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Let k be a field of positive
characteristic p. One considers some categories, whose objects are given classes of
finite p-groups, and morphisms are given classes of k-virtual bisets, i.e.,
linear combinations of bisets with coefficients in k. The category of k-linear
functors from such a category to the category of k-vector spaces is abelian,
and one can try to classify and describe its simple objects, or its projective
objects.
By specific subfunctors of the Burnside functor, which have a unique simple
quotient SQ,k, one will get some estimates on the k-dimension of the evaluations of
these simple functors. These evaluations are equalities for abelian p-groups, and for
such groups P the result is even stronger, since it provides some explicit k-bases for
the evaluations SQ,k(P).
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