A Local Strategy to Decide the Alperin and Dade Conjectures

Abstract

We present a new strategy which exploits both the maximal and p-local subgroup structure of a given finite simple group in order to decide the Alperin and Dade conjectures for this group. We demonstrate the computational effectiveness of this approach by using it to verify these conjectures for the Conway simple group Co . ᮊ

The Alperin and Uno Conjectures for the Fischer Simple Group Fi22

“…In [3] and [4], a modified local subgroup strategy was developed to classify the radical subgroups R. We review this method here. Suppose M is a subgroup of G such that N M (R) = N G (R).…”

“…Its maximal subgroups were constructed using the details supplied in [9] and the black-box algorithms of Wilson [20]. We also made extensive use of the algorithm described in [10] to construct random elements, and the procedures described in [3] and [4] The computations reported in this paper were carried out using Magma V.2.6-2 on a Sun UltraSPARC Enterprise 4000 server.…”

“…In [3] and [4], we presented a local subgroup strategy to decide the Alperin and Dade conjectures for the finite simple groups and demonstrated its computational effectiveness by using it to verify the Alperin and Dade conjectures for the Conway simple group Co 2 and the Fischer simple group Fi 23 . In this paper, we apply the strategy to verify the Alperin and Uno conjectures for the Fischer simple group Fi 22 .…”

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AbstractUsing the local subgroup strategy of [3] and [4], we classify the radical subgroups and chains of the Fischer simple group Fi 22 and verify the Alperin weight conjecture and the Uno reductive conjecture for this group; the latter is a refinement of the Dade reductive and Isaacs-Navarro conjectures.

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The Alperin Weight Conjecture and Uno's Conjecture for the Baby Monster B, p Odd

“…In Section 3, we explain how to construct faithful permutation representations of the thirteen maximal p-local subgroups. In Section 4, we recall the modified local strategy [4,5]; we also explain how we applied it to determine the radical subgroups of each maximal subgroup, and how to determine the fusion of the radical subgroups in B. In Section 5, we classify radical p-subgroups of B, and verify the Alperin weight conjecture.…”