  We determine the composition factors of the tensor product $S(E)\otimes S(E)$
of two copies of the symmetric algebra of the natural module $E$ of a general
linear group over an algebraically closed field of positive characteristic. Our
main result may be regarded as a substantial generalisation of the tensor
product theorem of Krop and Sullivan, on composition factors of $S(E)$. We
earlier answered the question of which polynomially injective modules are
infinitesimally injective in terms of the "divisibility index". We are now able
to give an explicit description of the divisibility index for polynomial
modules for general linear groups of degree at most $3$.
