  In the representation theory of finite groups, there is a well-known and
important conjecture, due to Brou\'e saying that for any prime p, if a p-block
A of a finite group G has an abelian defect group P, then A and its Brauer
corresponding block B of the normaliser N_G(P) of P in G are derived
equivalent. We prove in this paper, that Brou\'e's abelian defect group
conjecture, and even Rickard's splendid equivalence conjecture are true for the
faithful 3-block A with an elementary abelian defect group P of order 9 of the
double cover 2.HS of the Higman-Sims sporadic simple group. It then turns out
that both conjectures hold for all primes p and for all p-blocks of 2.HS.
