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AbstractSuppose thatpis 3, 5 or 7. In this paper, faithful permutation representations of maximalp-local subgroups are constructed, and the radicalp-chains of the Baby Monster B are classified. Hence, the Alperin weight conjecture and the Uno reductive conjecture can be verified for B, the latter being a refinement of Dade's reductive conjecture and the Isaacs-Navarro conjecture.doi:10.1112/s1461157000001078 fatcat:5sclyu7eira7ppc65klcxva6la