LOEWY LENGTHS OF BLOCKS WITH ABELIAN DEFECT GROUPS.

We consider 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. Using this, we show that if B is a 2-block of a finite group with abelian defect group D ≅= C2a1 × ·· ·×C2ar × (C2)8, where ai > 1 for all i and r ≥ 0, then d < LL(B) ≤ 2a1 + · · · + 2ar + 2s - r + 1, where |D| = 2d. When s = 1 the upper bound can be improved to 2a1 +· · ·+ 2ar + 2 - r. Together these give sharp upper bounds for every isomorphism type of D. A consequence is that when D is an abelian 2-group the Loewy length is bounded above by |D| except when D is a Klein-four group and B is Morita equivalent to the principal block of A5. We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal 3-blocks. [ABSTRACT FROM AUTHOR]