On quadratic conjecture

TL;DR

This paper proves the quadratic conjecture for certain $p$-groups of maximal class, extending the known cases and confirming the conjecture for all such groups when $p \\le 7$.

Contribution

It establishes the quadratic conjecture for $p$-groups of maximal class with specific order bounds, advancing understanding of the conjecture's validity.

Findings

Quadratic conjecture holds for $p$-groups of maximal class with $n \\le 8$.

It also holds when $n \\ge \\max\{2p-6,p+2\\}$.

The conjecture is confirmed for all such groups with $p \\le 7$.

Abstract

Quadratic conjecture is a strengthening of oliver's $p$-group conjecture. Let $G$ be a $p$-group of maximal class of order $p^n$. We prove that if $n\le 8$ or $n\ge \max\{2p-6,p+2\}$ then $G$ satisfies Quadratic Conjecture. Hence quadratic conjecture holds if $G$ is a $p$-group of maximal class where $p\le 7$.