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Let Pk be the graded polynomial algebra F2[x1, x2, . . . , xk] with the degree of each generator xi being 1, where F2 denote the prime field of two elements, and let GLk be the general linear group over F2 which acts regularly on Pk.
We study the algebraic transfer constructed by Singer  using the technique of the Peterson hit problem. This transfer is a homomorphism from the homology of the mod-2 Steenrod algebra A, TorAk,k+d(F2, F2), to the subspace of F2xAPk consisting of all the GLk-invariant classes of degree d.
In this paper, by using the results on the Peterson hit problem we present the proof of the fact that the Singer algebraic transfer is an isomorphism for k <= 3. This result has been proved by Singer in  for k <= 2 and by Boardman in  for k = 3. We show that the fourth Singer transfer is also an isomorphism in certain internal degrees. This result is new and it is different from the ones of Bruner, Ha and Hung , Chon and Ha , Ha , Hung and Quynh , Nam .