Lie Representations and Groups of Prime Power Order

Let p be a prime, P a finite non-cyclic p-gvoup and (xd)Q>(P) of the factor group P/$(P), and the map 61-+B is a homomorphism from the automorphism group Aut P of P to the automorphism group of P/$(P). When P/O(P) is regarded as a vector space over the field of order p, the restriction of Aut P to P/(P) is a group of linear transformations of this vector space. Our main result is to establish that every linear group arises in this way.