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

TL;DR

This paper constructs small $p$-groups with automorphism groups inducing large maximal subgroups on their Frattini quotients, providing new insights into the relationship between group size and automorphism actions.

Contribution

It introduces a method to construct $p$-groups with automorphism groups matching large maximal subgroups on the Frattini quotient, achieving minimal order and nilpotency class.

Findings

Constructed $p$-groups with automorphism groups inducing large maximal subgroups

Groups have minimal order among all with similar automorphism actions

Groups exhibit exponent $p$ and minimal nilpotency class

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 Kov\'acs. 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.