Abstract
Let M(k) be the Mitchell-Priddy summand, which is constructed by Mitchell and Priddy [1]. Mitchell and Priddy [1] also showed that M(k) homeomorphic to L(k) wedge L(k-1). We show that cohomology of L(k) is a free E-module for k=2,3, where E is the exterior algebra generated by Q_0 and Q_1. Then the connective K-theory of L(k) splits into copies of HZ/2, the Z/2 Eilenberg-Mac Lane spectrum, for k=2,3. Hence M(3) splits into copies of HZ/2.