guiding future developments in the structural study of block algebras.