  Let V be a mod 2 vector space of rank k. W. Singer defined a transfer
homomorphism from the GL(k,2) coinvariants of the primitives in the homology of
BV to the cohomology of the Steenrod algebra, as an algebraic version of the
geometric transfer from the stable homotopy of BV to the stable homotopy of
spheres. It has been shown that the algebraic transfer is highly nontrivial
and, more precisely, that it is an isomorphism for k=1, 2, or 3. However,
Singer showed that it is not an epimorphism for k=5. In this paper, we prove
that it also fails to be an epimorphism when k=4. Precisely, it does not detect
the non zero elements in the g family, in stems 20, 44, 92, and in general,
12*2^s - 4, for each s > 0. The transfer still fails to be an epimorphism even
after inverting Sq^0, thereby giving a negative answer to a prediction by
Minami.
