ON p-GROUPS WITH AUTOMORPHISM GROUPS RELATED TO THE CHEVALLEY GROUP G(2)(p)

摘要

Let p be an odd prime. We construct a p-group P of nilpotency class two, rank seven and exponent p, such that Aut(P) induces N-GL(7;p)(G2(p)) = Z(GL(7; p))G2(p) on the Frattini quotient P=Phi(P). The constructed group P is the smallest p-group with these properties, having order p(14), and when p = 3 our construction gives two nonisomorphic p-groups. To show that P satisfies the specified properties, we study the action of G(2)(q) on the octonion algebra over F-q, for each power q of p, and explore the reducibility of the exterior square of each irreducible seven-dimensional F-q[G2(q)]-module.