Centralizers of maximal order and nilpotent normal subgroups

Abstract

We give a negative answer to Kourovka Problem 14.67. We construct a finite group $G$ of order $660602880$, an involution $a\in G$, and a nilpotent normal subgroup $H$ of order $16384$ such that $|C_G(a)|=41287680\ge |C_G(g)|$ for every $g\ne1$, but $H\nleq C_G(a)$. The group $H$ is a bilinear central extension of two copies of $\mathbb{F}_2^4$ by $\bigwedge^2\mathbb{F}_2^4$. The ambient group is obtained by adjoining the natural action of $\mathrm{GL}_4(2)$ and an involution interchanging the two copies. The centralizer comparison follows from fixed-space bounds on the exterior square and elementary orbit calculations.