We complete in this Note the description of Singer's fourth transfer, already studied by many authors. More precisely, we show that each element of the family {p_i | i ≥ 0} belongs to the image of this fourth transfer. Combining this with previous results by R. Bruner, L.M. Hà, T.N. Nam and the first author, we deduce that the image of the algebraic transfer contains all the elements of the families {d_i | i ≥ 0}, {e_i | i ≥ 0}, {f_i | i ≥ 0} and {p_i | i ≥ 0}, but none from the families {g_i | i ≥ 1}, {D₃ (i) | i ≥ 0} and {p_i^′ | i ≥ 0}. The method used to prove that elements are in the transfer's image can be applied not only to the family of p_i's but to the families of d_i's, e_i's and f_i's as well. To cite this article: N.H.V. Hu'ng, V.T.N. Quỳnh, C. R. Acad. Sci. Paris, Ser. I 347 (2009). © 2009 Académie des sciences.

Hu'ng N.H.V., Quynh V.T.N.

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