Weights and Nilpotent Subgroups

In a finite group $G$, we consider nilpotent weights and prove a $\pi $-version of the Alperin Weight Conjecture for certain $\pi $-separable groups. This widely generalizes an earlier result by I. M. Isaacs and the 1st author.

The Inductive Blockwise Alperin Weight Condition for PSp4(q)

Let G be a finite group and ℓ be any prime dividing [Formula: see text]. The blockwise Alperin weight conjecture states that the number of the irreducible Brauer characters in an ℓ-block B of G equals the number of the G-conjugacy classes of ℓ-weights belonging to B. Recently, this conjecture has been reduced to the simple groups, which means that to prove the blockwise Alperin weight conjecture, it suffices to prove that all non-abelian simple groups satisfy the inductive blockwise Alperin weight condition. In this paper, we verify this inductive condition for the finite simple groups [Formula: see text] and non-defining characteristic, where q is a power of an odd prime.

The Inductive Blockwise Alperin Weight Condition for PSL(3, q)

The blockwise Alperin weight conjecture assets that for any finite group G and any prime l, the number of the Brauer characters in an l-block B equals the number of the G-conjugacy classes of l-weights belonging to B. Recently, the inductive blockwise Alperin weight condition has been introduced such that the blockwise Alperin weight conjecture holds if all non-abelian simple groups satisfy these conditions. We will verify the inductive blockwise Alperin weight condition for the finite simple groups PSL(3, q) in this paper.

A reduction theorem for the blockwise Alperin weight conjecture

We show that the blockwise version of the Alperin weight conjecture is true if for every finite non-abelian simple group a set of conditions holds. Furthermore we prove that several series of simple groups satisfy these assumptions. This refines recent work of Navarro and Tiep, who proved an analogous reduction theorem for the non-blockwise version of the Alperin weight conjecture.

The Alperin weight conjecture and Uno’s conjecture for the Monster 𝕄, p odd

Suppose thatpis 3,5,7,11 or 13. We classify the radicalp-chains of the Monster 𝕄 and verify the Alperin weight conjecture and Uno’s reductive conjecture for 𝕄, the latter being a refinement of Dade’s reductive conjecture and the Isaacs–Navarro conjecture.