Abstract. We consider functorial decompositions of $OmegaSigma X$ in the case where $X$ is a $p$-torsion suspension. By means of a geometric realization theorem, we show that the problem can be reduced to the one obtained by applying homology: that of finding natural coalgebra decompositions of tensor algebras. We solve the algebraic problem and give properties of the piece $A^{mathrm {min}} (V)$ of the decomposition of $T(V)$ which contains $V$ itself, including verification of the Cohen conjecture that in characteristic $p$ the primitives of $A^{mathrm {min}} (V)$ are concentrated in degrees of the form $p^t$. The results tie in with the representation theory of the symmetric group and in particular produce the maximum projective submodule of the important $S_n$-module $mathrm {Lie} (n)$.