Maximal linear groups induced on the Frattini quotient of a p-group

2016/03/17 by Bamberg, John, Glasby, S. P., Morgan, Luke +1
#20B25 #20D15 #20D45 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1603.05384

Abstract

Let p>3 be a prime. For each maximal subgroup H\leqslantGL(d,p) with |H| \geqslant p^3d+1, we construct a d-generator finite p-group G with the property that Aut(G) induces H on the Frattini quotient G/Φ(G) and |G| \leqslant p^(d4)/(2). A significant feature of this construction is that |G| is very small compared to |H|, shedding new light upon a celebrated result of Bryant and Kovács. The groups G that we exhibit have exponent p, and of all such groups G with the desired action of H on G/Φ(G), the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.

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