  Let $\Sigma_d$ denote the symmetric group of degree $d$ and let $K$ be a
field of positive characteristic $p$. For $p>2$ we give an explicit description
of the first cohomology group $H^1(\Sigma_d,{\rm{Sp}}(\lambda))$, of the Specht
module ${\rm{Sp}}(\lambda)$ over $K$, labelled by a partition $\lambda$ of $d$.
We also give a sufficient condition for the cohomology to be non-zero for $p=2$
and we find a lower bound for the dimension. Our method is to proceed by
comparison with the cohomology for the general linear group $G(n)$ over $K$ and
then to reduce to the calculation of ${\rm{Ext}}^1_{B(n)}(S^d E,K_\lambda)$,
where $B(n)$ is a Borel subgroup of $G(n)$, $S^dE$ denotes the $d$th symmetric
power of the natural module $E$ for $G(n)$ and $K_\lambda$ denotes the one
dimensional $B(n)$-module with weight $\lambda$. The main new input is the
description of module extensions by: extensions sequences, coherent triples of
extension sequences and coherent multi-sequences of extension sequences, and
the detailed calculation of the possibilities for such sequences. These
sequences arise from the action of divided powers elements in the negative part
of the hyperalgebra of $G(n)$.
