Nilpotent subgroups of class $2$ in finite groups

TL;DR

This paper proves that every finite group has a large nilpotent subgroup of class at most 2, with size growing as a power of the group size, answering a question of Pyber and showing near-optimal bounds.

Contribution

It establishes the existence of large nilpotent subgroups of class at most 2 in finite groups, providing bounds that are nearly tight and resolving a question posed by Pyber.

Findings

Every finite group of size at least 3 has a nilpotent subgroup of class at most 2.

The size of such a subgroup is at least |G|^{1/32 log log |G|}.

The result is essentially optimal.

Abstract

We show that every finite group $G$ of size at least $3$ has a nilpotent subgroup of class at most $2$ and size at least $|G|^{1/32\log\log|G|}$. This answers a question of Pyber, and is essentially best possible.