[Paper Review] Block fusion systems of the alternating groups

Martin Jacobsen|arXiv (Cornell University)|Apr 12, 2012

Advanced Algebra and Geometry1 references3 citations

TL;DR

This paper establishes that the block fusion system of any block in an alternating group is isomorphic to the group fusion system of an alternating group. Using a group-theoretic condition on elements of order coprime to the characteristic, the authors prove that certain central idempotents in the group ring vanish, enabling a structural analysis of defect groups and centric subgroups. The key result is that every block fusion system of an alternating group arises as the fusion system of an alternating group on a suitable subset of the underlying set.

ABSTRACT

We describe a purely group-theoretic condition on an element g of a finite group G which implies that g has coefficient zero in every central idempotent element of the group ring RG, provided that R is a ring of prime characteristic. We use this condition to prove that the fusion system associated to a block of an alternating group is always isomorphic to the group fusion system of an alternating group.

Motivation & Objective

• To determine whether block fusion systems of alternating groups are isomorphic to group fusion systems of alternating groups.
• To extend known results on symmetric groups to the alternating groups.
• To establish a group-theoretic criterion for elements to have zero coefficient in central idempotents over rings of prime characteristic.
• To analyze the structure of defect groups and centric subgroups in alternating group blocks.
• To show that every block fusion system of an alternating group arises as the fusion system of an alternating group on a subset of the index set.

Proposed method

• Develop a group-theoretic condition on elements g in a finite group G such that g has coefficient zero in all central idempotents of RG when R has prime characteristic.
• Apply this condition to symmetric and alternating groups to restrict possible defect groups.
• Analyze the centric subgroups of defect groups and their automorphism groups in the context of block fusion systems.
• Use Alperin's fusion theorem to reduce the problem to verifying automorphism group equality on centric subgroups.
• Leverage Brauer pairs and the Brauer homomorphism to compare fusion systems between symmetric and alternating groups.
• Construct a subset L of {1,…,n} such that the block fusion system is realized as the fusion system of A_L.

Experimental results

Research questions

• RQ1 Does every block fusion system of an alternating group arise as the fusion system of an alternating group on a subset of the index set?
• RQ2 What group-theoretic condition ensures that an element of a finite group has zero coefficient in all central idempotents over a ring of prime characteristic?
• RQ3 How do the automorphism groups of centric subgroups in defect groups of alternating group blocks compare to those in symmetric and alternating groups?
• RQ4 Under what conditions does the fusion system of a block in an alternating group coincide with the fusion system of a symmetric group or an alternating group?
• RQ5 Can the block fusion system of an alternating group be realized as the fusion system of an alternating group on a proper subset of the original index set?

Key findings

• The block fusion system of any block in an alternating group is isomorphic to the fusion system of an alternating group on a suitable subset of the index set.
• For any block idempotent e of A_n with defect group P, there exists a subset L ⊆ {1,…,n} such that P is a Sylow p-subgroup of A_L and the block fusion system on P equals F_P(A_L).
• When the fusion system is isomorphic to F_P(S_M), the complement of M in {1,…,n} must contain at least two elements, allowing the construction of a larger alternating group A_L with the same fusion system.
• If the block fusion system is isomorphic to F_P(S_M), then adding two elements to M to form L ensures that F_P(A_L) = F_P(S_M), so the system is realized as a group fusion system of A_L.
• In all cases, the fusion system is realized as F_P(A_L) for some L, proving that block fusion systems of alternating groups are always group fusion systems of alternating groups.
• The proof relies on analyzing the action of elements in the symmetric group on fixed points of the defect group and using the Brauer homomorphism to control the number of Brauer pairs associated to a block.