Loewy lengths of blocks with abelian defect groups

Abstract

We consider p p -blocks with abelian defect groups and in the first part prove a relationship between its Loewy length and that for blocks of normal subgroups of index p p . Using this, we show that if B B is a 2 2 -block of a finite group with abelian defect group D ≅ C 2 a 1 × ⋯ × C 2 a r × ( C 2 ) s D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s , where a i > 1 a_i > 1 for all i i and r ≥ 0 r \geq 0 , then d > LL ⁡ ( B ) ≤ 2 a 1 + ⋯ + 2 a r + 2 s − r + 1 d > \operatorname {LL}(B) \leq 2^{a_1}+\cdots +2^{a_r}+2s-r+1 , where | D | = 2 d |D|=2^d . When s = 1 s=1 the upper bound can be improved to 2 a 1 + ⋯ + 2 a r + 2 − r 2^{a_1}+\cdots +2^{a_r}+2-r . Together these give sharp upper bounds for every isomorphism type of D D . A consequence is that when D D is an abelian 2 2 -group the Loewy length is bounded above by | D | |D| except when D D is a Klein-four group and B B is Morita equivalent to the principal block of A 5 A_5 . We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal 3 3 -blocks.