Maximal linear groups induced on the Frattini quotient of a $p$group
Abstract
Let $p>3$ be a prime. For each maximal subgroup $H\leqslant\mathrm{GL}(d,p)$ with $H \geqslant p^{3d+1}$, we construct a $d$generator finite $p$group $G$ with the property that $\mathrm{Aut}(G)$ induces $H$ on the Frattini quotient $G/\Phi(G)$ and $G \leqslant p^{\frac{d^4}{2}}$. 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 $G/\Phi(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.
 Publication:

arXiv eprints
 Pub Date:
 March 2016
 DOI:
 10.48550/arXiv.1603.05384
 arXiv:
 arXiv:1603.05384
 Bibcode:
 2016arXiv160305384B
 Keywords:

 Mathematics  Group Theory;
 20D45;
 20D15;
 20B25
 EPrint:
 24 pages, 2 figures, 2 tables Typos corrected. Acknowledgement extended. To appear J. Pure. Appl. Algebra