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

Abstract

Suppose that p is 3, 5 or 7. In this paper, faithful permutation representations of maximal p-local subgroups are constructed, and the radical p-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.

Base sizes for sporadic simple groups

“…As observed in the proof of [22, 1.3], a c-tuple in Ω fails to be a base for G if and only if it is fixed by some x ∈ G of prime order, and the probability that a random c-tuple is fixed by x is at most fpr(x, Ω) c . Since G is transitive, fpr(x, Ω) = fpr(y, Ω) for all y ∈ x G (see (1)) and thus…”

“…The representations for (3)-(5) and(7)were constructed by An and Wilson (see Section 3 of[1]). Cases (6), (8) and (9) are straightforward.…”

The principal p‐blocks with four irreducible characters

“…By [14,39] it is enough to check the following four cases for G. 2 where the four groups are the Thompson simple group, the derived subgroup of Fischer's 24 group, the baby monster and the monster, respectively. So we get the assertion for these by GAP for the Th for p = 7, [2][3][4], respectively. Lemma 3.4.…”

The principal $p$-blocks with small numbers of characters

“…(1)Th and |D| = 7 2 , (2) Fi 24 and |D| = 5 2 , (3) B and |D| = 7 2 , (4) M and |D| = 11 2 where the four groups are the Thompson simple group, the derived subgroup of Fischer's 24 group, the baby monster and the monster, respectively. So we get the assertion for these by GAP for the Th for p = 7, [3], [4], [5], respectively. Lemma 2.11.…”