Abstract.This paper proves the Alperin's weight conjecture for the finite unitary groups when the characteristic r of modular representation is odd. Moreover, this paper proves the conjecture for finite odd dimensional special orthogonal groups and gives a combinatorial way to count the number of weights, block by block, for finite symplectic and even dimensional special orthogonal groups when r and the defining characteristic of the groups are odd.

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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

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We modify the local strategy of [2] and use it to classify the radical subgroups and chains of the Fischer simple group Fi 23. We verify the Alperin weight conjecture and the Dade final conjecture for this group.

This paper is part of a program to study the conjecture of E. C. Dade on counting characters in blocks for several finite groups of Lie type. The local structures of certain radical chains of Chevalley groups of type G2 are given and the ordinary conjecture is confirmed for the groups when the characteristic of the modular representation is distinct from the defining characteristic of the groups.