| Summary: | Consider the following situation: k will be an algebraically closed field of characteristic p and G will be a finite p-group, V will be a non-projective, indecomposable kG-module which is free on restriction to some maximal subgroup of G. Our purpose in doing this is to investigate Chouinard's theorem - all the proofs of which have been cohomological in nature - in a representation-theoretic way. This theorem may be shown to be equivalent to saying that, if G is not elementary abelian, V cannot be free on restriction to all the maximal subgroups of G. It is shown how to construct an exact sequence: O → V → P → P → V → O with P projective. From this an almost split sequence, O → V → X → V → O is constructed. It is shown that X can have at most two indecomposable summands. If φ denotes the Frattini subgroup of G, then V is free on restriction to φ. We may regard the set of φ-fixed points of V, V̄, as a module for Ḡ =G/φ. But Ḡ is elementary abelian, so we may consider the Carlson variety, Y(V̄) - this may be regarded as a subset of J/J² where J denotes the augmentation ideal of kG. It is shown that Y(V̄) is always a line. We define YG to be the union of all the lines Y(V̄) as V runs over all the kG-modules with the properties above. It is shown that YG is the whole of J/J² if and only if G is elementary abelian. It is also shown that, when G is one of a particular class of p-groups - the pseudo-special groups - which form the minimal counter-examples to Chouinard's theorem, that YG is the set of zeros of a sequence of homogeneous polynomials with coefficients in the field of p elements. Indeed, a specific construction for these polynomials is given.
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